diff --git a/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/2301.02868v1.pdf.txt b/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/2301.02868v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3bc75862e876bd11492624da40b229bcd84a9dd --- /dev/null +++ b/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/2301.02868v1.pdf.txt @@ -0,0 +1,773 @@ +Crucial role of Fe in determining the hard magnetic properties of Nd2Fe14B +Juba Bouaziz∗ +Department of Physics, University of Warwick, Coventry CV4 7AL, UK and +Peter Gr¨unberg Institut and Institute for Advanced Simulation, +Forschungszentrum J¨ulich & JARA, D-52425 J¨ulich, Germany +Christopher E. Patrick +Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK +Julie B. Staunton† +Department of Physics, University of Warwick, Coventry CV4 7AL, UK +(Dated: January 10, 2023) +Nd2Fe14B’s unsurpassed, hard magnetic properties for a wide range of temperatures result from +a combination of a large volume magnetization from Fe and a strong single-ion anisotropy from +Nd. Here, using finite temperature first-principles calculations, we focus on the other crucial roles +played by the Fe atoms in maintaining the magnetic order on the Nd sublattices, and hence the large +magnetic anisotropy, and directly generating significant uniaxial anisotropy at high temperatures. +We identify effective spins for atomistic modelling from the material’s interacting electrons and +quantify pairwise and higher order, non-pairwise magnetic interactions among them. We find the +Nd spins couple most strongly to spins on sites belonging to two specific Fe sublattices, 8j1, 8j2. +Moreover the Fe 8j1 sublattice also provides the electronic origin of the unusual, nonmonotonic +temperature dependence of the anisotropy of Y2Fe14B. Our work provides atomic-level resolution +of the properties of this fascinating magnetic material. +The elemental lanthanides show remarkable magnetic +properties deriving from their partially-filled shells of +atomic-like 4f electrons. +However, direct exploitation +of these properties is hindered by low magnetic order- +ing temperatures. +No elemental lanthanide retains its +magnetism at room temperature, with the highest Curie +temperature Tc being 292 K for Gd [1]. +Combining +the lanthanides with other elements can strengthen the +magnetic interactions and allow ordering to persist to +higher temperatures. +The most successful example of +this paradigm is the rare-earth/transition-metal (RE- +TM) family of permanent magnets [2]. Specifically, Nd- +Fe-B demonstrates exceptional magnetic strength over a +wide range of temperatures. Having revolutionized com- +puter hard disk technology in the last century, Nd-Fe-B is +again under intense investigation owing to its use in elec- +tric vehicle motors and renewable energy turbines [3]. +The RE-TM magnetic interactions are most simply de- +scribed in terms of the exchange field Bexch. +In this +picture, the TM-3d electrons produce an effective mag- +netic field which couples to the spin magnetic moments +of the RE ions. A minimal model to describe the RE ions +and the high magnetic anisotropy they generate combines +this exchange field with the interaction with an external +field Bext and the crystal field ˆVCF, which describes the +(predominantly) electrostatic interaction of the 4f charge +cloud with its environment [4–7]: +HRE = 2µB ˆS · Bexch + µB (ˆL + 2ˆS) · Bext + ˆVCF. (1) +ˆS and ˆL are the total spin and orbital angular mo- +mentum operators. +Values of the exchange field can +be extracted by fitting Eq. 1 to experimental data ob- +tained in inelastic neutron scattering (INS) or magneti- +zation measurements. Experimental estimates of Bexch +are far stronger than fields achievable in the laboratory +(µBBexch/kB >∼ 300 K [8], i.e. Bexch >∼ 450 T) as required +to maintain magnetic order above room temperature. +Going beyond a phenomenological understanding of +RE ordering requires an atomistic picture of the mag- +netic interactions among effective spins. Nd2Fe14B has a +tetragonal crystal structure with 68 atoms per unit cell +([8] [9]). The RE atoms occupy two crystallographically +distinct sites (RE4f and RE4g), which (together with Fe4c +and B4g atoms) form planes encapsulating the remain- +ing 5 Fe sublattices (4e, 8j1, 8j2, 16k1, 16k2). For the +Nd sites the spins come from the localized f-electrons +but for the TM sites the local effective spins, or local +moments, emerge from the material’s itinerant electron +fluid [10]. +Spin-polarized regions at atomic sites form +from co-operative behavior of the valence electrons and +at finite temperatures their orientations fluctuate on rel- +atively long time scales compared to the remaining elec- +tronic degrees of freedom. These local magnetic moments +are the pertinent, effective spins for the TM aspect of the +atomistic modelling. +A conceptually simple model assumes interactions only +between pairs of spins (ij) according to the classical +Heisenberg model, −Jij ˆSi · ˆ +Sj where ˆSi represents an +effective spin. Previous works [12, 13] calculate such Jij +parameters from first principles within density-functional +theory (DFT) [12, 13], and use them directly in atomistic +spin dynamics simulations. With the TM magnetocrys- +talline anisotropy (MCA) modelled as a sum of single +arXiv:2301.02868v1 [cond-mat.mtrl-sci] 7 Jan 2023 + +2 +c) +(c) +(b) +Nd spin +Nd+Fe spin +Nd orb +Fe orb +Exp +(a) +FIG. 1. (a) Nd2Fe14B’s magnetization versus (T/Tc) from DLM-DFT calculations compared to experiment [11]. (b) Sub-lattice +resolved magnetic order parameters and (c) Weiss fields. The dots indicate the full DLM-DFT results, dashed lines from a +pair-wise interaction model and continuous lines from a fit of the DLM-DFT results to the model discussed in the text Eq. (3). +ion-like terms, assumed to be substantial, and RE crystal +field coefficients taken from experiment, the simulations +can reproduce the magnetization behavior of Nd2Fe14B, +including the spin reorientation transition at low tem- +perature, and represent the current state-of-the-art in +modelling these magnets [14]. Although such a pair-wise +Heisenberg model is computationally straightforward to +implement, it is nonetheless a clear presumption for a +magnetic metal like Nd-Fe-B. Despite the huge technical +importance of the material, the role of “beyond Heisen- +berg” itinerant electron spin features has yet to be elu- +cidated for Nd-Fe-B. Moreover the MCA from the spin- +orbit coupling of the itinerant d-electrons is also not guar- +anteed to be single-ion like [15]. In this letter we quantify +the significance of both these aspects and propose ways +to improve atomistic spin modelling. +The disordered local moment (DLM) picture imple- +mented within DFT provides an appropriate ab initio +framework [10, 15, 16]. The approach combines statisti- +cal mechanics of the effective spins (local moments, {ei}) +and DFT, to describe the complementary evolution of +electronic and magnetic structure as a material’s tem- +perature is varied. Strongly correlated 4f-electron effects +are treated with a parameter free, self interaction correc- +tion (SIC) approach [17, 18] which incorporates Hund’s +rules naturally [19]. +As such DLM-DFT can describe +temperature-dependent magnetic properties of perma- +nent magnets as shown recently for the RECo5 fam- +ily [7]. The crucial RE contribution to the anisotropy is +accounted by crystal field theory, calculating the CF coef- +ficients within DFT using a robust numerical method [16] +so that the modelling is independent of any prior fit of +phenomenological parameters. +Here we investigate the nature of magnetic order in +Nd2Fe14B, sublattice-resolved, and describe the mag- +netic interactions among the effective spins associated +with both RE and TM sites. We show that the interac- +tions among the TM spins are influenced by the global +magnetic order and its impact and link with the spin- +polarised electrons of the system. This is in essence a +multi-spin coupling effect. We find significant diversity +in the behavior of the Fe local moments depending on +their location in the unit cell. While most TM spins are +ferromagnetically-coupled, some interact antiferromag- +netically with each other. This leads to some frustration +and a peculiar strong suppression of magnetic order on +the 8j1 sites which are located roughly midway between +the Nd-containing layers. We also find that the Nd spins +couple most strongly to spins on sites belonging to this +Fe sublattice along with those on another (8j2). Further- +more we discover a link between this 8j1 sublattice and +the unusual non-monotonic temperature dependence of +the non-RE MCA of the isostructural material Y2Fe14B, +resolving a longstanding a puzzle [11, 20]. Finally our +calculation of the anisotropy field of Nd2Fe14B across a +range of temperatures agrees well with experiment and +confirms the vital role played by the Fe spins for the +functionality of this champion magnet. +Apart from the local moments themselves, the cen- +tral quantities in DLM-DFT theory are Weiss fields {hi} +which, analogously to the exchange field of Eq.1, drive +the ordering of the local moments. +However, unlike +Bexch, the Weiss fields are not phenomenological, but +instead are rigorously defined by thermal averages over +the local moment orientational configurations {ei} of the +magnetic energy Ω{ei} [10, 15], i.e. +hi = +� +3 +4π ⟨Ω⟩ei;T dei . +(2) +where ⟨X⟩ei denotes the average of X with the restric- +tion that the orientation of the moment on site i is fixed +as ei and the order parameters of the local moments, +{mi} are the averages {⟨ei⟩} [10, 15]. ⟨Ω⟩�ei;T is calcu- +lated from DFT [10]. Crucially no prior prescription is +assumed for the form of the magnetic interactions inher- +ent in the first-principles Ω. For a pairwise Heisenberg +model, the magnetic energy Ω{ei} = −1/2 � +ij Jijei · ej +with Weiss fields linear in the {mi}, hi = � +j Jijmj. + +3 +16k1 +16k2 +8j1 +8j2 +f g 4e 4c +FIG. 2. The relative strengths of interactions between sites in +the unit cell (boron sites not included), Jij, (Eq. 3) highlight- +ing the RE-TM ones (sites 48–51 and 52–55 correspond to 4f +and 4g respectively). Numerical values in meV are given in [9] +along with specific site coordinates. Red/blue color indicates +FM/AF interactions. +Consequently beyond-pairwise terms are clearly identi- +fied in DLM-DFT theory from the non-linear dependence +of the Weiss fields on the {mi} [21–25]. The {mi} order +parameters, describing an equilibrium state at a temper- +ature T, are given by the self-consistent solution of Eq.2 +and mi = (−1/βhi + coth βhi), the Langevin function, +(β = 1/kBT). +Figure 1(a) shows the magnetization as a function of +T compared to experiment and resolved into the RE and +TM spin and orbital components. +The magnetization +is directed along θ = 45◦ in the (xz)-plane. +Full cal- +culational details are given in the Supplemental Mate- +rial [9] and references [16, 26–28]. The contribution from +a particular site i is found by multiplying its local mo- +ment magnitude, µi, by the order parameter mi(T). The +Fe and Nd spin moments interact antiferromagnetically +(AF) and order in an anti-parallel alignment in a fer- +rimagnetic state, but the large orbital moment of Nd, +pointing opposite to its spin, leads to overall ferromag- +netic (FM) order. +The Fe orbital moments are small +(∼ 0.05µB/atom). The calculated Tc is 1058 K, which, +although an overestimate of 473 K in comparison to the +experiment [11], is reasonable for a first-principles theory +which uses a mean field approximation for the statistical +mechanics of the effective spins [28]. +On each of the six Fe and two Nd sublattices ([8, 9] +the magnetic order varies from complete, {mi = 1}, +at T = 0K to zero above Tc, {mi = 0}. +Figure 1(b) +shows how the temperature dependence of magnetic or- +der varies across the sublattices. The Nd sublattices dis- +order more quickly than all the Fe sublattices except +the 8j1 one. +Complementary information in Fig. 1(c) +shows that Weiss fields, {hi}, promote strong ordering +when large and have considerable sublattice variation, +notably the factor ∼4 difference between the 8j1 and +8j2 sites. Analysis of {hi}, Eq.2, reveals the presence +and importance of interactions that fall outside those of +a Heisenberg-like model. For such a pairwise model the +Jij interactions (Fig. 2), directly obtained from the Weiss +fields for small values of the {mi}, are used to construct +the model’s Weiss fields and {mi} at all T (dashed lines +in Fig. 1(c)). There are large discrepancies from the full +ab initio DLM-DFT data away from Tc, leading us to +propose a more realistic representation of the interac- +tions which is straightforward to incorporate into atom- +istic spin modelling of the magnet’s properties. It leads +to a magnetic energy per unit cell +¯Ω = −1 +2 +� +ij +Jijmi · mj − 1 +4 +� +i +BI(mi · M)2, +(3) +where i, j run over the sites in the unit cell, I denotes one +of the 8 sub-lattices to which the site i belongs and M +is the total magnetization per unit cell, M = � +i µimi +where the order parameters on the RE sites are anti- +parallel to the TM sites for the ferrimagnetic state. The +second, higher order term captures the effect of the over- +all spin-polarization of the electronic structure on the +effective interactions between the local moments. Com- +puting Weiss fields from this expression fits the DLM- +DFT calculations very well as shown by the full curves in +Fig. 1(c) and ¯Ω closely approximates ⟨Ω⟩T . Table I lists +the BI parameters that measure the sublattice-dependent +size of these higher order, multi-spin terms. +System +4c +4e +8j1 +8j2 16k1 16k2 +Rf +Rg +Nd2Fe14B -15.42 14.31 -5.06 -1.38 3.82 4.44 -2.53 -1.41 +Y2Fe14B +-13.68 9.91 -4.07 1.27 4.25 4.07 +0.0 +0.0 +TABLE I. Effective, multi-spin interaction constants (in µeV), +BI, for Nd2Fe14B and Y2Fe14B. +Fig. 2 shows the relative strengths of the Jij interac- +tions between pairs of sites. They are represented on a +64 × 64 grid (56 Fe sites and 8 RE sites and arranged +according to sublattice). Numerical values are given as +Supplemental Material [9]. Assuming a range less than +roughly 5˚A, they can be directly used in atomistic spin +simulations together with the terms from Table I. The +figure illustrates the vital importance on the RE mag- +netic ordering of the hexagonal nets of Fe atoms [8, 9] +from the k1, k2 and notably sites on the j1 and j2 sublat- +tices. Indeed the largest contributions to the Weiss fields +at the RE sites originate from the j1 and j2 sublattices. +The TM-TM interactions are particularly varied rang- +ing from FM (red) for the majority to AF (blue). The +j1 sites have AF interactions with e, c and RE sites and +strong FM ones with j2 sites. +This frustration drives +this sublattice’s aversion to magnetic order. The diver- +sity of the interactions stems from the profound effect + +60 +40 +30 +40 +20 +Site +10 +20 +0 +-10 +20 +40 +60 +Site i4 +that atom coordination and spacing of Fe atoms in a +metallic material has on its magnetism. The archetype +for this quintessentially itinerant electron effect is fcc +Fe where squeezing the lattice turns ferromagnetic order +anti-ferromagnetic and then destroys it [29, 30]. +The diverse nature of the magnetic order on the six +Fe sub-lattices also has an impact on the intrinsic MCA +generated by the system’s itinerant valence electrons. As +found for other TM metal magnets [15, 26], a simple fit in +terms of a single ion model is unsatisfactory and, as found +for other itinerant electron magnets, two-ion type terms +should be included in the model [15, 31, 32]. Further- +more, on general grounds, modelling the MCA as a sum +of single ion anisotropy terms must be done extremely +carefully. The various Fe sites have different crystallo- +graphic point symmetries, and their unique symmetry +axes do not necessarily point along the c direction [33]. +There are important implications for atomistic spin dy- +namics simulations [14, 31, 34] where it is not correct +to assign a single ion anisotropy to each Fe atom with +the same angular dependence and same symmetry axes. +Rather a simpler and more rigorous alternative would +be to compute an anisotropy energy based on the vector +sum of all moments in the same sublattice so that a single +symmetry axis is recovered [5]. +While significantly smaller than from the f-electrons of +the Nd atoms, the primarily TM component of the MCA +grows in importance with rising T. +As the RE MCA +drops swiftly along with the magnetic order +[35, 36], +the TM contribution can actually increase as shown ex- +plicitly in measurements on Y2Fe14B [11, 20]. +Such +non-monotonic temperature variation is puzzling, and +has been attributed to a magnetostructural effect from +an anisotropic expansion of the crystal lattice, compet- +ing single-ion-like contributions [20] or competing single +and two-ion MCA using atomistic spin dynamics simula- +tions [31, 32] Since fully relativistic effects such as spin- +orbit coupling are included in our DLM-DFT theory we +investigate the MCA temperature dependence directly +and show our results in Fig. 3 for Y2Fe14B. +Using the highly accurate, full potential (FP) KKR +code [37, 38], we first calculate the MCA at T = 0K to +be ≈ 0.9 meV/formula unit (FU) which agrees well with +experimental values [11]. The same rapid loss of magnetic +order with increasing temperature which we find for the +Fe 8j1 sites in Nd2Fe14B (Fig. 1(b)) is also evident in +Y2Fe14B [9] and this points to a significant role for this +sublattice in the anomalous MCA T-dependence. +We +therefore carry out further FP MCA calculations where +now the Fe 8j1 sites are constrained to be magnetically +disordered (m8j1 = 0) via an equal weighting of local +moments on each of these sites along the ±x and ±z +directions. The effect on the computed MCA is striking +- it increases greatly to ≈ 1.7 meV/FU - and we infer +that the much faster decrease of 8j1 magnetic order with +temperature relative to that on the other Fe sublattices +is the key driver for the MCA T-dependence. +To test this proposition, we calibrate DLM-DFT MCA +values against our T = 0K FP MCA calculations, given +the current implementation with an atomic sphere ap- +proximation (ASA). Although the ASA values are smaller +than the FP ones, the same large increase of the value +when the Fe 8j1 sites are magnetically disordered is +found. In Fig. 3 we show the DLM-DFT temperature de- +pendent MCA both using the ASA (red curve) and also +scaled by the ratio between the FP and our ASA T = 0K +values (green). The increase with temperature is evident, +peaking at T/Tc = 50% in line with experiment [11] con- +firming our proposition. Since the calculations are for a +fixed lattice structure, we can exclude thermal expansion +as a cause of the non-monotonic behavior. We also show +the effect on the MCA of forcing the Fe8j1 sublattice to +remain magnetically disordered at all T, i.e. m8j1 = 0. +The resulting unscaled MCA, shown in black in Fig. 3, is +dramatically altered - the peak has gone and the MCA +decays linearly with temperature and the T = 0K value +is enhanced significantly. Clearly, establishment of mag- +netic order on the Fe8j1 sublattice correlates with a sub- +stantial drop in the (uniaxial) MCA. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T/TC +0.0 +0.5 +1.0 +1.5 +2.0 +K1 (meV/FU) +Exp +Th +Th-8j1 +Th-scl +FIG. 3. The T-dependence of the leading anisotropy constant, +K1, of Y2Fe14B from DLM-DFT theory (red curve) and ex- +periment (blue) [11]. The green curve shows the theory values, +scaled to account for the difference between FP [37] and ASA +([9]) at T = 0K. The black curve shows K1 (unscaled) if the +Fe 8j1 sublattice is constrained to be disordered magnetically. +Our ultimate goal is to describe Nd2Fe14B’s large mag- +netic anisotropy and its temperature variation. So to the +TM MCA we add the dominant RE components. These +are calculated [6, 39] from the solution of Eq. 1 where the +crystal field coefficients [9] are determined from first prin- +ciples [16], and exchange field, Bexch provided directly by +the DLM-DFT Weiss field for each Nd site (Fig. 1) di- +vided by the computed Nd spin moment of 3.66 µB. +Our calculated exchange fields of 699 and 725 T for +the RE f and g sites respectively are somewhat larger +than those used in fits of experimental magnetization + +5 +0.4 +0.6 +0.8 +1.0 +T/Tc +0 +5 +10 +µ0HA(T) +Computed +Computed-scl +Exp (Hirosawa 1986) +Exp (Grossinger 1986) +FIG. 4. Evolution of the anisotropy field, HA, versus T/Tc +from theory compared to experimental measurements (from +Refs. [42], black curve, +[11], blue curve). The agreement is +good above Tc/2. The red and green curves use the non-RE +MCA taken from the red and green curves of Fig. 3. +data (450–520 T [40]), but as pointed out in Ref. [8], +the large number of parameters in Eq. 1 introduce sig- +nificant uncertainties. In principle, INS data would pro- +vide a direct measure of the exchange fields but are not +available for Nd-Fe-B. Our proposed values are, however, +supported by the good agreement between INS experi- +ments [41] and our DLM-DFT calculations for the re- +lated Gd2Fe14B magnet (324 T vs 307/319 T). The Gd +exchange fields are substantially smaller than those cal- +culated for Nd. The relative difference (∼2) mirrors that +of the spin moments (7.46 vs. 3.66 µB) and reflects the +similar Weiss fields we calculate for the two materials. +Using the method of Ref. [39] we calculate effective +anisotropy constants K1(T), K2(T) and anisotropy field, +µ0HA = 2K1/M ab initio which is shown in Figure 4. +The red/green curves show µ0HA which includes the non- +RE contribution to the MCA of the red/green plots in +Fig. 3. Fig. 4 also shows the experimental measurements +from Refs. [11, 42]. Below T/Tc ∼ 0.5 there is some dis- +crepancy between the two sets of experimental data, but +above there is consistency between both the experiments +and our calculations. +The calculations show the clear +importance of the Fe-dominated MCA to the anisotropy +field at high temperatures - the red curve is over 1 T less +than the green one over a range of temperatures despite +the contributions from the non-RE MCAs differing by +less than 30 µeV per Fe atom. +Nd2Fe14B’s spin reorientation transition (SRT) at +135K [8, 14, 43] is not described by our calculations owing +to an underestimate of the high order crystal field coeffi- +cients [14, 44, 45]. This shortcoming exemplifies a more +general challenge for theory modelling of low T strongly +correlated f-electron effects to construct a robust way to +significantly enhance the value of these coefficients [46]. +Around room temperature and above, however, the ef- +fects on the MCA from these high order terms are small. +This is also the temperature regime where the tenets of +our DLM-DFT theory are valid. +Nd2Fe14B and the RE-TM permanent magnet family +to which it belongs have a compelling set of attributes. +Their technological value is enormous and growing and +their magnetic properties, at a fundamental level, come +from a rich and subtle combination of RE, localized, and +TM, itinerant electron, effects. +To enhance magnetic +functionality and extract pointers for the development of +even better materials, multiple interrelated aspects have +to be accounted for. 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Deriu, 3 d and 4 f magnetism in nd2fe14- x co +x b and y2fe14- x co x b compounds, Journal of applied +physics 61, 5369 (1987). +[50] K. Stevens, Matrix elements and operator equivalents +connected with the magnetic properties of rare earth ions, +Proceedings of the Physical Society. Section A 65, 209 +(1952). +[51] J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, +M. Du�lak, +L. Ferrighi, +J. Gavnholt, +C. Glinsvad, +V. Haikola, H. Hansen, et al., Electronic structure cal- +culations with gpaw: a real-space implementation of the +projector augmented-wave method, Journal of physics: +Condensed matter 22, 253202 (2010). +[52] D. S. G. Bauer, Development of a relativistic full- +potential +first-principles +multiple +scattering +Green +function method applied to complex magnetic textures +of nanostructures at surfaces, Forschungszentrum J¨ulich +http://publications.rwth-aachen.de/record/229375 +(2014). +[53] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin- +dependent electron liquid correlation energies for local +spin density calculations: a critical analysis, Canadian +Journal of Physics 58, 1200 (1980). + diff --git a/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/load_file.txt b/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7dce228276f1a1c3c153b3634ab03aa4c8fc74b7 --- /dev/null +++ b/-9E1T4oBgHgl3EQfCwJj/content/tmp_files/load_file.txt @@ -0,0 +1,616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf,len=615 +page_content='Crucial role of Fe in determining the hard magnetic properties of Nd2Fe14B Juba Bouaziz∗ Department of Physics, University of Warwick, Coventry CV4 7AL, UK and Peter Gr¨unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ulich & JARA, D-52425 J¨ulich, Germany Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Patrick Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Julie B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Staunton† Department of Physics, University of Warwick, Coventry CV4 7AL, UK (Dated: January 10, 2023) Nd2Fe14B’s unsurpassed, hard magnetic properties for a wide range of temperatures result from a combination of a large volume magnetization from Fe and a strong single-ion anisotropy from Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Here, using finite temperature first-principles calculations, we focus on the other crucial roles played by the Fe atoms in maintaining the magnetic order on the Nd sublattices, and hence the large magnetic anisotropy, and directly generating significant uniaxial anisotropy at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We identify effective spins for atomistic modelling from the material’s interacting electrons and quantify pairwise and higher order, non-pairwise magnetic interactions among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We find the Nd spins couple most strongly to spins on sites belonging to two specific Fe sublattices, 8j1, 8j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Moreover the Fe 8j1 sublattice also provides the electronic origin of the unusual, nonmonotonic temperature dependence of the anisotropy of Y2Fe14B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Our work provides atomic-level resolution of the properties of this fascinating magnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The elemental lanthanides show remarkable magnetic properties deriving from their partially-filled shells of atomic-like 4f electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' However, direct exploitation of these properties is hindered by low magnetic order- ing temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' No elemental lanthanide retains its magnetism at room temperature, with the highest Curie temperature Tc being 292 K for Gd [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Combining the lanthanides with other elements can strengthen the magnetic interactions and allow ordering to persist to higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The most successful example of this paradigm is the rare-earth/transition-metal (RE- TM) family of permanent magnets [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Specifically, Nd- Fe-B demonstrates exceptional magnetic strength over a wide range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Having revolutionized com- puter hard disk technology in the last century, Nd-Fe-B is again under intense investigation owing to its use in elec- tric vehicle motors and renewable energy turbines [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The RE-TM magnetic interactions are most simply de- scribed in terms of the exchange field Bexch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' In this picture, the TM-3d electrons produce an effective mag- netic field which couples to the spin magnetic moments of the RE ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' A minimal model to describe the RE ions and the high magnetic anisotropy they generate combines this exchange field with the interaction with an external field Bext and the crystal field ˆVCF, which describes the (predominantly) electrostatic interaction of the 4f charge cloud with its environment [4–7]: HRE = 2µB ˆS · Bexch + µB (ˆL + 2ˆS) · Bext + ˆVCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' (1) ˆS and ˆL are the total spin and orbital angular mo- mentum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Values of the exchange field can be extracted by fitting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1 to experimental data ob- tained in inelastic neutron scattering (INS) or magneti- zation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Experimental estimates of Bexch are far stronger than fields achievable in the laboratory (µBBexch/kB >∼ 300 K [8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Bexch >∼ 450 T) as required to maintain magnetic order above room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Going beyond a phenomenological understanding of RE ordering requires an atomistic picture of the mag- netic interactions among effective spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Nd2Fe14B has a tetragonal crystal structure with 68 atoms per unit cell ([8] [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The RE atoms occupy two crystallographically distinct sites (RE4f and RE4g), which (together with Fe4c and B4g atoms) form planes encapsulating the remain- ing 5 Fe sublattices (4e, 8j1, 8j2, 16k1, 16k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' For the Nd sites the spins come from the localized f-electrons but for the TM sites the local effective spins, or local moments, emerge from the material’s itinerant electron fluid [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Spin-polarized regions at atomic sites form from co-operative behavior of the valence electrons and at finite temperatures their orientations fluctuate on rel- atively long time scales compared to the remaining elec- tronic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' These local magnetic moments are the pertinent, effective spins for the TM aspect of the atomistic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' A conceptually simple model assumes interactions only between pairs of spins (ij) according to the classical Heisenberg model, −Jij ˆSi · ˆ Sj where ˆSi represents an effective spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Previous works [12, 13] calculate such Jij parameters from first principles within density-functional theory (DFT) [12, 13], and use them directly in atomistic spin dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' With the TM magnetocrys- talline anisotropy (MCA) modelled as a sum of single arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='02868v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='mtrl-sci] 7 Jan 2023 2 c) (c) (b) Nd spin Nd+Fe spin Nd orb Fe orb Exp (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' (a) Nd2Fe14B’s magnetization versus (T/Tc) from DLM-DFT calculations compared to experiment [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' (b) Sub-lattice resolved magnetic order parameters and (c) Weiss fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The dots indicate the full DLM-DFT results, dashed lines from a pair-wise interaction model and continuous lines from a fit of the DLM-DFT results to the model discussed in the text Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' ion-like terms, assumed to be substantial, and RE crystal field coefficients taken from experiment, the simulations can reproduce the magnetization behavior of Nd2Fe14B, including the spin reorientation transition at low tem- perature, and represent the current state-of-the-art in modelling these magnets [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Although such a pair-wise Heisenberg model is computationally straightforward to implement, it is nonetheless a clear presumption for a magnetic metal like Nd-Fe-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Despite the huge technical importance of the material, the role of “beyond Heisen- berg” itinerant electron spin features has yet to be elu- cidated for Nd-Fe-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Moreover the MCA from the spin- orbit coupling of the itinerant d-electrons is also not guar- anteed to be single-ion like [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' In this letter we quantify the significance of both these aspects and propose ways to improve atomistic spin modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The disordered local moment (DLM) picture imple- mented within DFT provides an appropriate ab initio framework [10, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The approach combines statisti- cal mechanics of the effective spins (local moments, {ei}) and DFT, to describe the complementary evolution of electronic and magnetic structure as a material’s tem- perature is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Strongly correlated 4f-electron effects are treated with a parameter free, self interaction correc- tion (SIC) approach [17, 18] which incorporates Hund’s rules naturally [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' As such DLM-DFT can describe temperature-dependent magnetic properties of perma- nent magnets as shown recently for the RECo5 fam- ily [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The crucial RE contribution to the anisotropy is accounted by crystal field theory, calculating the CF coef- ficients within DFT using a robust numerical method [16] so that the modelling is independent of any prior fit of phenomenological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Here we investigate the nature of magnetic order in Nd2Fe14B, sublattice-resolved, and describe the mag- netic interactions among the effective spins associated with both RE and TM sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We show that the interac- tions among the TM spins are influenced by the global magnetic order and its impact and link with the spin- polarised electrons of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' This is in essence a multi-spin coupling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We find significant diversity in the behavior of the Fe local moments depending on their location in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' While most TM spins are ferromagnetically-coupled, some interact antiferromag- netically with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' This leads to some frustration and a peculiar strong suppression of magnetic order on the 8j1 sites which are located roughly midway between the Nd-containing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We also find that the Nd spins couple most strongly to spins on sites belonging to this Fe sublattice along with those on another (8j2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Further- more we discover a link between this 8j1 sublattice and the unusual non-monotonic temperature dependence of the non-RE MCA of the isostructural material Y2Fe14B, resolving a longstanding a puzzle [11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Finally our calculation of the anisotropy field of Nd2Fe14B across a range of temperatures agrees well with experiment and confirms the vital role played by the Fe spins for the functionality of this champion magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Apart from the local moments themselves, the cen- tral quantities in DLM-DFT theory are Weiss fields {hi} which, analogously to the exchange field of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='1, drive the ordering of the local moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' However, unlike Bexch, the Weiss fields are not phenomenological, but instead are rigorously defined by thermal averages over the local moment orientational configurations {ei} of the magnetic energy Ω{ei} [10, 15], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' hi = � 3 4π ⟨Ω⟩ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='T dei .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' (2) where ⟨X⟩ei denotes the average of X with the restric- tion that the orientation of the moment on site i is fixed as ei and the order parameters of the local moments, {mi} are the averages {⟨ei⟩} [10, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' ⟨Ω⟩�ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='T is calcu- lated from DFT [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Crucially no prior prescription is assumed for the form of the magnetic interactions inher- ent in the first-principles Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' For a pairwise Heisenberg model, the magnetic energy Ω{ei} = −1/2 � ij Jijei · ej with Weiss fields linear in the {mi}, hi = � j Jijmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3 16k1 16k2 8j1 8j2 f g 4e 4c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The relative strengths of interactions between sites in the unit cell (boron sites not included), Jij, (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3) highlight- ing the RE-TM ones (sites 48–51 and 52–55 correspond to 4f and 4g respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Numerical values in meV are given in [9] along with specific site coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Red/blue color indicates FM/AF interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Consequently beyond-pairwise terms are clearly identi- fied in DLM-DFT theory from the non-linear dependence of the Weiss fields on the {mi} [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The {mi} order parameters, describing an equilibrium state at a temper- ature T, are given by the self-consistent solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='2 and mi = (−1/βhi + coth βhi), the Langevin function, (β = 1/kBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Figure 1(a) shows the magnetization as a function of T compared to experiment and resolved into the RE and TM spin and orbital components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The magnetization is directed along θ = 45◦ in the (xz)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Full cal- culational details are given in the Supplemental Mate- rial [9] and references [16, 26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The contribution from a particular site i is found by multiplying its local mo- ment magnitude, µi, by the order parameter mi(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The Fe and Nd spin moments interact antiferromagnetically (AF) and order in an anti-parallel alignment in a fer- rimagnetic state, but the large orbital moment of Nd, pointing opposite to its spin, leads to overall ferromag- netic (FM) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The Fe orbital moments are small (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='05µB/atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The calculated Tc is 1058 K, which, although an overestimate of 473 K in comparison to the experiment [11], is reasonable for a first-principles theory which uses a mean field approximation for the statistical mechanics of the effective spins [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' On each of the six Fe and two Nd sublattices ([8, 9] the magnetic order varies from complete, {mi = 1}, at T = 0K to zero above Tc, {mi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Figure 1(b) shows how the temperature dependence of magnetic or- der varies across the sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The Nd sublattices dis- order more quickly than all the Fe sublattices except the 8j1 one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Complementary information in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1(c) shows that Weiss fields, {hi}, promote strong ordering when large and have considerable sublattice variation, notably the factor ∼4 difference between the 8j1 and 8j2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Analysis of {hi}, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='2, reveals the presence and importance of interactions that fall outside those of a Heisenberg-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' For such a pairwise model the Jij interactions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 2), directly obtained from the Weiss fields for small values of the {mi}, are used to construct the model’s Weiss fields and {mi} at all T (dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' There are large discrepancies from the full ab initio DLM-DFT data away from Tc, leading us to propose a more realistic representation of the interac- tions which is straightforward to incorporate into atom- istic spin modelling of the magnet’s properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' It leads to a magnetic energy per unit cell ¯Ω = −1 2 � ij Jijmi · mj − 1 4 � i BI(mi · M)2, (3) where i, j run over the sites in the unit cell, I denotes one of the 8 sub-lattices to which the site i belongs and M is the total magnetization per unit cell, M = � i µimi where the order parameters on the RE sites are anti- parallel to the TM sites for the ferrimagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The second, higher order term captures the effect of the over- all spin-polarization of the electronic structure on the effective interactions between the local moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Com- puting Weiss fields from this expression fits the DLM- DFT calculations very well as shown by the full curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1(c) and ¯Ω closely approximates ⟨Ω⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Table I lists the BI parameters that measure the sublattice-dependent size of these higher order, multi-spin terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' System 4c 4e 8j1 8j2 16k1 16k2 Rf Rg Nd2Fe14B -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='42 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='31 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='06 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='44 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='53 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='41 Y2Fe14B 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='68 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='91 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Effective, multi-spin interaction constants (in µeV), BI, for Nd2Fe14B and Y2Fe14B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 2 shows the relative strengths of the Jij interac- tions between pairs of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' They are represented on a 64 × 64 grid (56 Fe sites and 8 RE sites and arranged according to sublattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Numerical values are given as Supplemental Material [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Assuming a range less than roughly 5˚A, they can be directly used in atomistic spin simulations together with the terms from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The figure illustrates the vital importance on the RE mag- netic ordering of the hexagonal nets of Fe atoms [8, 9] from the k1, k2 and notably sites on the j1 and j2 sublat- tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Indeed the largest contributions to the Weiss fields at the RE sites originate from the j1 and j2 sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The TM-TM interactions are particularly varied rang- ing from FM (red) for the majority to AF (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The j1 sites have AF interactions with e, c and RE sites and strong FM ones with j2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' This frustration drives this sublattice’s aversion to magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The diver- sity of the interactions stems from the profound effect 60 40 30 40 20 Site 10 20 0 10 20 40 60 Site i4 that atom coordination and spacing of Fe atoms in a metallic material has on its magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The archetype for this quintessentially itinerant electron effect is fcc Fe where squeezing the lattice turns ferromagnetic order anti-ferromagnetic and then destroys it [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The diverse nature of the magnetic order on the six Fe sub-lattices also has an impact on the intrinsic MCA generated by the system’s itinerant valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' As found for other TM metal magnets [15, 26], a simple fit in terms of a single ion model is unsatisfactory and, as found for other itinerant electron magnets, two-ion type terms should be included in the model [15, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Further- more, on general grounds, modelling the MCA as a sum of single ion anisotropy terms must be done extremely carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The various Fe sites have different crystallo- graphic point symmetries, and their unique symmetry axes do not necessarily point along the c direction [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' There are important implications for atomistic spin dy- namics simulations [14, 31, 34] where it is not correct to assign a single ion anisotropy to each Fe atom with the same angular dependence and same symmetry axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Rather a simpler and more rigorous alternative would be to compute an anisotropy energy based on the vector sum of all moments in the same sublattice so that a single symmetry axis is recovered [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' While significantly smaller than from the f-electrons of the Nd atoms, the primarily TM component of the MCA grows in importance with rising T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' As the RE MCA drops swiftly along with the magnetic order [35, 36], the TM contribution can actually increase as shown ex- plicitly in measurements on Y2Fe14B [11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Such non-monotonic temperature variation is puzzling, and has been attributed to a magnetostructural effect from an anisotropic expansion of the crystal lattice, compet- ing single-ion-like contributions [20] or competing single and two-ion MCA using atomistic spin dynamics simula- tions [31, 32] Since fully relativistic effects such as spin- orbit coupling are included in our DLM-DFT theory we investigate the MCA temperature dependence directly and show our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3 for Y2Fe14B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Using the highly accurate, full potential (FP) KKR code [37, 38], we first calculate the MCA at T = 0K to be ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='9 meV/formula unit (FU) which agrees well with experimental values [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The same rapid loss of magnetic order with increasing temperature which we find for the Fe 8j1 sites in Nd2Fe14B (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1(b)) is also evident in Y2Fe14B [9] and this points to a significant role for this sublattice in the anomalous MCA T-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We therefore carry out further FP MCA calculations where now the Fe 8j1 sites are constrained to be magnetically disordered (m8j1 = 0) via an equal weighting of local moments on each of these sites along the ±x and ±z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The effect on the computed MCA is striking it increases greatly to ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='7 meV/FU - and we infer that the much faster decrease of 8j1 magnetic order with temperature relative to that on the other Fe sublattices is the key driver for the MCA T-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' To test this proposition, we calibrate DLM-DFT MCA values against our T = 0K FP MCA calculations, given the current implementation with an atomic sphere ap- proximation (ASA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Although the ASA values are smaller than the FP ones, the same large increase of the value when the Fe 8j1 sites are magnetically disordered is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3 we show the DLM-DFT temperature de- pendent MCA both using the ASA (red curve) and also scaled by the ratio between the FP and our ASA T = 0K values (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The increase with temperature is evident, peaking at T/Tc = 50% in line with experiment [11] con- firming our proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Since the calculations are for a fixed lattice structure, we can exclude thermal expansion as a cause of the non-monotonic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' We also show the effect on the MCA of forcing the Fe8j1 sublattice to remain magnetically disordered at all T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' m8j1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The resulting unscaled MCA, shown in black in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3, is dramatically altered - the peak has gone and the MCA decays linearly with temperature and the T = 0K value is enhanced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Clearly, establishment of mag- netic order on the Fe8j1 sublattice correlates with a sub- stantial drop in the (uniaxial) MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 T/TC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 K1 (meV/FU) Exp Th Th-8j1 Th-scl FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The T-dependence of the leading anisotropy constant, K1, of Y2Fe14B from DLM-DFT theory (red curve) and ex- periment (blue) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The green curve shows the theory values, scaled to account for the difference between FP [37] and ASA ([9]) at T = 0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The black curve shows K1 (unscaled) if the Fe 8j1 sublattice is constrained to be disordered magnetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Our ultimate goal is to describe Nd2Fe14B’s large mag- netic anisotropy and its temperature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' So to the TM MCA we add the dominant RE components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' These are calculated [6, 39] from the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1 where the crystal field coefficients [9] are determined from first prin- ciples [16], and exchange field, Bexch provided directly by the DLM-DFT Weiss field for each Nd site (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1) di- vided by the computed Nd spin moment of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='66 µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Our calculated exchange fields of 699 and 725 T for the RE f and g sites respectively are somewhat larger than those used in fits of experimental magnetization 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='0 T/Tc 0 5 10 µ0HA(T) Computed Computed-scl Exp (Hirosawa 1986) Exp (Grossinger 1986) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Evolution of the anisotropy field, HA, versus T/Tc from theory compared to experimental measurements (from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' [42], black curve, [11], blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The agreement is good above Tc/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The red and green curves use the non-RE MCA taken from the red and green curves of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' data (450–520 T [40]), but as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' [8], the large number of parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 1 introduce sig- nificant uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' In principle, INS data would pro- vide a direct measure of the exchange fields but are not available for Nd-Fe-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Our proposed values are, however, supported by the good agreement between INS experi- ments [41] and our DLM-DFT calculations for the re- lated Gd2Fe14B magnet (324 T vs 307/319 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The Gd exchange fields are substantially smaller than those cal- culated for Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The relative difference (∼2) mirrors that of the spin moments (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='46 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='66 µB) and reflects the similar Weiss fields we calculate for the two materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Using the method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' [39] we calculate effective anisotropy constants K1(T), K2(T) and anisotropy field, µ0HA = 2K1/M ab initio which is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The red/green curves show µ0HA which includes the non- RE contribution to the MCA of the red/green plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' 4 also shows the experimental measurements from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' [11, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Below T/Tc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='5 there is some dis- crepancy between the two sets of experimental data, but above there is consistency between both the experiments and our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The calculations show the clear importance of the Fe-dominated MCA to the anisotropy field at high temperatures - the red curve is over 1 T less than the green one over a range of temperatures despite the contributions from the non-RE MCAs differing by less than 30 µeV per Fe atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Nd2Fe14B’s spin reorientation transition (SRT) at 135K [8, 14, 43] is not described by our calculations owing to an underestimate of the high order crystal field coeffi- cients [14, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' This shortcoming exemplifies a more general challenge for theory modelling of low T strongly correlated f-electron effects to construct a robust way to significantly enhance the value of these coefficients [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Around room temperature and above, however, the ef- fects on the MCA from these high order terms are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' This is also the temperature regime where the tenets of our DLM-DFT theory are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Nd2Fe14B and the RE-TM permanent magnet family to which it belongs have a compelling set of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Their technological value is enormous and growing and their magnetic properties, at a fundamental level, come from a rich and subtle combination of RE, localized, and TM, itinerant electron, effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' To enhance magnetic functionality and extract pointers for the development of even better materials, multiple interrelated aspects have to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' Our ab initio DLM-DFT modelling has shown the importance of describing accurately the rich and complex itinerant electron magnetism associ- ated with the Fe sites and valence electrons generally for the production of the robust exchange field acting on the RE atoms, the higher order effective spin interactions and the nature of the non-f electron MCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The modifi- cations proposed here should be incorporated into future atomistic, effective spin and micromagnetic modelling to correctly describe these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' The work was supported by EPSRC (UK) Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' EP/M028941/1 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=') and Royal Society Re- search Grant RGS\\R1\\201151 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content=' ∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='bouaziz@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfCwJj/content/2301.02868v1.pdf'} +page_content='de † j.' metadata={'source': 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This paper +presents the modeling, control and simulation of a new type of hybrid micro-small UAVs, +coined as lifting-wing quadcopters. +The airframe orientation of the lifting wing needs to +tilt a specific angle often within 45 degrees, neither nearly 90 nor approximately 0 degrees. +Compared with some convertiplane and tail-sitter UAVs, the lifting-wing quadcopter has a +highly reliable structure, robust wind resistance, low cruise speed and reliable transition flight, +making it potential to work fully-autonomous outdoor or some confined airspace indoor. In the +modeling part, forces and moments generated by both lifting wing and rotors are considered. +Based on the established model, a unified controller for the full flight phase is designed. The +controller has the capability of uniformly treating the hovering and forward flight, and enables +a continuous transition between two modes, depending on the velocity command. What is more, +by taking rotor thrust and aerodynamic force under consideration simultaneously, a control +allocation based on optimization is utilized to realize cooperative control for energy saving. +Finally, comprehensive Hardware-In-the-Loop (HIL) simulations are performed to verify the +advantages of the designed aircraft and the proposed controller. +Nomenclature +𝑜e𝑥e𝑦e𝑧e += +Earth-Fixed Coordinate Frame(eF ) +𝑜b𝑥b𝑦b𝑧b += +Quadcopter-Body Coordinate Frame(bF ) +𝑜l𝑥l𝑦l𝑧l += +Lifting-Wing Coordinate Frame(lF ) +𝑜w𝑥w𝑦w𝑧w += +Wind Coordinate Frame(wF ) +ep += +Position in eF +ev += +Velocity in eF +bva, lva += +Airspeed vector in bF and lF , respectively +evw += +Wind velocity in eF +𝑉a += +Airspeed +∗Corresponding Author, is with the School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China. +Email: qq_buaa@buaa.edu.cn. +arXiv:2301.00730v1 [cs.RO] 2 Jan 2023 + +𝜙, 𝜃, 𝜓 += +Euler angles in bF +𝜔𝑥b, 𝜔𝑦b, 𝜔𝑧b += +Angular velocity in bF +𝛼 += +Angle of attack in lF +𝛽 += +Sideslip angle in lF +𝐶𝐿 += +Aerodynamic lift coefficient +𝐶𝐷 += +Aerodynamic drag coefficient +𝐶𝑚 += +Aerodynamic pitch moment coefficient +𝐶𝑌 += +Aerodynamic lateral force coefficient +𝐶𝑙 += +Aerodynamic roll moment coefficient +𝐶𝑛 += +Aerodynamic yaw moment coefficient +𝜅 += +Installation angle of the lifting wing +𝜂 += +Installation angle of the motor +𝑐 += +Mean chord of the lifting wing +𝑏 += +Wingspan of the lifting wing +𝑆 += +Area of the lifting wing +I. Introduction +A. Why Lifting-wing Quadcopter +Unmanned aerial vehicles (UAVs) have attracted lots of recent attention due to their outstanding performances in +many fields, such as aerial photography, precision farming, and unmanned cargo. According to [1], UAV platforms are +currently dominated by three types: fixed-wing UAV, rotorcraft UAV, and their hybrid that integrates the advantages of +the first two. The hybrid UAVs have the capability of Vertical Take-off and Landing (VTOL), which enables more +accessible grounding or holding by hovering. This might be mandated by authorities in high traffic areas such as lower +altitudes in the urban airspace. Furthermore, hybrid UAVs are categorized into two types: convertiplane and tail-sitter. +A convertiplane maintains its airframe orientation in all flight modes, but a tail-sitter is an aircraft that takes off and +lands vertically on its tail, and the entire airframe needs to tilt nearly 90◦ to accomplish forward flight [1, 2]. +In March 2015, Google announced that the tail-sitter UAV for a packet delivery service was scrapped, because it is +still too difficult to control in a reliable and robust manner according to the conclusion came by the project leader [3]. +Some studies try to remedy this by newly designed controllers [4]. However, unlike this way, we will study a new type +of hybrid UAV, coined as the lifting-wing quadcopter [5, 6], to overcome the difficulty Google’s Project Wing faced. A +lifting-wing quadcopter is a quadcopter [7, 8] with a lifting wing installed at a specific mounting angle. During the +flight, the quadcopter will provide thrust upward and forward simultaneously; and the lifting wing also contributes a +2 + +a) VertiKUL 2 +b) Vespertilio +c) RflyLW2 +d) Prime Air +Fig. 1 +Prototypes of lifting-wing quadcopters. +lifting force partially. +As shown in Fig. 1, as far as we know, some prototypes of lifting-wing quadcopters in public can be found, such as +VertiKUL2 by the University of Leuven (Fig. 1 (a), Sept 2015)[9], Vespertilio by the VOLITATION company(Fig. 1 +(b))[10], the latest version of the Prime Air delivery drone unveiled by Amazon(Fig. 1(d), Jun 2019)[11] and, RflyLW2 +by us(Fig. 1 (c)) [5, 6]. +The lifting-wing quadcopter is a new type of hybrid UAV because convertiplane and tail-sitter UAVs in their cruise +phase work as fixed-wing UAVs, so the airframe orientation is defined as the head of the fixed-wing. But, the lifting-wing +quadcopter is more like a quadcopter. The airframe orientation of the lifting wing needs to tilt a specific angle often +within 45◦, neither nearly 90◦ (corresponding to tail-sitter UAVs) nor approximately 0◦(corresponding to convertiplane +UAVs). Fig. 2 shows the full flight phase of the three VTOL UAVs. The design and performance evaluation of +lifting-wing quadcopters have been studied extensively in [5, 6]. In order to make this paper self-contained, we briefly +introduce the advantages of the lifting-wing quadcopter compared with some convertiplane and tail-sitter UAVs. +• Highly reliable structure. It does not require extra transition actuators. This is a reliable structure by eliminating +the need for complicated control. +• Robust wind resistance. It has a shorter lifting wing compared with the corresponding fixed wing of convertiplanes +and tail-sitter UAVs, because rotors can share the lift. Moreover, as shown in Fig. 4, it does not have a vertical +rudder. Instead, this function is replaced by the yaw control of the quadcopter component. In order to improve the +3 + +融天 +福阳融天 +MOITATIJOVAsmirgc9iocobret① +② +③ +② +② +② +③ +④ +④ +① +① +② +③ +④ +② +(c) Lifting-wing quadcopter +(b) Convertiplane UAV +(a) Tail-sitter UAV +① Vertical Take-off +② Transition +③ Level Flight +④ Vertical Landing +a) Tail-sitter UAV +b) Convertiplane UAV +c) Lifting-wing quadcopter +① +② +③ +② +④ +② +③ +② +① +② +③ +② +④ +① Vertical Take-off +② Transition +③ Forward Flight +④ Vertical Landing +④ +① +Fig. 2 +Different flight modes of some VTOL UAVs. +yaw control ability, the axes of rotors do not point only upward anymore as shown in Fig. 4(a). This implies that +the thrust component by rotors can change the yaw directly rather than merely counting on the reaction torque of +rotors. From the above, the wind interference is significantly reduced on the one hand; on the other hand, the yaw +control ability is improved. As a result, it has better maneuverability and hover control to resist the disturbance of +wind than those by tail-sitter and convertiplane UAVs. +• Low cruise speed. It can make a cruise at a lower speed than that by convertiplanes and tail-sitter UAVs, +meanwhile saving energy compared with corresponding quadcopters. This is very useful when a UAV flies in +confined airspace such as a tunnel, where the high speed is very dangerous. Although current hybrid UAVs can +have a big or long wing for low cruise speed, they cannot work in many confined airspace due to their long +wingspan. However, the lifting-wing quadcopter can be small. +• Reliable transition flight. When a tail-sitter UAV performs transition flight, the velocity and angle of attack will +change dramatically, leading to complicated aerodynamics even stall. Unlike tail-sitter UAVs, the lifting-wing +quadcopter only has to tilt a specific angle often smaller than 45◦ rather than 90◦. The airflow on the lifting wing +is stable, and lift and drag are changed linearly with the angle of attack. These will avoid great difficulty (or say +danger) in controlling within the full flight envelope. +With the four features above, it is potential for the lifting-wing quadcopter to work fully-autonomously outdoors or +in some confined airspace indoors replacing with corresponding quadcopters. The further comparisons with multicopter, +4 + +noitiansTno-seT IeoinsV +gnibasI IeoinsV +A ELTable 1 +Comparison of different VTOL UAVs. +Endurance +Reliability +Wind Resistance at Hover +Flight Range +Multicopter tilt-rotor/wing convertiplane +4 +2 +2 +4 +Multicopter dual-system convertiplane +3 +4 +2 +3 +Multicopter tail-sitter +4 +4 +1 +4 +Lifting-wing multicopter +2 +4 +4 +2 +Multicopter +1 +5 +5 +1 +Note: bigger number implies better. +Endurance +Flight Range +Wind Resistance at Hover +Reliability +Multicopter tilt-rotor/wing +convertiplane +Multicopter dual-system +convertiplane +Multicopter tail-sitter +Lifting-wing Multicopter +Multicopter +Fig. 3 +Comparison of different VTOL UAVs. +tilt-rotor/wing convertiplane, multicopter dual-system convertiplane and multicopter tail-sitter [2] are summarized in +Tab. 1 and Fig. 3. As shown, the lifting-wing quadcopter possesses the feature between current hybrid UAVs and +quadcopters, and it is more like a quadcopter. +B. Control of Current Hybrid UAVs +The control of the lifting-wing quadcopter has the following two distinguishing features. +• Unified control for full flight phases. Hybrid UAVs often have three different flight modes, including the hover, +the transition flight, and the forward flight. By taking the multicopter tilt-rotor/wing convertiplane, multicopter +dual-system convertiplane and multicopter-tail sitter, for example, their take-off and landing are controlled only by +the quadcopter component, while the forward flight is controlled like a fixed-wing aircraft. The two control ways +are very different, so the transition flight is challenging due to the nonlinearities and uncertainties. However, a full +flight phase of the lifting-wing quadcopter always involves thrust by the quadcopter and aerodynamic force by the +5 + +19i2list1oto1itluM-- +9li-1oitlMliniwo-lititl +19voH1690n6t2i291bniW +豪0 +....p!! +2 +Euqnlgucelifting wing. Therefore, the lifting-wing quadcopter can be considered under only the transition flight mode in the +full flight phase (hover control here also will take the aerodynamic force into consideration due to wind on the +lifting wing). As a result, a unified control is needed. Fortunately, the lifting-wing quadcopter only needs to tilt a +specific angle often smaller than 45◦, rather than 90◦ like tail sitter UAVs. This reduces the possibility of having a +stall. +• Cooperative control for energy saving. The transition flight for current hybrid UAVs is very short, so not +too much attention needs to pay to energy consumption in practice. However, it should be considered for the +lifting-wing quadcopter as it is under the transition flight mode in the full flight phase. Cooperative control for +energy saving is feasible. For example, roll control can be performed by both the quadcopter component and the +ailerons by the lifting wing. Obviously, the aileron control is more energy-saving. +Among the control phase of a hybrid UAV, the transition control is the most challenging issue [12], especially for +tail-sitter UAVs. Since the actuators of tail-sitter UAVs are like those of lifting-wing quadcopters, the existing transition +control of tail-sitter UAVs can be used for reference. +(i) Trajectory open-loop control for transition flight. +The open-loop trajectory tracking control is very straightforward. The principle is to make the UAV enter into +another mode’s condition by focusing on controlling some variables like altitude other than the trajectory, +then switch to the controller of the next mode. For example, increasing thrust and reducing its pitch angle at +the same time can make a tail-sitter UAV enter into forwarding flight [13, 14]. The aim is to keep the altitude +the same [14]. Because the transition time for tail-sitter UAVs is short, the trajectory will not change too +much. Obviously, this method is inapplicable to lifting-wing quadcopters. +(ii) Trajectory closed-loop control for transition flight. +• Linearization method based on optimization. According to the trajectory and the model, the reference +state and feedforward are derived by optimization in advance [15, 16]. Based on them, the linearization +can be performed. With the resulting linear model, existing controllers against uncertainties and +disturbance are designed [17, 18]. As for the lifting-wing quadcopter, this method is applicable when +the model and transition trajectory are known prior. Furthermore, cooperative control of the quadcopter +component or the ailerons of the lifting wing can be performed by taking energy-saving into optimization. +However, in practice, the model is often uncertain as the payload is often changed, such as parcel +delivery. Also, this method is not very flexible due to that the trajectory has to be known a priori. +• Nonlinear control method. One way is to take all aerodynamic forces as disturbances, and only the +quadcopter component works for the flight transition [19, 20]. This requires that the quadcopter has +a strong control ability to reject the aerodynamic force. Another way takes the aerodynamic force +into consideration explicitly to generate a proper attitude [21, 22]. How to cooperatively control the +6 + +quadcopter component and the actuators of fixed-wing is not found so far. This is because, we guess, +the transition flight is often short, and more attention is paid to making the UAV stable by reducing the +possibility of stall rather than optimization. +As shown above, the linearization method based on optimization is somewhat not flexible, but cooperative control for +energy saving can be performed in an open-loop manner. The nonlinear control method is flexible but not considering +how to control cooperatively for energy saving. +C. Our Work and Contributions +In this paper, we will consider designing a unified controller for the full flight phase of a lifting-wing quadcopter. +What is more, the quadcopter component and the ailerons of the lifting wing work cooperatively to save energy. First, +we build the model of the lifting-wing quadcopter. Unlike the tail-sitter UAV, it does not have a rudder, and its tilted +rotors will generate force components on the XY-plane in the quadcopter-body coordinate frame (traditional quadcopters +do not have the force component on the XY-plane). Because of this, the translational dynamic involves five control +variables, namely three-dimensional force in the quadcopter-body coordinate frame and two Euler angles (pitch and roll +angles), further by considering the aerodynamic force determined by Euler angles. However, it is difficult and a bit +too early to determine the five control variables according to the three-dimensional desired acceleration, because it is +hard to obtain the bounds of these control variables. An improper choice may not be realized by actuators. To this +end, we only choose the 𝑜b𝑧b force (the main force component) in the quadcopter-body coordinate frame and two Euler +angles (pitch and roll angles) to determine the desired acceleration uniquely, leaving the other two force components as +a lumped disturbance. This adopts the controlling idea of quadcopters [7, 23], but the computation method is different +due to the existence of the aerodynamic force. With the determined Euler angles, moments are further determined in the +lifting wing coordinate frame. So far, the unified control for the full flight phase is accomplished. Finally, we will utilize +the control allocation to realize cooperative control for energy saving. The 𝑜b𝑧b force and three-dimensional moments +will be realized by four rotors and two ailerons. This is why we have the freedom to optimize the allocation for saving +energy. The principle behind this is to make the aerodynamic force (two ailerons) undertake the control task as much as +possible because aileron control is more energy-saving than rotor control. As a result, cooperative control for energy +saving is accomplished. +The contributions of this paper are: (i) establish the model of a lifting-wing quadcopter for the first time; (ii) a +unified controller design for the full flight phase of the lifting-wing quadcopter; (iii) control allocation for energy-saving +performance. Comprehensive HIL simulation experiments are performed to show (i) the proposed lifting-wing +quadcopter is more energy-saving with aileron; (ii) synthesizing the angular rate command from the coordinated turn in +high-speed flight can reduce sideslip; (iii) the transition phase of the proposed lifting-wing quadcopter is significantly +better than the tail-sitter and UAVs. +7 + +ex +ey +bx +by + +c +b +x +d +y +d +1 +2 +3 +4 +a) Front view +b) Left view +c) Top view +d) 3D view +ey +ez +bz +by + + +CoG +ex +lx +lz +bz + + +bx +ar + +al + +CoG +CoG + +al + +ar + +Fig. 4 +Coordinate frames and nomenclatures. +II. Coordinate Frame +A lifting-wing quadcopter is divided into two components, the lifting-wing component and the quadcopter component +as shown in Fig. 4. According to these, the following coordinate frames are defined. +A. Earth-Fixed Coordinate Frame (eF ) +The earth-fixed coordinate frame 𝑜e𝑥e𝑦e𝑧e is an inertial frame. The 𝑜e𝑧e axis points perpendicularly to the ground, +and the 𝑜e𝑥e axis points to a certain direction in the horizontal plane. Then, the 𝑜e𝑦e axis is determined according to the +right-hand rule. This frame is fixed, the initial position of the lifting-wing quadcopter or the center of the Earth is often +set as the coordinate origin 𝑜e. +B. Quadcopter-Body Coordinate Frame (bF ) +The quadcopter-body coordinate frame 𝑜b𝑥b𝑦b𝑧b is fixed to the quadcopter component of a lifting-wing quadcopter. +The Center of Gravity (CoG) of the lifting-wing quadcopter is chosen as the origin 𝑜b of bF . The 𝑜b𝑥b axis points to +the nose direction in the symmetric plane of the quadcopter. The 𝑜b𝑧b axis is in the symmetric plane of the quadcopter, +pointing downward, perpendicular to the 𝑜b𝑥b axis. The 𝑜b𝑦b axis is determined according to the right-hand rule. +8 + +C. Lifting-Wing Coordinate Frame (lF ) +The lifting-wing coordinate frame 𝑜l𝑥l𝑦l𝑧l is fixed to the lifting-wing component. The origin 𝑜l of lF is also set at +the CoG of the lifting-wing quadcopter. The 𝑜l𝑥l axis is in the symmetric plane pointing to the nose of the lifting wing. +The 𝑜l𝑧l axis is in the symmetric plane of the lifting wing, pointing downward, perpendicular to the 𝑜l𝑥l axis, and the +𝑜l𝑦l axis is determined according to the right-hand rule. The installation angle of the lifting wing, that is the angle +between the 𝑜l𝑥l axis and the 𝑜l𝑥l𝑦l plane, is denoted by 𝜅 ∈ R as shown in Fig. 4(b). +D. Wind Coordinate Frame (wF ) +The origin 𝑜w of the wind coordinate frame 𝑜w𝑥w𝑦w𝑧w is also at the CoG of the lifting-wing quadcopter. The 𝑜w𝑥w +axis is aligned with the airspeed vector. The 𝑜w𝑧w axis is in the symmetric plane of the lifting wing, pointing downward, +perpendicular to the 𝑜w𝑥w axis, and the 𝑜w𝑦w axis is determined according to the right-hand rule. The angle of attack +(AoA), denoted by 𝛼 ∈ R, is defined as the angle between the projection of the airspeed vector on the 𝑜l𝑥l𝑧l plane and +the 𝑜l𝑥l as shown in Fig. 4(b). The sideslip angle, denoted by 𝛽 ∈ R, is defined as the angle between the airspeed vector +and the 𝑜l𝑥l𝑧l plane as shown in Fig. 4(c). +To convert the aerodynamic forces and moments acting on frame wF and lF to bF respectively, two rotation +matrices are defined as followed: +Rb +w(𝜆) = +�������� +cos 𝜆 cos 𝛽 +− cos 𝜆 sin 𝛽 +sin 𝜆 +sin 𝛽 +cos 𝛽 +0 +− sin 𝜆 cos 𝛽 +sin 𝜆 sin 𝛽 +cos 𝜆 +�������� +, Rb +l = +�������� +cos 𝜅 +0 +sin 𝜅 +0 +1 +0 +− sin 𝜅 +0 +cos 𝜅 +�������� +, +where 𝜆 = 𝜅 − 𝛼. And the rotation matrix Re +b maps a vector from frame bF to eF , defined by +Re +b = +�������� +cos 𝜃 cos 𝜓 − sin 𝜃 sin 𝜙 sin 𝜓 +− sin 𝜓 cos 𝜙 +cos 𝜓 sin 𝜃 + cos 𝜃 sin 𝜙 cos 𝜓 +sin 𝜃 sin 𝜙 cos 𝜓 + cos 𝜃 sin 𝜓 +cos 𝜙 cos 𝜓 +sin 𝜓 sin 𝜃 − cos 𝜙 cos 𝜃 sin 𝜙 +− cos 𝜙 sin 𝜃 +sin 𝜙 +cos 𝜙 cos 𝜃 +�������� +. +III. MODELING +A. Assumptions +For the sake of model simplicity, the following assumptions are made: +Assumption 1. The body structure is rigid and symmetric about the 𝑜l𝑥l𝑦l plane. +Assumption 2. The mass and the moments of inertia are constant. +Assumption 3. The geometric center of the lifting-wing quadcopter is the same as the CoG. +Assumption 4. The aircraft is only subjected to gravity, aerodynamic forces, and the forces generated by rotors. +9 + +B. Flight Control Rigid Model +By Assumptions 1-2, the Newton’s equation of motion is applied to get the translational motion as follows +e �p = ev +e�v = Re +b +bf +𝑚 +(1) +where ep = +� +𝑝𝑥e 𝑝𝑦e 𝑝𝑧e +�T and ev = +� +𝑣𝑥e 𝑣𝑦e 𝑣𝑧e +�T are the position and velocity expressed in frame eF respectively; +𝑚 is the mass, bf is the total force acting on the airframe expressed in frame bF . +To facilitate attitude control and combine the control characteristics of the rotor and lifting wing, the rotational +dynamics is carried out in frame lF . It is given by Euler’s equation of motion as +�Re +l = Re +l +�l𝝎 +� +× +J · l �𝝎 = lm − l𝝎 × (J·l𝝎) +(2) +where the rotational matrix is derived by Re +l = Re +b Rb +l , Rb +l being a constant matrix; lm is the total moment acting +on the airframe expressed in frame lF , l𝝎 = +� +𝜔𝑥l 𝜔𝑦l 𝜔𝑧l +�T is the angular velocity in frame lF , +�l𝝎 +� +× denotes the +skew-symmetric matric +�l𝝎 +� +× = +�������� +0 +−𝜔𝑧l +𝜔𝑦l +𝜔𝑧l +0 +−𝜔𝑥l +−𝜔𝑦l +𝜔𝑥l +0 +�������� +, +and J ∈ R3×3 is the inertia matrix given by +J = +�������� +𝐽𝑥 +0 +−𝐽𝑥𝑧 +0 +𝐽𝑦 +0 +−𝐽𝑥𝑧 +0 +𝐽𝑧 +�������� +. +C. Forces and Moments +By Assumptions 3-4, the total forces and moments acting on the UAV are decomposed into three parts: the +aerodynamic forces and moments acting on the airframe (fa and ma), the forces and moments generated by rotors (fr and +mr), and the gravitational forces f𝑔, where f𝑔 = [0 0 𝑔]T, 𝑔 is the gravitational acceleration. The front two types of +forces and moments will be described detailly in the following two subsections. +10 + +Lf +Df +Yf +a +v +x +av +z +av +y +av +1T +2T +3T +4T + +lo +ly +lx +lz + + +Fig. 5 +Forces act on a lifting-wing quadcopter. +1. Forces and Moments in the Quadcopter Component +In the quadcopter part, the thrust and torque produced by one rotor are given by +𝑇𝑖 = 𝐾 𝑓 𝜛𝑖2, 𝑀𝑖 = 𝐾𝑚𝜛𝑖2 = 𝐾𝑚 +𝐾 𝑓 +𝑇𝑖 +(3) +where 𝐾 𝑓 > 0 is the lift force coefficient, 𝐾𝑚 > 0 is the drag torque coefficient, and 𝜛𝑖 is the angular rate of the ith +rotor, 𝑖 = 1, 2, 3, 4. In order to improve the controllability during performing yaw, an installation angle 𝜂 is set as shown +in Fig. 4(a). The left motors tilt to the positive left and right motors tilt to the positive right. +Because of the installation angle 𝜂, the forces and moments produced by rotors are expressed by the thrust on each +propeller 𝑇𝑖 as +������������� +𝑓𝑟𝑦 +𝑓𝑟𝑧 +𝑚𝑟𝑥 +𝑚𝑟𝑦 +𝑚𝑟𝑧 +������������� += +������������� +sin 𝜂 +− sin 𝜂 +− sin 𝜂 +sin 𝜂 +− cos 𝜂 +− cos 𝜂 +− cos 𝜂 +− cos 𝜂 +−𝑑𝑦 cos 𝜂 +𝑑𝑦 cos 𝜂 +𝑑𝑦 cos 𝜂 +−𝑑𝑦 cos 𝜂 +𝑑𝑥 cos 𝜂 +−𝑑𝑥 cos 𝜂 +𝑑𝑥 cos 𝜂 +−𝑑𝑥 cos 𝜂 +𝐾1 +𝐾1 +−𝐾1 +−𝐾1 +������������� +���������� +𝑇1 +𝑇2 +𝑇3 +𝑇4 +���������� +, +(4) +where 𝐾1 = 𝐾𝑚 +�𝐾 𝑓 + 𝑑𝑥 sin 𝜂, 𝑑𝑥 and 𝑑𝑦 are the components of the distance from the center of the lifting-wing +quadcopter to a propeller on the 𝑜b𝑥b𝑦b plane, as shown in Fig. 4(c). +2. Forces and Moments in the Lifting-wing Component +The aerodynamic forces and moments acting on the lifting wing are mainly generated by the lifting wing itself and +the ailerons at the trailing edge, as shown in Fig. 5. +11 + +Let evw be the wind velocity in eF . Then +bva = bv − (Re +b)T · evw. +(5) +Thus, the airspeed vector lva = +� +𝑣a𝑥 𝑣a𝑦 𝑣a𝑧 +�T and airspeed 𝑉𝑎 are defined as +lva = (Rb +l )T · bva, +(6) +𝑉a = +√︃ +𝑣2a𝑥 + 𝑣2a𝑦 + 𝑣2a𝑧. +(7) +The aerodynamic angles 𝛼 and 𝛽 are defined as +𝛼 = tan−1( 𝑣a𝑧 +𝑣a𝑥 +), 𝛽 = sin−1( +𝑣a𝑦 +𝑉a +). +(8) +In the longitudinal plane, lift, drag, and pitching moment acting on the lifting-wing body are given by +𝑓𝐿 = 𝑄𝑆(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒) +𝑓𝐷 = 𝑄𝑆(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒) +𝑚 = 𝑄𝑆𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒). +(9) +The lateral force and the roll and yaw moments acting on the lifting-wing body are given by +𝑓𝑌 = 𝑄𝑆(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) +𝑙 = 𝑄𝑆𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎) +𝑛 = 𝑄𝑆𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎) +(10) +where 𝑄 = 1 +2 𝜌𝑉2 +𝑎; 𝐶𝐿, 𝐶𝐷, 𝐶𝑚,𝐶𝑌 , 𝐶𝑙 and 𝐶𝑛 are nondimensional aerodynamic coefficients, 𝐶𝐿 𝛿𝑒, 𝐶𝑚𝛿𝑒, 𝐶𝐷 𝛿𝑒,𝐶𝑌 𝛿𝑎, +𝐶𝑛𝛿𝑎 and 𝐶𝑙𝛿𝑎 are control derivative; 𝑆 is the area of the lifting wing, 𝑐 is the mean chord of the lifting wing, 𝑏 is the +wingspan of the lifting-wing aircraft, 𝛿𝑒 and 𝛿𝑎 are calculated using the right and the left aileron (𝛿𝑎𝑟 and 𝛿𝑎𝑙, as shown +in Fig. 4(d)) as +� +𝛿𝑒 +𝛿𝑎 +� += +� +1 +1 +−1 +1 +� � +𝛿𝑎𝑟 +𝛿𝑎𝑙 +� +. +(11) +The external forces and moments are summarized as +12 + +Table 2 +Lifting-wing quadcopter structure parameters, lift force and drag torque coefficients +𝑚 +Aircraft mass +1.92 kg +𝜅 +Installation angle of lifting wing +34 deg +𝜂 +Installation angle of motor +10 deg +𝑑𝑥 +The distance from 𝑜𝑏𝑥𝑏 to a propeller +0.25 m +𝑑𝑦 +The distance from 𝑜𝑏𝑦𝑏 to a propeller +0.2125 m +[𝐽𝑥𝑥 𝐽𝑦𝑦 𝐽𝑧𝑧] +Moment of inertia +[5.12 5.54 7.6] × 10−2 kg · m2 +𝑏 +Wingspan of the lifting-wing aircraft +0.94 m +𝑐 +Mean chord of the lifting wing +0.17 m +𝐾𝑚 +Drag moment coefficient +5.875e-07 kg · m2 +𝐾 𝑓 +Lift force coefficient +2.824e-05 kg · m2 +bf = +�������� +0 +𝑓𝑟𝑦 +𝑓𝑟𝑧 +�������� ++ 𝑄𝑆Rb +w +�������� +−(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒) +(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) +−(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒) +�������� ++ 𝑚Rb +ef𝑔 +(12) +lm = Rl +b +�������� +𝑚𝑟𝑥 +𝑚𝑟𝑦 +𝑚𝑟𝑧 +�������� ++ 𝑄𝑆 +�������� +𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎) +𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒) +𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎) +�������� +. +(13) +The structure parameters, lift force and drag torque coefficients of the lifting-wing quadcopter are given in Tab.2. +IV. CONTROLLER DESIGN +The successive loop closure is a common control architecture for UAVs [24], which consists of an outer-loop +controlling the position and an inner-loop for attitude, as illustrated in Fig. 6. The basic idea behind successive +loop closure is to close several simple feedback loops in succession around the open-loop plant dynamics rather than +designing a single control system. The position controller receives the desired position and then computes the desired +acceleration. Then the desired acceleration is mapped to the collective thrust and attitude. The attitude command is sent +to the inner-loop, while the thrust command skips directly to the control allocation. To facilitate the control experiment +step by step, the attitude can also be commanded by the pilot. Furthermore, the attitude controller receives the desired +attitude and generates the desired moment. Finally, the control allocation algorithm distributes the moment command +from the inner-loop and the direct force command from the outer-loop to corresponding ailerons and rotors. +13 + +Position +Controller +Attitude +Controller +Control +Allocation +d +m +Lifting-wing +Quadcopter +Dynamics +, +T δ +d +p +Mannual +Inputs +Signal +Distribution +d +Θ +df +d +Θ +df +df +d +Θ +a +, +p v +av +q +Fig. 6 +Control structure +A. Controller Design Model +Since 𝛼, 𝛽 are not easy to obtain, we consider that 𝛼 ≈ 𝜅 + 𝜃 and 𝛽 ≈ 0. The translational dynamic involves five +control variables, namely three-dimensional forces in frame bF and two Euler angles (pitch and roll angles). However, it +is a bit too early to determine the five control variables according to the three-dimensional desired acceleration because +it is hard to obtain the bounds of these control variables. An improper choice may not be realized by ailerons. To +this end, we only choose 𝑓𝑧 (the main force component) in frame bF and two Euler angles (pitch and roll angles) to +determine the desired acceleration uniquely, and the desired yaw angle is directly specified, leaving 𝑓𝑦 as a disturbance. +This adopts the controlling idea of quadcopters[7, 23], but the computation method is different due to the existence of +the aerodynamic force. According to the idea above, we rewrite the system Eqs. (1) and (2) in the form as +e �p = ev +e�v = u + g + d1 +�Re +l = Re +l +�l𝝎 +� +× +l �𝝎 = J−1 · lm + d2 +. +(14) +Here +u = +Re +b +𝑚 +���� +� +�������� +0 +0 +− 𝑓𝑧 +�������� ++ 𝑄𝑆Rb +w +�������� +−𝐶𝐷 +0 +−𝐶𝐿 +�������� +���� +� +, +and d1, d2 are disturbances, where +d1 = +Re +b +𝑚 +���� +� +�������� +0 +𝑓𝑦 +0 +�������� ++ 𝑄𝑆Rb +w +�������� +−𝐶𝐷 𝛿𝑒𝛿𝑒 +(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) +−𝐶𝐿 𝛿𝑒𝛿𝑒 +�������� +���� +� +d2 = −J−1 · l𝝎 × (J · l𝝎). +14 + +B. Position Control +Given a twice differentiable trajectory pd(𝑡), in order to satisfy lim +𝑡→∞ ∥ep(𝑡) − pd(𝑡)∥ = 0, the desired ud for Eq. (14) +can be designed as a PID controller in the form +ud = −g + �pd − KPd (ev − �pd) − KPp (ep − pd) − KPi +∫ +(ep − pd) d𝑠 +(15) +where KPp, KPi, KPd ∈ R3×3 are diagonal matrices acting as control gains. The left work is to determine desired thrust +by rotors 𝑓d ∈ Ω 𝑓 and 𝜃d, 𝜙d ∈ Ω𝑎 such that +( 𝑓d, 𝜃d, 𝜙d) = +arg min +𝑓𝑧 ∈Ω 𝑓 ,𝜃,𝜙∈Ω𝑎 +∥u ( 𝑓𝑧, 𝜃, 𝜙) − ud∥ +(16) +where Ω 𝑓 is a set to confine the force, and Ω𝑎 is a set to confine the pitch and roll. In order to reduce drag, the vehicle’s +nose should be consistent with the current direction of the vehicle velocity, that is +𝜓d = tan−1 +� 𝑣𝑦e +𝑣𝑥e +� +. +(17) +The attitude command can also be given by the pilot, in case the position controller fails with GPS denied, as shown +in Fig. 6. Finally, the desired attitude is given as 𝚯d = [𝜙d 𝜃d 𝜓d]T. +C. Attitude Control +The attitude controller generates the desired moment from the output of the position controller or the attitude given +by the pilot, as shown in Fig. 6. As far as we know, studies about the hybrid UAV hardly consider the lateral control, +such as turning right or left. As for the considered UAV, the control on yaw is quite different between the multicopter +mode and the fixed-wing mode. To establish a unified control, the attitude control is performed on the lifting-wing +frame lF , so l𝚯d = [𝜙d 𝜃d + 𝜅 𝜓d]T. +1. Basic Attitude Control +The attitude error is presented in the form of quaternion based on which the corresponding controller is designed. +This can guarantee a uniform and good convergence rate for all initial attitude errors [25] +qe = q∗ +d ⊗ qe +l = +� +𝑞e0 𝑞e1 𝑞e2 𝑞e3 +�T, +(18) +15 + +where qd is transformed from l𝚯d with ‘ZXY’ rotational sequence, (·)∗ is the conjugate of a quaternion, and ⊗ is the +quaternion product. Then qe is transformed into the axis-angle form qe = +� +cos 𝜗 +2 𝝃T +e sin 𝜗 +2 +�T by +𝜗 = wrap𝜋 +�2acos �𝑞e0 +�� +𝝃e = +� +[0 0 0]T, +𝜃 = 0 +sign(𝑞e0) +𝜗 +sin 𝜗/2 +� +𝑞e1 𝑞e2 𝑞e3 +�T, +𝜃 ≠ 0. +(19) +The function wrap𝜋(𝜗) constrains the 𝜗 in [−𝜋 𝜋] to ensure the shortest rotation path. To eliminate the attitude error, +the attitude control is designed as +l𝝎ac = sat �K𝚯p𝝃e, 𝝎min, 𝝎max +� +(20) +where K𝚯p ∈ R3×3 is the diagonal matrix acting as the control gain, 𝝎min and 𝝎max ∈ R3 are the minimum and maximum +angular control rates, the function sat (x, xmin, xmax) is defined as +sat (x, xmin, xmax) ≜ +�������� +sat �𝑥1, 𝑥1,min, 𝑥1,max +� +... +sat �𝑥𝑛, 𝑥𝑛,min, 𝑥𝑛,max +� +�������� +, sat �𝑥𝑘, 𝑥𝑘,min, 𝑥𝑘,max +� ≜ +����� +����� +𝑥𝑘,min, +𝑥𝑘 < 𝑥𝑘,min +𝑥𝑘,max, +𝑥𝑘 > 𝑥𝑘,max +𝑥𝑘, +else +. +(21) +2. Lateral Compensation +When the UAV is in high-speed flight, a roll command given to track a specified trajectory will cause a lateral skid. +In order to reduce the sideslip angle when the UAV turns at a high speed, the coordinated turn should be considered. In +lF frame, if it is assumed that there is no wind, the coordinated turn equation is expressed as [24] +�𝜓d = 𝑔 tan 𝜙 +𝑉𝑎 +. +(22) +It should be noted that the Euler angles are the attitude presentation between lF . So the desired yaw rate generated by +coordinated turn is +l𝜔ct = �𝜓d cos 𝜃 cos 𝜙. +(23) +3. Angular Rate Command Synthesis +In the lifting-wing coordinated turn, 𝑉𝑎 being zero makes no sense. In addition, considering that coordinated turn +should not be used when airspeed is small, a weight coefficient related to airspeed is added. So the desired angular rates +are rewritten as +l𝜔d = +� +𝜔dac,𝑥 𝜔dac,𝑦 +� +𝜔dac,𝑧 + 𝑤·l𝜔ct +�� +(24) +16 + +where 𝑤 = sat +� +𝑉𝑎−𝑉min +𝑉max−𝑉min , 0, 1 +� +. When the airspeed is slower than 𝑉min, the desired yaw rate is completely decided by +the basic attitude controller. In contrast, when l𝜔d𝑧 = 0 and the airspeed reaches the specified value 𝑉max, the desired +yaw rate is completely decided by the coordinated turn. +4. Attitude Rate Control +To eliminate the attitude rate error, the controller is designed as +lmd = sat +� +J(−K𝝎p(l𝝎 − l𝝎d) − md,I − K𝝎d(l �𝝎 − l �𝝎d)), −md,max, md,max +� +(25) +where md,I = sat +� +K𝝎i +∫ +(l𝝎 − l𝝎d)d𝑠, − md,Imax, md,Imax +� +, K𝜔p, K𝜔i, K𝝎d ∈ R3×3 are diagonal matrices acting as +control gains, md,Imax is maximum amplitude of integral action, md,max is the maximum moment generated by actuators. +So far, we have obtained 𝑓d and lmd, which will be further realized by 𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙. +D. Control Allocation +The lifting-wing quadcopter is an over-actuated aircraft, which provides six independent control inputs, namely +𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙 to meet a specific thrust 𝑓d ∈ R and moment demand lmd ∈ R3. A method of control allocation +based on optimization is proposed. Recalling the control in translational dynamic, if we determined the three-dimensional +force in bF and two Euler angles (pitch and roll angles) by an optimization before, then the six control variables (plus +desired yaw angle) will be determined by six actuators uniquely. If so, however, the optimization in the control of the +translational dynamic is not related to energy-saving directly. This is why we only choose the 𝑜b𝑧b force (the main force +component) and two Euler angles (pitch and roll angles) to determine the desired acceleration as in Eq.(16). +First, Eqs. (12) and (13) are rearranged as +���������� +𝑓𝑧 +l𝑚𝑥 − 𝑄𝑆𝑏𝐶𝑙 +l𝑚𝑦 − 𝑄𝑆𝑐𝐶𝑚 +l𝑚𝑧 − 𝑄𝑆𝑏𝐶𝑛 +���������� +���������������������������������������� +uv += +���������� +− cos 𝜂 +− cos 𝜂 +− cos 𝜂 +− cos 𝜂 +0 +0 +−𝐾2 +𝐾3 +𝐾2 +−𝐾3 +−𝑄𝑆𝑏𝐶𝑙 𝛿𝑎 +𝑄𝑆𝑏𝐶𝑙 𝛿𝑎 +𝑑𝑥 cos 𝜂 +−𝑑𝑥 cos 𝜂 +𝑑𝑥 cos 𝜂 +−𝑑𝑥 cos 𝜂 +𝑄𝑆𝑐𝐶𝑚𝛿𝑒 +𝑄𝑆𝑐𝐶𝑚𝛿𝑒 +−𝐾5 +𝐾4 +𝐾5 +−𝐾4 +−𝑄𝑆𝑏𝐶𝑛𝛿𝑎 +𝑄𝑆𝑏𝐶𝑛𝛿𝑎 +���������� +������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������ +B +���������������� +𝑇1 +𝑇2 +𝑇3 +𝑇4 +𝛿𝑎𝑟 +𝛿𝑎𝑙 +���������������� +���������� +𝜹 +(26) +where 𝐾2 = 𝑑𝑦 cos 𝜂 cos 𝜅 +𝐾1 sin 𝜅, 𝐾3 = 𝑑𝑦 cos 𝜂 cos 𝜅 −𝐾1 sin 𝜅, 𝐾4 = 𝑑𝑦 cos 𝜂 sin 𝜅 +𝐾1 cos 𝜅, 𝐾5 = 𝑑𝑦 cos 𝜂 sin 𝜅 − +𝐾1 cos 𝜅. The value 𝜹 is the control input of the actuator , uv is the virtual control, B is the control efficiency matrix. +As shown in the Eq.(26), rank (B) = 4, the dimension of 𝜹 is 6, which is higher than that of uv, so Eq.(26) has a +minimum norm solution. To take the control priority of the uv components, actuator priority of 𝜹 and actuator saturation +17 + +under consideration, the control allocation is formulated to be an optimization problem as +min +��Wu +�B𝜹 − uv,d +���2 + 𝛾 +��W𝜹 +�𝜹 − 𝜹p +���2 +s.t. +𝜹− ⩽ 𝜹 ⩽ ¯𝜹 +(27) +where uv,d = +� +𝑓d l𝑚d𝑥 − 𝑄𝑆𝑏𝐶𝑙 l𝑚d𝑦 − 𝑄𝑆𝑐𝐶𝑚 l𝑚d𝑧 − 𝑄𝑆𝑏𝐶𝑛 +�T is the desired virtual control, 𝜹p is the preferred +control vector which will be specified later, Wu ∈ R6×6 is a positive definite weighting matrix that prioritizes the +commands in case the desired virtual input uv,d cannot be achieved, W𝜹 ∈ R6×6 is a positive definite weighting matrix +that prioritizes the different actuators, 𝜹− = max(𝜹min, 𝜹l − Δ𝜹) , ¯𝜹 = min(𝜹max, 𝜹l + Δ𝜹) are lower and upper bounds at +each sampling instant of actuators, 𝜹min and 𝜹max ∈ R6 are actuator position limits, Δ𝜹 ∈ R6 is the rate limit, and 𝜹l is +the last value of 𝜹. +There are two optimization objectives in Eq.(27), namely +��Wu +�B𝜹 − uv,d +���2 and +��W𝜹 +�𝜹 − 𝜹p +���2. The first one is +the primary objective of minimizing the slack variables weighted by Wu, so the weighting factor 𝛾 is often chosen to be +a small value. In many cases, the preferred control vector is set to the last value of 𝜹, 𝜹p = 𝜹l. But, in order to save +energy, we prefer to use the aerodynamic force because the change of rotor force implies the motor’s rotational speed +change, which is power-hungry. According to the consideration above, we can give more weight to 𝛿𝑎𝑟 and 𝛿𝑎𝑙 and set +first four elements of 𝜹p to 𝑇1+𝑇2+𝑇3+𝑇4 +4 +, and last two elements to that of 𝜹l. +The hardware resources of the flight control are very limited, so it is very important to ensure the real-time operation +of the proposed algorithm. Several different algorithms, like redistributed pseudoinverse, interior-point methods, and +active set methods have been proposed to solve the constrained quadratic programming problem. Among them, active +set methods[26] perform well in the considered control allocation problems, because they have the advantage that their +initialization can take advantage of the solution from the previous sample (known as the warm start), which is often a +good guess for the optimal solution at the current sample. This can reduce the number of iterations needed to find the +optimal solution in many cases. The study [27] shows that the computational complexity of the active set methods is +similar to the redistributed pseudoinverse method and the fixed-point algorithm, but the active set methods produce +solutions with better accuracy. +V. Simulation Experiments +In order to verify the three main advantages of the designed aircraft: control performance, energy saving and reliable +transition flight, several experiments are conducted. The verification of these performances is mainly determined by two +factors, namely, with and without ailerons, and with and without coordinated turn. +Therefore, experiments are primarily composed of two parts. One is the comparison of control performance, energy +saving and transition flight with and without aileron. And the other is the comparison of control performance with and +18 + +without coordinated turn, but both with aileron. In addition, the transition flight of three different VTOL vehicles, as +shown in Fig. 2, is analyzed in the HIL simulation environment. +A. Simulation Platform +The HIL simulation is carried out in the RflySim platform [28–30], which provides CopterSim simulator and +supports Pixhawk/PX4 autopilot.When perform the HIL simulation, CopterSim sends sensor data such as accelerometer, +barometer, magnetometer, which is generated by a mathematical model, to the Pixhawk system via a USB serial port. +The Pixhawk/PX4 autopilot will receive the sensors data for state estimation by the EKF2 filter and send the estimated +states to the controller through the internal uORB message bus as the feedback. Then the controller sends the control +signal of each motor as output back to CopterSim. Thereby a close-loop is established in the HIL simulation. Compared +with the numerical simulation, the control algorithm is deployed and run in a real embedded system as the real flight +does. After HIL simulation, the controller will be directly deployed to a real vehicle and further verification experiments +will be performed. +PixHawk Autopilot Hardware System +HIL Real-Time Simulation System +Control +Signals +Sensor +Data +Multicopter Simulation + 3D Display + Fault +Injection +Fig. 7 +HIL simulation of RflyLW2. +The lifting-wing quadcopter model is built in MATLAB/Simulink according to Eqs. (1),(2) (12), (13). Besides, the +dynamics of the motor and servo are modeled as first-order transfers as follows: +𝜛 = +1 +𝑇m𝑠 + 1 (𝐶m𝜎𝑖 + 𝜛b), 𝛿a = +1 +𝑇a𝑠 + 1 (𝐶a𝜎𝑗 + 𝛿b), 𝑖 = 1, 2, 3, 4, 𝑗 = 5, 6 +(28) +where 𝜎𝑖 ∈ [0, 1] is 𝑖th throttle command, 𝑇m and 𝑇a are the time constant parameter of motor and servo response, 𝐶m, +𝐶a, 𝛿b and 𝜛b are constant parameters. The sensors used in HIL simulation include IMU, magnetometer, barometer, +GPS and airspeed meter, are modeled reference to [7]. +Due to the vehicle’s VTOL capabilities, the aerodynamic parameters must not only be modeled up to the stall AoA, +19 + +L +inrgps中 +9 0 +0 +1 [0 +1 4= [0 +X0 +EX+ BI (CCIT) +3 + x2* 14 +[e000 rry +[30 +(+9 +30 +3 + xg sa +玉净大量 +(去+)的内 +To +#4 +F¥4 +[880 xten, +0* -[o +50 +SB +30u +% +20 +O +COWBOr eBOFb +BRITYBIE AFTEBI +一VMWIE0.00 +000 +00.0 +0:00 +0:00 +000OMES +128 +MAd +LETEW +LETEWS +2MUCH +CAIAIR32 +26K11D2V +NE +LMEbut also in the post-stall region, in order to cover the entire flight envelope. The full angle aerodynamic parameters +are obtained by combining the small angle aerodynamic parameters obtained by CFD simulation and the empirical +aerodynamic model [31]. The aerodynamic characteristics at low and large AoA are approximated as +����� +����� +𝐶𝐿𝑆 (𝛼) = +0.5𝑐2 +2 +(𝑐2 − 𝑐3)cos2(𝛼) + 𝑐3 +sin(2𝛼) +𝐶𝐷𝑆 (𝛼) = 𝑐0 + +𝑐2𝑐3 +(𝑐2 − 𝑐3) cos2 (𝛼) + 𝑐3 +sin2 (𝛼) +, +���� +���� +𝐶𝐿𝐿 (𝛼) = 𝑐1 sin (2𝛼) +𝐶𝐷𝐿 (𝛼) = 𝑐0 + 2𝑐1sin2 (𝛼) +, +and a pseudo-sigmoid function +𝜎 (𝛼0, 𝑘, 𝛼) = +1 + tanh �𝑘𝛼2 +0 − 𝑘𝛼2� +1 + tanh +� +𝑘𝛼2 +0 +� +, 𝛼 ∈ [−𝜋, 𝜋) +is used to blend low and large AoA regions together +���� +���� +𝐶𝐿 (𝛼) = 𝐶𝐿𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐿, 𝛼) + 𝐶𝐿𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐿, 𝛼)] +𝐶𝐷 (𝛼) = 𝐶𝐷𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐷, 𝛼) + 𝐶𝐷𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐷, 𝛼)] +. +(29) +The low and large AOA parameters 𝑐0, 𝑐1, 𝑐2, 𝑐3, and blending parameters 𝛼0, 𝑘𝐿, 𝑘𝐷 are turned according to CFD results, +and their values are set to 𝑐0 = 0.055, 𝑐1 = 0.9, 𝑐2 = 13.0, 𝑐3 = 3.3, 𝛼0 = 3deg, 𝑘𝐿 = 38, 𝑘𝐷 = 48. Corresponding +Aerodynamic curves are shown in Fig. 8. +(deg) + +(deg) + +L +C +D +C +and +L +C +D +C +L +D +C +C +Fig. 8 +Aerodynamic parameters obtained by CFD +B. Simulation Experiments +1. Verifying the Effectiveness of Aileron +In order to show the effectiveness of ailerons, two groups of comparative experiments are carried out. The attitude +control and position control are the same, but the control allocation of the first group uses the aileron, while the second +20 + +021- +-J00 +20 +0 +J00 +J20 +500 +2.0 +2.0 +2.0500 +021- +-J00 +-20 +0 +J00 +J20 +500 +-J2 +-JO +2- +0 +2 +JO +J2Time(s) +Throttle×1000 +Power(W) +Fitted Power +Measured Power +Fig. 9 +Throttle command and identification result of the motor +group does not. The second group only depends on the quadcopter control allocation. In this experiment, the aircraft +tracks the specified trajectory, as shown in Fig. 10(a), which is a straight line plus a circle with a radius of 200m. +As shown in Fig. 10(b), after adding the aileron, the control amplitude of the four motors is almost the same trend, +especially when the aircraft transits between straight and circular trajectories, indicating that the attitude control is +mainly realized by the aileron, and motors are more like a thruster on a fixed-wing in this case. However, when the +attitude changes sharply, as shown in Fig. 10(b) during 𝑡 = 22 ∼ 23s, the motor and aileron will participate in the +control at the same time to improve the control performance. When the state of the UAV is in the slow adjustment, the +aileron undertakes the control as much as possible to save energy. +In order to quantify the influence of ailerons, the relationship between throttle command and energy consumption of +the motor and actuator is identified. A sinusoidal signal with the constant amplitude and linear increasing frequency +is designed, and the energy consumption test experiments are carried out on the servo and motor at the same time. +Measurement and identification results of the motor are shown in the Fig.9, in which the amplitude of the throttle +decreases as the frequency increasing, because a low-pass filter in Eq.(28) is applied to it. We find that the motor power +has a quadratic function relationship with a fixed throttle, and the power is different when the motor accelerates and +decelerates. When the change frequency of the throttle command becomes faster, the power is increased. Therefore, the +following formula is established to fit the relationship between throttle command and motor power +𝑃𝑇 = +4 +∑︁ +𝑖=1 +𝑝1𝜎𝑖2 + 𝑝2𝜎𝑖 + 𝑝3 �𝜎 𝑝4 +𝑢𝑝,𝑖 + 𝑝5 �𝜎 𝑝6 +𝑑𝑜𝑤𝑛,𝑖 + 𝑝7 +(30) +21 + +180 +500 +SSO +J40 +seo +580 +300 +400 +420 +eoo +Q20180 +500 +SSO +540 +seo +580 +300 +20 +100 +J20 +50OWith the given lifting-wing quadcopter platform, 𝑝1 = 563.7, 𝑝2 = −147.4, 𝑝3 = 15, 𝑝4 = 1.05, 𝑝5 = 4, 𝑝6 = 1, 𝑝7 = +0.05538. With respect to the servo, its power is negligible, because the power of servo is only 0.2W, far less than the +motor’s, even if the 100g weight is suspended on the two ailerons. +The trajectory tracking experiment is performed three times, with the forward flight speed of 5m/s, 10m/s and 20m/s, +respectively. The powers in three flight speeds are shown in Fig. 10(c). It can be observed that when the speed is 5m/s, +the power is the same, because the aileron is not used at slow flight speed. When the speed increases to 10m/s, the +power with the aileron is slightly smaller. Further when the speed increases to 20m/s, the power with the aileron is +greatly smaller because the aerodynamic force is stronger at high speed. +Time(s), without aileron +T1~T4(N) +Time(s), with aileron +b) Control output with and without aileron when the forward flight speed is 20m/s +c) Control indexs in different forward flight speeds +a) Flight path +Time(s), 5m/s +x(m) +y(m) +-z(m) +Time(s), 10m/s +Time(s), 20m/s +Power(W) +without aileron +with aileron +ar +al + and +(deg) + + +Fig. 10 +The attitude control performance and mixer output with and without aileron +2. Verifying the Effectiveness of Coordinated Turn +As for the lateral control, experiments are carried out in hover and high-speed forward flight, and the results are +shown in Fig. 11. In the high-speed forward flight, as shown in Fig. 11 yellow region, when a turn command is given +for the first time during 30.8 ∼ 42.2s, the lifting-wing quadcopter with coordinated turn has no sideslip angle and has a +better lateral tracking performance, as shown in Fig.11(a.1 vs. b.1, and a.5 vs. b.5). In this state, the control of the +lifting-wing quadcopter is similar to the fixed-wing, and the desired yaw rate is generated by Eq.(23) as the feedforward. +When the turn command is given for the second time during 60.8 ∼ 72.2s, the lifting-wing quadcopter is in low-speed +flight, and the control of the lifting-wing quadcopter is similar to the quadcopter, so a big sideslip angle appears both in +experiments with and without coordinated turn. +22 + +0 +0 +J00 +J20 +S00 +5O +0 +5O0 +0 +J00 +J20 +S00 +0 +2 +10 +J2400 +e00 +0 +500 +Q00 +400 +5OO +0- +-05 +40- +eo- +80-50 +JI +53 +54 +52 +e'812 +5O +52 +30 +32 +40 +42 +20 +J20 +500 +520 +30012 +50 +52 +30 +32 +40 +42 +300 +400 +200J2 +50 +52 +30 +32 +40 +42 +20 +J20F +500 +520 +3000 +20 +J00 +J20 +500 +- +0 +I0 +0 +J00 +J20 +S00 +2 +10 +J2Time(s) +b) Vehicle states with coordinated turn +Time(s) +a) Vehicle states without coordinated turn +Desired +Response +>10m/s +6~10m/s +6~10m/s +>10m/s +(deg) + +(deg) + +30.8-42.2 +60.8-72.2 +(a.1) +(a.2) +(a.3) +(a.4) +(a.5) +(b.1) +(b.2) +(b.3) +(b.4) +(b.5) +Desired Yaw Rate +Yaw Response +(rad) + +a(m s) +V +Fig. 11 +The control response with and without coordinated turn +3. Transition Flight +The transition flight is often defined as the aircraft changing from hover to level flight. To quantify the phase, the +transition process is defined as the time it takes for the aircraft to start a transitional flight to the airspeed greater than +18m/s. The quality of the transition is mainly reflected in the transition time and control accuracy of altitude. In the +simulation, the aircraft will take off to a certain altitude, and then a step signal of −30◦ is given to the pitch channel. +This is because the wind CFD test results show that the maximum energy efficiency is obtained when 𝛼 = 4◦. As a +comparison, the same experiment is performed on a tail-sitter quadcopter. The model of the tail-sitter quadcopter is built +on the lifting-wing quadcopter with the installation angle of the lifting-wing from 34◦ to 90◦, as shown in Fig. 2(a). +Fig.12 shows the response curves of the pitch angle, altitude and airspeed of the lifting-wing quadcopter and +tail-sitter UAV in the transition phrase in the HIL simulation. It can be observed that during the transition phase of +23 + +0 +O +40 +eo +80 +J00 +-50 +0 +5O5O +40 +eo +80 +J00 +-50 +0 +5O5O +40 +e +80 +J00 +-30F +50 +-J0 +050 +40 +eo +80 +J00 +-40 +50 +00 +O +40 +eo +80 +0 +JO +SO5O +40 +e +80 +J00 +JO +5O0 +5O +40 +eo +80 +30 +-50 +-JO +00 +5O +40 +eo +80 +0 +5O +40 +eo0 +SO +40 +eo +80 +2.I- +-J +2.0- +050 +40 +eo +80 +J00 +-J +0Time(s) +Va(m/s) +RflyLW2 +Tail-sitter-60° +Tail-sitter-70° +Take +off +Hovering Transition +Flight +Time(s) +Time(s) +Take +off +Hovering Transition +Flight +Take +off +Hovering Transition +Flight +RflyLW2 +Tail-sitter-60° +Tail-sitter-70° +RflyLW2 +Tail-sitter-60° +Tail-sitter-70° +Pitch(rad) +Altitude(m) +a) Airspeed response +b) Pitch response +c) Altitude response +Fig. 12 +Transition flight of lifting-wing quadcopter and tail-sitter UAV. +the lifting-wing quadcopter, the time for the pitch angle to reach the desired value is 1.1s, as shown in Fig.12(b), and +the time it takes for the airspeed to reach the set 18m/s is 4.7s as shown in Fig.12(a). So the transition time of the +lifting-wing quadcopter is 4.7s. Furthermore, the altitude decreases as soon as the transition starts, with a maximum +altitude error of 0.09 m at 𝑡 = 21.68 s. As for tail-sitter UAV, the transition time is 7.1s, when the desired pitch angle is +−70◦. But after the transition, the altitude drops sharply, as shown in Fig.12(c). When the desired pitch angle is −60◦, +the altitude is stable, but the transition time is 20.48s much longer than that of the RflyLW2. Thus, the transition phase of +the lifting-wing quadcopter is significantly better than the tail-sitter UAV, and does not require an additional controller. +VI. Conclusion +The modeling, controller design, and HIL simulation verification of the lifting-wing quadcopter—a novel type +of hybrid aircraft—are presented in this paper. The modeling portion takes into account the forces and moments +produced by the lifting wing and rotors. A unified controller for the entire flight phase is created based on the existing +model. Depending on the velocity command, the controller can regard hovering and forward flight equally and enable +a seamless transition between the two modes. The experimental results show that the proposed aircraft outperforms +the tail-sitter UAV in terms of tracking performance during the transition phase. In addition, the controller combines +the characteristics of the quadcopter and fixed-wing control law, allowing it to retain yaw during the hover phase and +decrease sideslip angle during the cruise phase. What is more, a control allocation based on optimization is utilized +to realize cooperative control for energy saving, by taking rotor thrust and aerodynamic force under consideration +simultaneously. Through identification, we find that compared with the motor, the aileron can reduce the energy +consumption when implementing high-frequency control inputs. +References +[1] Saeed, A. 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W., Small unmanned aircraft: Theory and practice, Princeton university press, 2012. +[25] Lyu, X., Gu, H., Zhou, J., Li, Z., Shen, S., and Zhang, F., “A hierarchical control approach for a quadrotor tail-sitter VTOL UAV +and experimental verification,” 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, +Vancouver, BC, 2017, pp. 5135–5141. +[26] Johansen, T. A., and Fossen, T. I., “Control allocation—A survey,” Automatica, Vol. 49, No. 5, 2013, pp. 1087–1103. +[27] Harkegard, O., “Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation,” +Proceedings of the 41st IEEE Conference on Decision and Control, 2002., Vol. 2, IEEE, 2002, pp. 1295–1300. +[28] Wang, S., Dai, X., Ke, C., and Quan, Q., “RflySim: A rapid multicopter development platform for education and research +based on Pixhawk and MATLAB,” 2021 International Conference on Unmanned Aircraft Systems (ICUAS), IEEE, 2021, pp. +1587–1594. +[29] Dai, X., Ke, C., Quan, Q., and Cai, K.-Y., “RFlySim: Automatic test platform for UAV autopilot systems with FPGA-based +hardware-in-the-loop simulations,” Aerospace Science and Technology, Vol. 114, 2021, p. 106727. +[30] rflysim.com, “rflysim,” https://rflysim.com/en/index.html, 2020. Accessed January 1, 2023. +[31] Pucci, D., Hamel, T., Morin, P., and Samson, C., “Nonlinear control of PVTOL vehicles subjected to drag and lift,” 2011 50th +IEEE Conference on Decision and Control and European Control Conference, IEEE, 2011, pp. 6177–6183. +26 + diff --git a/0dAyT4oBgHgl3EQf1PkS/content/tmp_files/load_file.txt b/0dAyT4oBgHgl3EQf1PkS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48a2fb499cc075cca35c3cea023c17f4e40f8cd1 --- /dev/null +++ b/0dAyT4oBgHgl3EQf1PkS/content/tmp_files/load_file.txt @@ -0,0 +1,887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf,len=886 +page_content='Lifting-wing Quadcopter Modeling and Unified Control Quan Quan∗, Shuai Wang, and Wenhan Gao Hybrid unmanned aerial vehicles (UAVs) integrate the efficient forward flight of fixed- wing and vertical takeoff and landing (VTOL) capabilities of multicopter UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This paper presents the modeling, control and simulation of a new type of hybrid micro-small UAVs, coined as lifting-wing quadcopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The airframe orientation of the lifting wing needs to tilt a specific angle often within 45 degrees, neither nearly 90 nor approximately 0 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Compared with some convertiplane and tail-sitter UAVs, the lifting-wing quadcopter has a highly reliable structure, robust wind resistance, low cruise speed and reliable transition flight, making it potential to work fully-autonomous outdoor or some confined airspace indoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In the modeling part, forces and moments generated by both lifting wing and rotors are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Based on the established model, a unified controller for the full flight phase is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The controller has the capability of uniformly treating the hovering and forward flight, and enables a continuous transition between two modes, depending on the velocity command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' What is more, by taking rotor thrust and aerodynamic force under consideration simultaneously, a control allocation based on optimization is utilized to realize cooperative control for energy saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Finally, comprehensive Hardware-In-the-Loop (HIL) simulations are performed to verify the advantages of the designed aircraft and the proposed controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Nomenclature 𝑜e𝑥e𝑦e𝑧e = Earth-Fixed Coordinate Frame(eF ) 𝑜b𝑥b𝑦b𝑧b = Quadcopter-Body Coordinate Frame(bF ) 𝑜l𝑥l𝑦l𝑧l = Lifting-Wing Coordinate Frame(lF ) 𝑜w𝑥w𝑦w𝑧w = Wind Coordinate Frame(wF ) ep = Position in eF ev = Velocity in eF bva, lva = Airspeed vector in bF and lF , respectively evw = Wind velocity in eF 𝑉a = Airspeed ∗Corresponding Author, is with the School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Email: qq_buaa@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='00730v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='RO] 2 Jan 2023 𝜙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜓 = Euler angles in bF 𝜔𝑥b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜔𝑦b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜔𝑧b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Angular velocity in bF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝛼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Angle of attack in lF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝛽 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Sideslip angle in lF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝐿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic lift coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝐷 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic drag coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic pitch moment coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝑌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic lateral force coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic roll moment coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐶𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Aerodynamic yaw moment coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝜅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Installation angle of the lifting wing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Installation angle of the motor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Mean chord of the lifting wing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Wingspan of the lifting wing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Area of the lifting wing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Why Lifting-wing Quadcopter Unmanned aerial vehicles (UAVs) have attracted lots of recent attention due to their outstanding performances in many fields, such as aerial photography, precision farming, and unmanned cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' According to [1], UAV platforms are currently dominated by three types: fixed-wing UAV, rotorcraft UAV, and their hybrid that integrates the advantages of the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The hybrid UAVs have the capability of Vertical Take-off and Landing (VTOL), which enables more accessible grounding or holding by hovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This might be mandated by authorities in high traffic areas such as lower altitudes in the urban airspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Furthermore, hybrid UAVs are categorized into two types: convertiplane and tail-sitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A convertiplane maintains its airframe orientation in all flight modes, but a tail-sitter is an aircraft that takes off and lands vertically on its tail, and the entire airframe needs to tilt nearly 90◦ to accomplish forward flight [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In March 2015, Google announced that the tail-sitter UAV for a packet delivery service was scrapped, because it is still too difficult to control in a reliable and robust manner according to the conclusion came by the project leader [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Some studies try to remedy this by newly designed controllers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, unlike this way, we will study a new type of hybrid UAV, coined as the lifting-wing quadcopter [5, 6], to overcome the difficulty Google’s Project Wing faced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A lifting-wing quadcopter is a quadcopter [7, 8] with a lifting wing installed at a specific mounting angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' During the flight, the quadcopter will provide thrust upward and forward simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' and the lifting wing also contributes a 2 a) VertiKUL 2 b) Vespertilio c) RflyLW2 d) Prime Air Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1 Prototypes of lifting-wing quadcopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' lifting force partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1, as far as we know, some prototypes of lifting-wing quadcopters in public can be found, such as VertiKUL2 by the University of Leuven (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1 (a), Sept 2015)[9], Vespertilio by the VOLITATION company(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1 (b))[10], the latest version of the Prime Air delivery drone unveiled by Amazon(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1(d), Jun 2019)[11] and, RflyLW2 by us(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1 (c)) [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The lifting-wing quadcopter is a new type of hybrid UAV because convertiplane and tail-sitter UAVs in their cruise phase work as fixed-wing UAVs, so the airframe orientation is defined as the head of the fixed-wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' But, the lifting-wing quadcopter is more like a quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The airframe orientation of the lifting wing needs to tilt a specific angle often within 45◦, neither nearly 90◦ (corresponding to tail-sitter UAVs) nor approximately 0◦(corresponding to convertiplane UAVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2 shows the full flight phase of the three VTOL UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The design and performance evaluation of lifting-wing quadcopters have been studied extensively in [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to make this paper self-contained, we briefly introduce the advantages of the lifting-wing quadcopter compared with some convertiplane and tail-sitter UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Highly reliable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It does not require extra transition actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is a reliable structure by eliminating the need for complicated control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Robust wind resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It has a shorter lifting wing compared with the corresponding fixed wing of convertiplanes and tail-sitter UAVs, because rotors can share the lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Moreover, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4, it does not have a vertical rudder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Instead, this function is replaced by the yaw control of the quadcopter component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to improve the 3 融天 福阳融天 MOITATIJOVAsmirgc9iocobret① ② ③ ② ② ② ③ ④ ④ ① ① ② ③ ④ ② (c) Lifting-wing quadcopter (b) Convertiplane UAV (a) Tail-sitter UAV ① Vertical Take-off ② Transition ③ Level Flight ④ Vertical Landing a) Tail-sitter UAV b) Convertiplane UAV c) Lifting-wing quadcopter ① ② ③ ② ④ ② ③ ② ① ② ③ ② ④ ① Vertical Take-off ② Transition ③ Forward Flight ④ Vertical Landing ④ ① Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2 Different flight modes of some VTOL UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' yaw control ability, the axes of rotors do not point only upward anymore as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This implies that the thrust component by rotors can change the yaw directly rather than merely counting on the reaction torque of rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' From the above, the wind interference is significantly reduced on the one hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' on the other hand, the yaw control ability is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As a result, it has better maneuverability and hover control to resist the disturbance of wind than those by tail-sitter and convertiplane UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Low cruise speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It can make a cruise at a lower speed than that by convertiplanes and tail-sitter UAVs, meanwhile saving energy compared with corresponding quadcopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is very useful when a UAV flies in confined airspace such as a tunnel, where the high speed is very dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Although current hybrid UAVs can have a big or long wing for low cruise speed, they cannot work in many confined airspace due to their long wingspan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, the lifting-wing quadcopter can be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Reliable transition flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When a tail-sitter UAV performs transition flight, the velocity and angle of attack will change dramatically, leading to complicated aerodynamics even stall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Unlike tail-sitter UAVs, the lifting-wing quadcopter only has to tilt a specific angle often smaller than 45◦ rather than 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The airflow on the lifting wing is stable, and lift and drag are changed linearly with the angle of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' These will avoid great difficulty (or say danger) in controlling within the full flight envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' With the four features above, it is potential for the lifting-wing quadcopter to work fully-autonomously outdoors or in some confined airspace indoors replacing with corresponding quadcopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The further comparisons with multicopter, 4 noitiansTno-seT IeoinsV gnibasI IeoinsV A ELTable 1 Comparison of different VTOL UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Endurance Reliability Wind Resistance at Hover Flight Range Multicopter tilt-rotor/wing convertiplane 4 2 2 4 Multicopter dual-system convertiplane 3 4 2 3 Multicopter tail-sitter 4 4 1 4 Lifting-wing multicopter 2 4 4 2 Multicopter 1 5 5 1 Note: bigger number implies better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Endurance Flight Range Wind Resistance at Hover Reliability Multicopter tilt-rotor/wing convertiplane Multicopter dual-system convertiplane Multicopter tail-sitter Lifting-wing Multicopter Multicopter Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 3 Comparison of different VTOL UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' tilt-rotor/wing convertiplane, multicopter dual-system convertiplane and multicopter tail-sitter [2] are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As shown, the lifting-wing quadcopter possesses the feature between current hybrid UAVs and quadcopters, and it is more like a quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Control of Current Hybrid UAVs The control of the lifting-wing quadcopter has the following two distinguishing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Unified control for full flight phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Hybrid UAVs often have three different flight modes, including the hover, the transition flight, and the forward flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' By taking the multicopter tilt-rotor/wing convertiplane, multicopter dual-system convertiplane and multicopter-tail sitter, for example, their take-off and landing are controlled only by the quadcopter component, while the forward flight is controlled like a fixed-wing aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The two control ways are very different, so the transition flight is challenging due to the nonlinearities and uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, a full flight phase of the lifting-wing quadcopter always involves thrust by the quadcopter and aerodynamic force by the 5 19i2list1oto1itluM-- 9li-1oitlMliniwo-lititl 19voH1690n6t2i291bniW 豪0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='.p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2 Euqnlgucelifting wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Therefore, the lifting-wing quadcopter can be considered under only the transition flight mode in the full flight phase (hover control here also will take the aerodynamic force into consideration due to wind on the lifting wing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As a result, a unified control is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Fortunately, the lifting-wing quadcopter only needs to tilt a specific angle often smaller than 45◦, rather than 90◦ like tail sitter UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This reduces the possibility of having a stall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Cooperative control for energy saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The transition flight for current hybrid UAVs is very short, so not too much attention needs to pay to energy consumption in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, it should be considered for the lifting-wing quadcopter as it is under the transition flight mode in the full flight phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Cooperative control for energy saving is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' For example, roll control can be performed by both the quadcopter component and the ailerons by the lifting wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Obviously, the aileron control is more energy-saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Among the control phase of a hybrid UAV, the transition control is the most challenging issue [12], especially for tail-sitter UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Since the actuators of tail-sitter UAVs are like those of lifting-wing quadcopters, the existing transition control of tail-sitter UAVs can be used for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (i) Trajectory open-loop control for transition flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The open-loop trajectory tracking control is very straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The principle is to make the UAV enter into another mode’s condition by focusing on controlling some variables like altitude other than the trajectory, then switch to the controller of the next mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' For example, increasing thrust and reducing its pitch angle at the same time can make a tail-sitter UAV enter into forwarding flight [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The aim is to keep the altitude the same [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Because the transition time for tail-sitter UAVs is short, the trajectory will not change too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Obviously, this method is inapplicable to lifting-wing quadcopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (ii) Trajectory closed-loop control for transition flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Linearization method based on optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' According to the trajectory and the model, the reference state and feedforward are derived by optimization in advance [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Based on them, the linearization can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' With the resulting linear model, existing controllers against uncertainties and disturbance are designed [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As for the lifting-wing quadcopter, this method is applicable when the model and transition trajectory are known prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Furthermore, cooperative control of the quadcopter component or the ailerons of the lifting wing can be performed by taking energy-saving into optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, in practice, the model is often uncertain as the payload is often changed, such as parcel delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Also, this method is not very flexible due to that the trajectory has to be known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Nonlinear control method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' One way is to take all aerodynamic forces as disturbances, and only the quadcopter component works for the flight transition [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This requires that the quadcopter has a strong control ability to reject the aerodynamic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Another way takes the aerodynamic force into consideration explicitly to generate a proper attitude [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' How to cooperatively control the 6 quadcopter component and the actuators of fixed-wing is not found so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is because, we guess, the transition flight is often short, and more attention is paid to making the UAV stable by reducing the possibility of stall rather than optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As shown above, the linearization method based on optimization is somewhat not flexible, but cooperative control for energy saving can be performed in an open-loop manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The nonlinear control method is flexible but not considering how to control cooperatively for energy saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Our Work and Contributions In this paper, we will consider designing a unified controller for the full flight phase of a lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' What is more, the quadcopter component and the ailerons of the lifting wing work cooperatively to save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' First, we build the model of the lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Unlike the tail-sitter UAV, it does not have a rudder, and its tilted rotors will generate force components on the XY-plane in the quadcopter-body coordinate frame (traditional quadcopters do not have the force component on the XY-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Because of this, the translational dynamic involves five control variables, namely three-dimensional force in the quadcopter-body coordinate frame and two Euler angles (pitch and roll angles), further by considering the aerodynamic force determined by Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, it is difficult and a bit too early to determine the five control variables according to the three-dimensional desired acceleration, because it is hard to obtain the bounds of these control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' An improper choice may not be realized by actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To this end, we only choose the 𝑜b𝑧b force (the main force component) in the quadcopter-body coordinate frame and two Euler angles (pitch and roll angles) to determine the desired acceleration uniquely, leaving the other two force components as a lumped disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This adopts the controlling idea of quadcopters [7, 23], but the computation method is different due to the existence of the aerodynamic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' With the determined Euler angles, moments are further determined in the lifting wing coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' So far, the unified control for the full flight phase is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Finally, we will utilize the control allocation to realize cooperative control for energy saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜b𝑧b force and three-dimensional moments will be realized by four rotors and two ailerons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is why we have the freedom to optimize the allocation for saving energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The principle behind this is to make the aerodynamic force (two ailerons) undertake the control task as much as possible because aileron control is more energy-saving than rotor control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As a result, cooperative control for energy saving is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The contributions of this paper are: (i) establish the model of a lifting-wing quadcopter for the first time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (ii) a unified controller design for the full flight phase of the lifting-wing quadcopter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (iii) control allocation for energy-saving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Comprehensive HIL simulation experiments are performed to show (i) the proposed lifting-wing quadcopter is more energy-saving with aileron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (ii) synthesizing the angular rate command from the coordinated turn in high-speed flight can reduce sideslip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (iii) the transition phase of the proposed lifting-wing quadcopter is significantly better than the tail-sitter and UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 7 ex ey bx by \uf079 c b x d y d 1 2 3 4 a) Front view b) Left view c) Top view d) 3D view ey ez bz by \uf066 \uf068 CoG ex lx lz bz \uf06b \uf071 bx ar \uf064 al \uf064 CoG CoG \uf079 al \uf064 ar \uf064 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4 Coordinate frames and nomenclatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Coordinate Frame A lifting-wing quadcopter is divided into two components, the lifting-wing component and the quadcopter component as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' According to these, the following coordinate frames are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Earth-Fixed Coordinate Frame (eF ) The earth-fixed coordinate frame 𝑜e𝑥e𝑦e𝑧e is an inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜e𝑧e axis points perpendicularly to the ground, and the 𝑜e𝑥e axis points to a certain direction in the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Then, the 𝑜e𝑦e axis is determined according to the right-hand rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This frame is fixed, the initial position of the lifting-wing quadcopter or the center of the Earth is often set as the coordinate origin 𝑜e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Quadcopter-Body Coordinate Frame (bF ) The quadcopter-body coordinate frame 𝑜b𝑥b𝑦b𝑧b is fixed to the quadcopter component of a lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The Center of Gravity (CoG) of the lifting-wing quadcopter is chosen as the origin 𝑜b of bF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜b𝑥b axis points to the nose direction in the symmetric plane of the quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜b𝑧b axis is in the symmetric plane of the quadcopter, pointing downward, perpendicular to the 𝑜b𝑥b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜b𝑦b axis is determined according to the right-hand rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Lifting-Wing Coordinate Frame (lF ) The lifting-wing coordinate frame 𝑜l𝑥l𝑦l𝑧l is fixed to the lifting-wing component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The origin 𝑜l of lF is also set at the CoG of the lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜l𝑥l axis is in the symmetric plane pointing to the nose of the lifting wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜l𝑧l axis is in the symmetric plane of the lifting wing, pointing downward, perpendicular to the 𝑜l𝑥l axis, and the 𝑜l𝑦l axis is determined according to the right-hand rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The installation angle of the lifting wing, that is the angle between the 𝑜l𝑥l axis and the 𝑜l𝑥l𝑦l plane, is denoted by 𝜅 ∈ R as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Wind Coordinate Frame (wF ) The origin 𝑜w of the wind coordinate frame 𝑜w𝑥w𝑦w𝑧w is also at the CoG of the lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜w𝑥w axis is aligned with the airspeed vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The 𝑜w𝑧w axis is in the symmetric plane of the lifting wing, pointing downward, perpendicular to the 𝑜w𝑥w axis, and the 𝑜w𝑦w axis is determined according to the right-hand rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The angle of attack (AoA), denoted by 𝛼 ∈ R, is defined as the angle between the projection of the airspeed vector on the 𝑜l𝑥l𝑧l plane and the 𝑜l𝑥l as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The sideslip angle, denoted by 𝛽 ∈ R, is defined as the angle between the airspeed vector and the 𝑜l𝑥l𝑧l plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To convert the aerodynamic forces and moments acting on frame wF and lF to bF respectively, two rotation matrices are defined as followed: Rb w(𝜆) = �������� cos 𝜆 cos 𝛽 − cos 𝜆 sin 𝛽 sin 𝜆 sin 𝛽 cos 𝛽 0 − sin 𝜆 cos 𝛽 sin 𝜆 sin 𝛽 cos 𝜆 �������� , Rb l = �������� cos 𝜅 0 sin 𝜅 0 1 0 − sin 𝜅 0 cos 𝜅 �������� , where 𝜆 = 𝜅 − 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' And the rotation matrix Re b maps a vector from frame bF to eF , defined by Re b = �������� cos 𝜃 cos 𝜓 − sin 𝜃 sin 𝜙 sin 𝜓 − sin 𝜓 cos 𝜙 cos 𝜓 sin 𝜃 + cos 𝜃 sin 𝜙 cos 𝜓 sin 𝜃 sin 𝜙 cos 𝜓 + cos 𝜃 sin 𝜓 cos 𝜙 cos 𝜓 sin 𝜓 sin 𝜃 − cos 𝜙 cos 𝜃 sin 𝜙 − cos 𝜙 sin 𝜃 sin 𝜙 cos 𝜙 cos 𝜃 �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' MODELING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Assumptions For the sake of model simplicity, the following assumptions are made: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The body structure is rigid and symmetric about the 𝑜l𝑥l𝑦l plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The mass and the moments of inertia are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The geometric center of the lifting-wing quadcopter is the same as the CoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The aircraft is only subjected to gravity, aerodynamic forces, and the forces generated by rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Flight Control Rigid Model By Assumptions 1-2, the Newton’s equation of motion is applied to get the translational motion as follows e �p = ev e�v = Re b bf 𝑚 (1) where ep = � 𝑝𝑥e 𝑝𝑦e 𝑝𝑧e �T and ev = � 𝑣𝑥e 𝑣𝑦e 𝑣𝑧e �T are the position and velocity expressed in frame eF respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝑚 is the mass, bf is the total force acting on the airframe expressed in frame bF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To facilitate attitude control and combine the control characteristics of the rotor and lifting wing, the rotational dynamics is carried out in frame lF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It is given by Euler’s equation of motion as �Re l = Re l �l𝝎 � × J · l �𝝎 = lm − l𝝎 × (J·l𝝎) (2) where the rotational matrix is derived by Re l = Re b Rb l , Rb l being a constant matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' lm is the total moment acting on the airframe expressed in frame lF , l𝝎 = � 𝜔𝑥l 𝜔𝑦l 𝜔𝑧l �T is the angular velocity in frame lF , �l𝝎 � × denotes the skew-symmetric matric �l𝝎 � × = �������� 0 −𝜔𝑧l 𝜔𝑦l 𝜔𝑧l 0 −𝜔𝑥l −𝜔𝑦l 𝜔𝑥l 0 �������� , and J ∈ R3×3 is the inertia matrix given by J = �������� 𝐽𝑥 0 −𝐽𝑥𝑧 0 𝐽𝑦 0 −𝐽𝑥𝑧 0 𝐽𝑧 �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Forces and Moments By Assumptions 3-4, the total forces and moments acting on the UAV are decomposed into three parts: the aerodynamic forces and moments acting on the airframe (fa and ma), the forces and moments generated by rotors (fr and mr), and the gravitational forces f𝑔, where f𝑔 = [0 0 𝑔]T, 𝑔 is the gravitational acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The front two types of forces and moments will be described detailly in the following two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10 Lf Df Yf a v x av z av y av 1T 2T 3T 4T \uf068 lo ly lx lz \uf061 \uf062 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 5 Forces act on a lifting-wing quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Forces and Moments in the Quadcopter Component In the quadcopter part, the thrust and torque produced by one rotor are given by 𝑇𝑖 = 𝐾 𝑓 𝜛𝑖2, 𝑀𝑖 = 𝐾𝑚𝜛𝑖2 = 𝐾𝑚 𝐾 𝑓 𝑇𝑖 (3) where 𝐾 𝑓 > 0 is the lift force coefficient, 𝐾𝑚 > 0 is the drag torque coefficient, and 𝜛𝑖 is the angular rate of the ith rotor, 𝑖 = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to improve the controllability during performing yaw, an installation angle 𝜂 is set as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The left motors tilt to the positive left and right motors tilt to the positive right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Because of the installation angle 𝜂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' the forces and moments produced by rotors are expressed by the thrust on each propeller 𝑇𝑖 as ������������� 𝑓𝑟𝑦 𝑓𝑟𝑧 𝑚𝑟𝑥 𝑚𝑟𝑦 𝑚𝑟𝑧 ������������� = ������������� sin 𝜂 − sin 𝜂 − sin 𝜂 sin 𝜂 − cos 𝜂 − cos 𝜂 − cos 𝜂 − cos 𝜂 −𝑑𝑦 cos 𝜂 𝑑𝑦 cos 𝜂 𝑑𝑦 cos 𝜂 −𝑑𝑦 cos 𝜂 𝑑𝑥 cos 𝜂 −𝑑𝑥 cos 𝜂 𝑑𝑥 cos 𝜂 −𝑑𝑥 cos 𝜂 𝐾1 𝐾1 −𝐾1 −𝐾1 ������������� ���������� 𝑇1 𝑇2 𝑇3 𝑇4 ���������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (4) where 𝐾1 = 𝐾𝑚 �𝐾 𝑓 + 𝑑𝑥 sin 𝜂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝑑𝑥 and 𝑑𝑦 are the components of the distance from the center of the lifting-wing quadcopter to a propeller on the 𝑜b𝑥b𝑦b plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Forces and Moments in the Lifting-wing Component The aerodynamic forces and moments acting on the lifting wing are mainly generated by the lifting wing itself and the ailerons at the trailing edge, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 11 Let evw be the wind velocity in eF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Then bva = bv − (Re b)T · evw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (5) Thus, the airspeed vector lva = � 𝑣a𝑥 𝑣a𝑦 𝑣a𝑧 �T and airspeed 𝑉𝑎 are defined as lva = (Rb l )T · bva, (6) 𝑉a = √︃ 𝑣2a𝑥 + 𝑣2a𝑦 + 𝑣2a𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (7) The aerodynamic angles 𝛼 and 𝛽 are defined as 𝛼 = tan−1( 𝑣a𝑧 𝑣a𝑥 ), 𝛽 = sin−1( 𝑣a𝑦 𝑉a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (8) In the longitudinal plane, lift, drag, and pitching moment acting on the lifting-wing body are given by 𝑓𝐿 = 𝑄𝑆(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒) 𝑓𝐷 = 𝑄𝑆(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒) 𝑚 = 𝑄𝑆𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (9) The lateral force and the roll and yaw moments acting on the lifting-wing body are given by 𝑓𝑌 = 𝑄𝑆(𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) 𝑙 = 𝑄𝑆𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎) 𝑛 = 𝑄𝑆𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎) (10) where 𝑄 = 1 2 𝜌𝑉2 𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝐶𝐿, 𝐶𝐷, 𝐶𝑚,𝐶𝑌 , 𝐶𝑙 and 𝐶𝑛 are nondimensional aerodynamic coefficients, 𝐶𝐿 𝛿𝑒, 𝐶𝑚𝛿𝑒, 𝐶𝐷 𝛿𝑒,𝐶𝑌 𝛿𝑎, 𝐶𝑛𝛿𝑎 and 𝐶𝑙𝛿𝑎 are control derivative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝑆 is the area of the lifting wing, 𝑐 is the mean chord of the lifting wing, 𝑏 is the wingspan of the lifting-wing aircraft, 𝛿𝑒 and 𝛿𝑎 are calculated using the right and the left aileron (𝛿𝑎𝑟 and 𝛿𝑎𝑙, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4(d)) as � 𝛿𝑒 𝛿𝑎 � = � 1 1 −1 1 � � 𝛿𝑎𝑟 𝛿𝑎𝑙 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (11) The external forces and moments are summarized as 12 Table 2 Lifting-wing quadcopter structure parameters, lift force and drag torque coefficients 𝑚 Aircraft mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='92 kg 𝜅 Installation angle of lifting wing 34 deg 𝜂 Installation angle of motor 10 deg 𝑑𝑥 The distance from 𝑜𝑏𝑥𝑏 to a propeller 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='25 m 𝑑𝑦 The distance from 𝑜𝑏𝑦𝑏 to a propeller 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2125 m [𝐽𝑥𝑥 𝐽𝑦𝑦 𝐽𝑧𝑧] Moment of inertia [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='6] × 10−2 kg · m2 𝑏 Wingspan of the lifting-wing aircraft 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='94 m 𝑐 Mean chord of the lifting wing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='17 m 𝐾𝑚 Drag moment coefficient 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='875e-07 kg · m2 𝐾 𝑓 Lift force coefficient 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='824e-05 kg · m2 bf = �������� 0 𝑓𝑟𝑦 𝑓𝑟𝑧 �������� + 𝑄𝑆Rb w �������� −(𝐶𝐷 + 𝐶𝐷 𝛿𝑒𝛿𝑒) (𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) −(𝐶𝐿 + 𝐶𝐿 𝛿𝑒𝛿𝑒) �������� + 𝑚Rb ef𝑔 (12) lm = Rl b �������� 𝑚𝑟𝑥 𝑚𝑟𝑦 𝑚𝑟𝑧 �������� + 𝑄𝑆 �������� 𝑏(𝐶𝑙 + 𝐶𝑙 𝛿𝑎𝛿𝑎) 𝑐(𝐶𝑚 + 𝐶𝑚𝛿𝑒𝛿𝑒) 𝑏(𝐶𝑛 + 𝐶𝑛𝛿𝑎𝛿𝑎) �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (13) The structure parameters, lift force and drag torque coefficients of the lifting-wing quadcopter are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' CONTROLLER DESIGN The successive loop closure is a common control architecture for UAVs [24], which consists of an outer-loop controlling the position and an inner-loop for attitude, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The basic idea behind successive loop closure is to close several simple feedback loops in succession around the open-loop plant dynamics rather than designing a single control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The position controller receives the desired position and then computes the desired acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Then the desired acceleration is mapped to the collective thrust and attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The attitude command is sent to the inner-loop, while the thrust command skips directly to the control allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To facilitate the control experiment step by step, the attitude can also be commanded by the pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Furthermore, the attitude controller receives the desired attitude and generates the desired moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Finally, the control allocation algorithm distributes the moment command from the inner-loop and the direct force command from the outer-loop to corresponding ailerons and rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 13 Position Controller Attitude Controller Control Allocation d m Lifting-wing Quadcopter Dynamics , T δ d p Mannual Inputs Signal Distribution d Θ df d Θ df df d Θ a , p v av q Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 6 Control structure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Controller Design Model Since 𝛼, 𝛽 are not easy to obtain, we consider that 𝛼 ≈ 𝜅 + 𝜃 and 𝛽 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The translational dynamic involves five control variables, namely three-dimensional forces in frame bF and two Euler angles (pitch and roll angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, it is a bit too early to determine the five control variables according to the three-dimensional desired acceleration because it is hard to obtain the bounds of these control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' An improper choice may not be realized by ailerons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To this end, we only choose 𝑓𝑧 (the main force component) in frame bF and two Euler angles (pitch and roll angles) to determine the desired acceleration uniquely, and the desired yaw angle is directly specified, leaving 𝑓𝑦 as a disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This adopts the controlling idea of quadcopters[7, 23], but the computation method is different due to the existence of the aerodynamic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' According to the idea above, we rewrite the system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (1) and (2) in the form as e �p = ev e�v = u + g + d1 �Re l = Re l �l𝝎 � × l �𝝎 = J−1 · lm + d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (14) Here u = Re b 𝑚 ���� � �������� 0 0 − 𝑓𝑧 �������� + 𝑄𝑆Rb w �������� −𝐶𝐷 0 −𝐶𝐿 �������� ���� � , and d1, d2 are disturbances, where d1 = Re b 𝑚 ���� � �������� 0 𝑓𝑦 0 �������� + 𝑄𝑆Rb w �������� −𝐶𝐷 𝛿𝑒𝛿𝑒 (𝐶𝑌 + 𝐶𝑌 𝛿𝑎𝛿𝑎) −𝐶𝐿 𝛿𝑒𝛿𝑒 �������� ���� � d2 = −J−1 · l𝝎 × (J · l𝝎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Position Control Given a twice differentiable trajectory pd(𝑡), in order to satisfy lim 𝑡→∞ ∥ep(𝑡) − pd(𝑡)∥ = 0, the desired ud for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (14) can be designed as a PID controller in the form ud = −g + �pd − KPd (ev − �pd) − KPp (ep − pd) − KPi ∫ (ep − pd) d𝑠 (15) where KPp, KPi, KPd ∈ R3×3 are diagonal matrices acting as control gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The left work is to determine desired thrust by rotors 𝑓d ∈ Ω 𝑓 and 𝜃d, 𝜙d ∈ Ω𝑎 such that ( 𝑓d, 𝜃d, 𝜙d) = arg min 𝑓𝑧 ∈Ω 𝑓 ,𝜃,𝜙∈Ω𝑎 ∥u ( 𝑓𝑧, 𝜃, 𝜙) − ud∥ (16) where Ω 𝑓 is a set to confine the force, and Ω𝑎 is a set to confine the pitch and roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to reduce drag, the vehicle’s nose should be consistent with the current direction of the vehicle velocity, that is 𝜓d = tan−1 � 𝑣𝑦e 𝑣𝑥e � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (17) The attitude command can also be given by the pilot, in case the position controller fails with GPS denied, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Finally, the desired attitude is given as 𝚯d = [𝜙d 𝜃d 𝜓d]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Attitude Control The attitude controller generates the desired moment from the output of the position controller or the attitude given by the pilot, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As far as we know, studies about the hybrid UAV hardly consider the lateral control, such as turning right or left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As for the considered UAV, the control on yaw is quite different between the multicopter mode and the fixed-wing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To establish a unified control, the attitude control is performed on the lifting-wing frame lF , so l𝚯d = [𝜙d 𝜃d + 𝜅 𝜓d]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Basic Attitude Control The attitude error is presented in the form of quaternion based on which the corresponding controller is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This can guarantee a uniform and good convergence rate for all initial attitude errors [25] qe = q∗ d ⊗ qe l = � 𝑞e0 𝑞e1 𝑞e2 𝑞e3 �T, (18) 15 where qd is transformed from l𝚯d with ‘ZXY’ rotational sequence, (·)∗ is the conjugate of a quaternion, and ⊗ is the quaternion product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Then qe is transformed into the axis-angle form qe = � cos 𝜗 2 𝝃T e sin 𝜗 2 �T by 𝜗 = wrap𝜋 �2acos �𝑞e0 �� 𝝃e = � [0 0 0]T, 𝜃 = 0 sign(𝑞e0) 𝜗 sin 𝜗/2 � 𝑞e1 𝑞e2 𝑞e3 �T, 𝜃 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (19) The function wrap𝜋(𝜗) constrains the 𝜗 in [−𝜋 𝜋] to ensure the shortest rotation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To eliminate the attitude error, the attitude control is designed as l𝝎ac = sat �K𝚯p𝝃e, 𝝎min, 𝝎max � (20) where K𝚯p ∈ R3×3 is the diagonal matrix acting as the control gain, 𝝎min and 𝝎max ∈ R3 are the minimum and maximum angular control rates, the function sat (x, xmin, xmax) is defined as sat (x, xmin, xmax) ≜ �������� sat �𝑥1, 𝑥1,min, 𝑥1,max � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' sat �𝑥𝑛, 𝑥𝑛,min, 𝑥𝑛,max � �������� , sat �𝑥𝑘, 𝑥𝑘,min, 𝑥𝑘,max � ≜ ����� ����� 𝑥𝑘,min, 𝑥𝑘 < 𝑥𝑘,min 𝑥𝑘,max, 𝑥𝑘 > 𝑥𝑘,max 𝑥𝑘, else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (21) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Lateral Compensation When the UAV is in high-speed flight, a roll command given to track a specified trajectory will cause a lateral skid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to reduce the sideslip angle when the UAV turns at a high speed, the coordinated turn should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In lF frame, if it is assumed that there is no wind, the coordinated turn equation is expressed as [24] �𝜓d = 𝑔 tan 𝜙 𝑉𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (22) It should be noted that the Euler angles are the attitude presentation between lF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' So the desired yaw rate generated by coordinated turn is l𝜔ct = �𝜓d cos 𝜃 cos 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (23) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Angular Rate Command Synthesis In the lifting-wing coordinated turn, 𝑉𝑎 being zero makes no sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In addition, considering that coordinated turn should not be used when airspeed is small, a weight coefficient related to airspeed is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' So the desired angular rates are rewritten as l𝜔d = � 𝜔dac,𝑥 𝜔dac,𝑦 � 𝜔dac,𝑧 + 𝑤·l𝜔ct �� (24) 16 where 𝑤 = sat � 𝑉𝑎−𝑉min 𝑉max−𝑉min , 0, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the airspeed is slower than 𝑉min, the desired yaw rate is completely decided by the basic attitude controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In contrast, when l𝜔d𝑧 = 0 and the airspeed reaches the specified value 𝑉max, the desired yaw rate is completely decided by the coordinated turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Attitude Rate Control To eliminate the attitude rate error, the controller is designed as lmd = sat � J(−K𝝎p(l𝝎 − l𝝎d) − md,I − K𝝎d(l �𝝎 − l �𝝎d)), −md,max, md,max � (25) where md,I = sat � K𝝎i ∫ (l𝝎 − l𝝎d)d𝑠, − md,Imax, md,Imax � , K𝜔p, K𝜔i, K𝝎d ∈ R3×3 are diagonal matrices acting as control gains, md,Imax is maximum amplitude of integral action, md,max is the maximum moment generated by actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' So far, we have obtained 𝑓d and lmd, which will be further realized by 𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Control Allocation The lifting-wing quadcopter is an over-actuated aircraft, which provides six independent control inputs, namely 𝑇1, · · · , 𝑇4, 𝛿𝑎𝑟, 𝛿𝑎𝑙 to meet a specific thrust 𝑓d ∈ R and moment demand lmd ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A method of control allocation based on optimization is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Recalling the control in translational dynamic, if we determined the three-dimensional force in bF and two Euler angles (pitch and roll angles) by an optimization before, then the six control variables (plus desired yaw angle) will be determined by six actuators uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' If so, however, the optimization in the control of the translational dynamic is not related to energy-saving directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is why we only choose the 𝑜b𝑧b force (the main force component) and two Euler angles (pitch and roll angles) to determine the desired acceleration as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' First, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (12) and (13) are rearranged as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑓𝑧 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='l𝑚𝑥 − 𝑄𝑆𝑏𝐶𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='l𝑚𝑦 − 𝑄𝑆𝑐𝐶𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='l𝑚𝑧 − 𝑄𝑆𝑏𝐶𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='uv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='− cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='− cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='− cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='− cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝐾2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐾3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐾2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝐾3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝑄𝑆𝑏𝐶𝑙 𝛿𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑄𝑆𝑏𝐶𝑙 𝛿𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑑𝑥 cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝑑𝑥 cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑑𝑥 cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝑑𝑥 cos 𝜂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑄𝑆𝑐𝐶𝑚𝛿𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑄𝑆𝑐𝐶𝑚𝛿𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝐾5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐾4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝐾5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝐾4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='−𝑄𝑆𝑏𝐶𝑛𝛿𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑄𝑆𝑏𝐶𝑛𝛿𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑇1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑇2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑇3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝑇4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝛿𝑎𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝛿𝑎𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='𝜹 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='where 𝐾2 = 𝑑𝑦 cos 𝜂 cos 𝜅 +𝐾1 sin 𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝐾3 = 𝑑𝑦 cos 𝜂 cos 𝜅 −𝐾1 sin 𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝐾4 = 𝑑𝑦 cos 𝜂 sin 𝜅 +𝐾1 cos 𝜅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝐾5 = 𝑑𝑦 cos 𝜂 sin 𝜅 − 𝐾1 cos 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The value 𝜹 is the control input of the actuator , uv is the virtual control, B is the control efficiency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As shown in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (26), rank (B) = 4, the dimension of 𝜹 is 6, which is higher than that of uv, so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (26) has a minimum norm solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To take the control priority of the uv components, actuator priority of 𝜹 and actuator saturation 17 under consideration, the control allocation is formulated to be an optimization problem as min ��Wu �B𝜹 − uv,d ���2 + 𝛾 ��W𝜹 �𝜹 − 𝜹p ���2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹− ⩽ 𝜹 ⩽ ¯𝜹 (27) where uv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='d = � 𝑓d l𝑚d𝑥 − 𝑄𝑆𝑏𝐶𝑙 l𝑚d𝑦 − 𝑄𝑆𝑐𝐶𝑚 l𝑚d𝑧 − 𝑄𝑆𝑏𝐶𝑛 �T is the desired virtual control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹p is the preferred control vector which will be specified later,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Wu ∈ R6×6 is a positive definite weighting matrix that prioritizes the commands in case the desired virtual input uv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='d cannot be achieved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' W𝜹 ∈ R6×6 is a positive definite weighting matrix that prioritizes the different actuators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹− = max(𝜹min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹l − Δ𝜹) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' ¯𝜹 = min(𝜹max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹l + Δ𝜹) are lower and upper bounds at each sampling instant of actuators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 𝜹min and 𝜹max ∈ R6 are actuator position limits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Δ𝜹 ∈ R6 is the rate limit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' and 𝜹l is the last value of 𝜹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' There are two optimization objectives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (27), namely ��Wu �B𝜹 − uv,d ���2 and ��W𝜹 �𝜹 − 𝜹p ���2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The first one is the primary objective of minimizing the slack variables weighted by Wu, so the weighting factor 𝛾 is often chosen to be a small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In many cases, the preferred control vector is set to the last value of 𝜹, 𝜹p = 𝜹l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' But, in order to save energy, we prefer to use the aerodynamic force because the change of rotor force implies the motor’s rotational speed change, which is power-hungry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' According to the consideration above, we can give more weight to 𝛿𝑎𝑟 and 𝛿𝑎𝑙 and set first four elements of 𝜹p to 𝑇1+𝑇2+𝑇3+𝑇4 4 , and last two elements to that of 𝜹l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The hardware resources of the flight control are very limited, so it is very important to ensure the real-time operation of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Several different algorithms, like redistributed pseudoinverse, interior-point methods, and active set methods have been proposed to solve the constrained quadratic programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Among them, active set methods[26] perform well in the considered control allocation problems, because they have the advantage that their initialization can take advantage of the solution from the previous sample (known as the warm start), which is often a good guess for the optimal solution at the current sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This can reduce the number of iterations needed to find the optimal solution in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The study [27] shows that the computational complexity of the active set methods is similar to the redistributed pseudoinverse method and the fixed-point algorithm, but the active set methods produce solutions with better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Simulation Experiments In order to verify the three main advantages of the designed aircraft: control performance, energy saving and reliable transition flight, several experiments are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The verification of these performances is mainly determined by two factors, namely, with and without ailerons, and with and without coordinated turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Therefore, experiments are primarily composed of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' One is the comparison of control performance, energy saving and transition flight with and without aileron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' And the other is the comparison of control performance with and 18 without coordinated turn, but both with aileron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In addition, the transition flight of three different VTOL vehicles, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2, is analyzed in the HIL simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Simulation Platform The HIL simulation is carried out in the RflySim platform [28–30], which provides CopterSim simulator and supports Pixhawk/PX4 autopilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='When perform the HIL simulation, CopterSim sends sensor data such as accelerometer, barometer, magnetometer, which is generated by a mathematical model, to the Pixhawk system via a USB serial port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The Pixhawk/PX4 autopilot will receive the sensors data for state estimation by the EKF2 filter and send the estimated states to the controller through the internal uORB message bus as the feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Then the controller sends the control signal of each motor as output back to CopterSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Thereby a close-loop is established in the HIL simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Compared with the numerical simulation, the control algorithm is deployed and run in a real embedded system as the real flight does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' After HIL simulation, the controller will be directly deployed to a real vehicle and further verification experiments will be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' PixHawk Autopilot Hardware System HIL Real-Time Simulation System Control Signals Sensor Data Multicopter Simulation + 3D Display + Fault Injection Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 7 HIL simulation of RflyLW2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The lifting-wing quadcopter model is built in MATLAB/Simulink according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (1),(2) (12), (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Besides, the dynamics of the motor and servo are modeled as first-order transfers as follows: 𝜛 = 1 𝑇m𝑠 + 1 (𝐶m𝜎𝑖 + 𝜛b), 𝛿a = 1 𝑇a𝑠 + 1 (𝐶a𝜎𝑗 + 𝛿b), 𝑖 = 1, 2, 3, 4, 𝑗 = 5, 6 (28) where 𝜎𝑖 ∈ [0, 1] is 𝑖th throttle command, 𝑇m and 𝑇a are the time constant parameter of motor and servo response, 𝐶m, 𝐶a, 𝛿b and 𝜛b are constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The sensors used in HIL simulation include IMU, magnetometer, barometer, GPS and airspeed meter, are modeled reference to [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Due to the vehicle’s VTOL capabilities, the aerodynamic parameters must not only be modeled up to the stall AoA, 19 L inrgps中 9 0 0 1 [0 1 4= [0 X0 EX+ BI (CCIT) 3 x2* 14 [e000 rry [30 (+9 30 3 xg sa 玉净大量 (去+)的内 To #4 F¥4 [880 xten, 0* -[o 50 SB 30u % 20 O COWBOr eBOFb BRITYBIE AFTEBI 一VMWIE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='00 000 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 0:00 0:00 000OMES 128 MAd LETEW LETEWS 2MUCH CAIAIR32 26K11D2V NE LMEbut also in the post-stall region, in order to cover the entire flight envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The full angle aerodynamic parameters are obtained by combining the small angle aerodynamic parameters obtained by CFD simulation and the empirical aerodynamic model [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The aerodynamic characteristics at low and large AoA are approximated as ����� ����� 𝐶𝐿𝑆 (𝛼) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='5𝑐2 2 (𝑐2 − 𝑐3)cos2(𝛼) + 𝑐3 sin(2𝛼) 𝐶𝐷𝑆 (𝛼) = 𝑐0 + 𝑐2𝑐3 (𝑐2 − 𝑐3) cos2 (𝛼) + 𝑐3 sin2 (𝛼) , ���� ���� 𝐶𝐿𝐿 (𝛼) = 𝑐1 sin (2𝛼) 𝐶𝐷𝐿 (𝛼) = 𝑐0 + 2𝑐1sin2 (𝛼) , and a pseudo-sigmoid function 𝜎 (𝛼0, 𝑘, 𝛼) = 1 + tanh �𝑘𝛼2 0 − 𝑘𝛼2� 1 + tanh � 𝑘𝛼2 0 � , 𝛼 ∈ [−𝜋, 𝜋) is used to blend low and large AoA regions together ���� ���� 𝐶𝐿 (𝛼) = 𝐶𝐿𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐿, 𝛼) + 𝐶𝐿𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐿, 𝛼)] 𝐶𝐷 (𝛼) = 𝐶𝐷𝑆 (𝛼) 𝜎 (𝛼0, 𝑘𝐷, 𝛼) + 𝐶𝐷𝐿 (𝛼) [1 − 𝜎 (𝛼0, 𝑘𝐷, 𝛼)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (29) The low and large AOA parameters 𝑐0, 𝑐1, 𝑐2, 𝑐3, and blending parameters 𝛼0, 𝑘𝐿, 𝑘𝐷 are turned according to CFD results, and their values are set to 𝑐0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='055, 𝑐1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='9, 𝑐2 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0, 𝑐3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='3, 𝛼0 = 3deg, 𝑘𝐿 = 38, 𝑘𝐷 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Corresponding Aerodynamic curves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (deg) \uf061 (deg) \uf061 L C D C and L C D C L D C C Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 8 Aerodynamic parameters obtained by CFD B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Simulation Experiments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Verifying the Effectiveness of Aileron In order to show the effectiveness of ailerons, two groups of comparative experiments are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The attitude control and position control are the same, but the control allocation of the first group uses the aileron, while the second 20 021- J00 20 0 J00 J20 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0500 021- J00 20 0 J00 J20 500 J2 JO 2- 0 2 JO J2Time(s) Throttle×1000 Power(W) Fitted Power Measured Power Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 9 Throttle command and identification result of the motor group does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The second group only depends on the quadcopter control allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In this experiment, the aircraft tracks the specified trajectory, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10(a), which is a straight line plus a circle with a radius of 200m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10(b), after adding the aileron, the control amplitude of the four motors is almost the same trend, especially when the aircraft transits between straight and circular trajectories, indicating that the attitude control is mainly realized by the aileron, and motors are more like a thruster on a fixed-wing in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' However, when the attitude changes sharply, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10(b) during 𝑡 = 22 ∼ 23s, the motor and aileron will participate in the control at the same time to improve the control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the state of the UAV is in the slow adjustment, the aileron undertakes the control as much as possible to save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In order to quantify the influence of ailerons, the relationship between throttle command and energy consumption of the motor and actuator is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A sinusoidal signal with the constant amplitude and linear increasing frequency is designed, and the energy consumption test experiments are carried out on the servo and motor at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Measurement and identification results of the motor are shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='9, in which the amplitude of the throttle decreases as the frequency increasing, because a low-pass filter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (28) is applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' We find that the motor power has a quadratic function relationship with a fixed throttle, and the power is different when the motor accelerates and decelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the change frequency of the throttle command becomes faster, the power is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Therefore, the following formula is established to fit the relationship between throttle command and motor power 𝑃𝑇 = 4 ∑︁ 𝑖=1 𝑝1𝜎𝑖2 + 𝑝2𝜎𝑖 + 𝑝3 �𝜎 𝑝4 𝑢𝑝,𝑖 + 𝑝5 �𝜎 𝑝6 𝑑𝑜𝑤𝑛,𝑖 + 𝑝7 (30) 21 180 500 SSO J40 seo 580 300 400 420 eoo Q20180 500 SSO 540 seo 580 300 20 100 J20 50OWith the given lifting-wing quadcopter platform, 𝑝1 = 563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='7, 𝑝2 = −147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='4, 𝑝3 = 15, 𝑝4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='05, 𝑝5 = 4, 𝑝6 = 1, 𝑝7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='05538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' With respect to the servo, its power is negligible, because the power of servo is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2W, far less than the motor’s, even if the 100g weight is suspended on the two ailerons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The trajectory tracking experiment is performed three times, with the forward flight speed of 5m/s, 10m/s and 20m/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The powers in three flight speeds are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It can be observed that when the speed is 5m/s, the power is the same, because the aileron is not used at slow flight speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the speed increases to 10m/s, the power with the aileron is slightly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Further when the speed increases to 20m/s, the power with the aileron is greatly smaller because the aerodynamic force is stronger at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Time(s), without aileron T1~T4(N) Time(s), with aileron b) Control output with and without aileron when the forward flight speed is 20m/s c) Control indexs in different forward flight speeds a) Flight path Time(s), 5m/s x(m) y(m) z(m) Time(s), 10m/s Time(s), 20m/s Power(W) without aileron with aileron ar al and (deg) \uf064 \uf064 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 10 The attitude control performance and mixer output with and without aileron 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Verifying the Effectiveness of Coordinated Turn As for the lateral control, experiments are carried out in hover and high-speed forward flight, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In the high-speed forward flight, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 11 yellow region, when a turn command is given for the first time during 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='8 ∼ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2s, the lifting-wing quadcopter with coordinated turn has no sideslip angle and has a better lateral tracking performance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='11(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In this state, the control of the lifting-wing quadcopter is similar to the fixed-wing, and the desired yaw rate is generated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' (23) as the feedforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the turn command is given for the second time during 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='8 ∼ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2s, the lifting-wing quadcopter is in low-speed flight, and the control of the lifting-wing quadcopter is similar to the quadcopter, so a big sideslip angle appears both in experiments with and without coordinated turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='J00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='J20 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='J2Time(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='b) Vehicle states with coordinated turn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Time(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='a) Vehicle states without coordinated turn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Desired ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='>10m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='6~10m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='6~10m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='>10m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='(deg) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='(deg) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='\uf066 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='8-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='8-72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='3) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='4) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='5) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='2) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='3) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='4) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='5) Desired Yaw Rate Yaw Response (rad) \uf062 a(m s) V Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 11 The control response with and without coordinated turn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Transition Flight The transition flight is often defined as the aircraft changing from hover to level flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' To quantify the phase, the transition process is defined as the time it takes for the aircraft to start a transitional flight to the airspeed greater than 18m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The quality of the transition is mainly reflected in the transition time and control accuracy of altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In the simulation, the aircraft will take off to a certain altitude, and then a step signal of −30◦ is given to the pitch channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' This is because the wind CFD test results show that the maximum energy efficiency is obtained when 𝛼 = 4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As a comparison, the same experiment is performed on a tail-sitter quadcopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The model of the tail-sitter quadcopter is built on the lifting-wing quadcopter with the installation angle of the lifting-wing from 34◦ to 90◦, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='12 shows the response curves of the pitch angle, altitude and airspeed of the lifting-wing quadcopter and tail-sitter UAV in the transition phrase in the HIL simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' It can be observed that during the transition phase of 23 0 O 40 eo 80 J00 50 0 5O5O 40 eo 80 J00 50 0 5O5O 40 e 80 J00 30F 50 J0 050 40 eo 80 J00 40 50 00 O 40 eo 80 0 JO SO5O 40 e 80 J00 JO 5O0 5O 40 eo 80 30 50 JO 00 5O 40 eo 80 0 5O 40 eo0 SO 40 eo 80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='I- J 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='0- 050 40 eo 80 J00 J 0Time(s) Va(m/s) RflyLW2 Tail-sitter-60° Tail-sitter-70° Take off Hovering Transition Flight Time(s) Time(s) Take off Hovering Transition Flight Take off Hovering Transition Flight RflyLW2 Tail-sitter-60° Tail-sitter-70° RflyLW2 Tail-sitter-60° Tail-sitter-70° Pitch(rad) Altitude(m) a) Airspeed response b) Pitch response c) Altitude response Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' 12 Transition flight of lifting-wing quadcopter and tail-sitter UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' the lifting-wing quadcopter, the time for the pitch angle to reach the desired value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1s, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='12(b), and the time it takes for the airspeed to reach the set 18m/s is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='7s as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' So the transition time of the lifting-wing quadcopter is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Furthermore, the altitude decreases as soon as the transition starts, with a maximum altitude error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='09 m at 𝑡 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='68 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' As for tail-sitter UAV, the transition time is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='1s, when the desired pitch angle is −70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' But after the transition, the altitude drops sharply, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='12(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' When the desired pitch angle is −60◦, the altitude is stable, but the transition time is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content='48s much longer than that of the RflyLW2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Thus, the transition phase of the lifting-wing quadcopter is significantly better than the tail-sitter UAV, and does not require an additional controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Conclusion The modeling, controller design, and HIL simulation verification of the lifting-wing quadcopter—a novel type of hybrid aircraft—are presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The modeling portion takes into account the forces and moments produced by the lifting wing and rotors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' A unified controller for the entire flight phase is created based on the existing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Depending on the velocity command, the controller can regard hovering and forward flight equally and enable a seamless transition between the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' The experimental results show that the proposed aircraft outperforms the tail-sitter UAV in terms of tracking performance during the transition phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' In addition, the controller combines the characteristics of the quadcopter and fixed-wing control law, allowing it to retain yaw during the hover phase and decrease sideslip angle during the cruise phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' What is more, a control allocation based on optimization is utilized to realize cooperative control for energy saving, by taking rotor thrust and aerodynamic force under consideration simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQf1PkS/content/2301.00730v1.pdf'} +page_content=' Through identification, we find that compared with the motor, the aileron can reduce the energy consumption when implementing high-frequency control inputs.' metadata={'source': 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b/0tFQT4oBgHgl3EQf0TbP/content/tmp_files/2301.13416v1.pdf.txt @@ -0,0 +1,1018 @@ +Structure Flow-Guided Network for Real Depth Super-Resolution +Jiayi Yuan*, Haobo Jiang*, Xiang Li, Jianjun Qian, Jun Li†, Jian Yang† +PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education +Jiangsu Key Lab of Image and Video Understanding for Social Security +School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China +{jiayiyuan, jiang.hao.bo, xiang.li.implus, csjqian, junli, csjyang}@njust.edu.cn +Real LR +Groundtruth +RGB image +FDSR +Ours +(c) +(e) +(d) +(f) +(g) +(h) +(i) +(j) +(k) +(l) +(a) +(b) +Synthetic LR +Figure 1: In this paper, we propose a novel structure flow-guided method for real-world DSR. Our method obtains better depth +edge recovery (g-h), compared to (e) and (f) using the SOTA method, FDSR (He et al. 2021). (a-b) Synthetic LR depth maps; +(c) Real LR depth map with the structural distortion; (d) Real LR depth map with the edge noise (e.g., holes); (i-j) Ground-truth +HR depth maps; (k-l) RGB image guidance. +Abstract +Real depth super-resolution (DSR), unlike synthetic settings, +is a challenging task due to the structural distortion and the +edge noise caused by the natural degradation in real-world +low-resolution (LR) depth maps. These defeats result in sig- +nificant structure inconsistency between the depth map and +the RGB guidance, which potentially confuses the RGB- +structure guidance and thereby degrades the DSR quality. In +this paper, we propose a novel structure flow-guided DSR +framework, where a cross-modality flow map is learned to +guide the RGB-structure information transferring for pre- +cise depth upsampling. Specifically, our framework consists +of a cross-modality flow-guided upsampling network (CFU- +Net) and a flow-enhanced pyramid edge attention network +(PEANet). CFUNet contains a trilateral self-attention module +combining both the geometric and semantic correlations for +reliable cross-modality flow learning. Then, the learned flow +maps are combined with the grid-sampling mechanism for +coarse high-resolution (HR) depth prediction. PEANet targets +*These authors contributed equally. +†corresponding authors +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +at integrating the learned flow map as the edge attention into +a pyramid network to hierarchically learn the edge-focused +guidance feature for depth edge refinement. Extensive exper- +iments on real and synthetic DSR datasets verify that our ap- +proach achieves excellent performance compared to state-of- +the-art methods. +Introduction +With the fast development of cheap RGB-D sensors, depth +maps have played a much more important role in a variety +of computer vision applications, such as object recognition +(Blum et al. 2012; Eitel et al. 2015), 3D reconstruction (Hou, +Dai, and Nießner 2019; Newcombe et al. 2011), and virtual +reality (Meuleman et al. 2020)). However, the defects (e.g., +low resolution and structural distortion) lying in the cheap +RGB-D sensors (e.g., Microsoft Kinect and HuaweiP30Pro), +still hinder their more extensive applications in real world. +Also, although the popular DSR methods (Song et al. 2020; +Kim, Ponce, and Ham 2021; Sun et al. 2021) have achieved +excellent DSR accuracy on synthetic LR depth maps, the +significant domain gap between the real and the synthetic +data largely degrades their DSR precision on the real data. +arXiv:2301.13416v1 [cs.CV] 31 Jan 2023 + +This domain gap is mainly caused by different genera- +tion mechanisms of the LR depth map. The synthetic LR +depth map is usually generated via artificial degradation +(e.g., down-sampling operation), while the real one is from +natural degradation (e.g., noise, blur, and distortion). Differ- +ent from the synthetic data, there are two challenges of the +real-data DSR as below. The first one is the severe structural +distortion (see Fig. 1 (c)), especially for the low-reflection +glass surface or the infrared-absorbing surface. The second +one is the edge noise even the holes (see Fig. 1 (d)), caused +by the physical limitations or the low processing power of +the depth sensors. Both of the challenges above present +a significant difference between the real and the synthetic +data, which inherently degrades the generalization precision +of the synthetic DSR methods to the real data. +In this paper, we develop a novel structure flow-guided +DSR framework to handle the above challenges. For the +structural distortion, we propose a cross-modality flow- +guided upsampling network (CFUNet) that learns a struc- +tured flow between the depth map and the RGB image to +guide their structure alignment for the recovery of the dis- +torted depth structure. It includes two key components: a +trilateral self-attention module and a cross-modality cross- +attention module. In detail, the former leverages the geomet- +ric and semantic correlations (i.e., coordinate distance, pixel +difference, feature difference) to guide the relevant depth- +feature aggregation into each depth feature to supplement +the missing depth-structure information. The latter utilizes +the enhanced depth feature and the RGB feature as the in- +put for their sufficient message passing and flow-map gen- +eration. Finally, we combine the flow map with the grid- +sampling mechanism for the coarse HR depth prediction. +For the edge noise, we present a flow-enhanced pyramid +edge attention network (PEANet) that integrates the learned +structure flow map as the edge attention into a pyramid net- +work to learn the edge-focused guidance feature for the edge +refinement of the coarse HR depth map predicted above. +Considering the structure clue (i.e., edge region tends to +own significant flow-value fluctuations) lying in the learned +flow map, we combine the flow map with the RGB fea- +ture to form the flow-enhanced RGB feature for highlighting +the RGB-structure region. Then, we feed the flow-enhanced +RGB feature into an iterative pyramid network for its edge- +focused guidance feature learning. The low-level guidance +features effectively filter the RGB-texture noise (guided by +the flow map), while the high-level guidance features exploit +the rich context information for more precise edge-feature +capture. Finally, we pass the learned guidance feature and +the depth feature into a decoder network to predict the edge- +refined HR depth map. Extensive experiments on challeng- +ing real-world datasets verify the effectiveness of our pro- +posed method (see examples in Fig. 1(g-h)). In summary, +our contributions are as follows: +• We propose an effective cross-modality flow-guided up- +sampling network (CFUNet), where a structure flow map +is learned to guide the structure alignment between the +depth map and the RGB image for the recovery of the +distorted depth edge. +• We present a flow-enhanced pyramid edge attention net- +work (PEANet) that integrates the flow map as edge at- +tention into a pyramid network to hierarchically learn the +edge-focused guidance feature for edge refinement. +• Extensive experiments on the real and synthetic datasets +verify the effectiveness of the proposed framework, and +we achieve state-of-the-art restoration performance on +multiple DSR dataset benchmarks. +Related Work +Synthetic Depth Super-Resolution +The synthetic depth super-resolution (DSR) architectures +can be divided into the pre-upsampling methods and the +progressive upsampling methods (Wang et al. 2020). The +pre-upsampling DSR methods first upsample the input depth +with interpolation algorithms (e.g., bicubic) from LR to HR, +and then feed it into depth recovery network layers. (Li +et al. 2016) introduce the first pre-upsampling network ar- +chitecture. As this method handles arbitrary scaling factor +depth, more and more similar approaches have been pre- +sented to further facilitate DSR task (Li et al. 2019; Lutio +et al. 2019; Zhu et al. 2018; Chen and Jung 2018; Hao et al. +2019; Su et al. 2019). However, upsampling in one step is +not suitable for large scaling factors simply because it usu- +ally leads to losing much detailed information. To tackle +these issues, a progressive upsampling structure is designed +in MSG-net(Tak-Wai, Loy, and Tang 2016), which gradually +upsamples the LR depth map by transposed convolution lay- +ers at different scale levels. Since then, various progressive +upsample-based methods have been proposed that greatly +promote the development of this domain(Tak-Wai, Loy, and +Tang 2016; Guo et al. 2019; He et al. 2021; Zuo et al. 2019). +Recently, the joint-task learning framework achieves im- +pressive performance, such as DSR & completion (Yan et al. +2022), depth estimation & enhancement (Wang et al. 2021) +and DSR & depth estimation (Tang et al. 2021; Sun et al. +2021). Inspired by these joint-task methods, we combine the +alignment task with the super-resolution task to distill the +cross-modality knowledge for robust depth upsampling. +Real-world Depth Super-Resolution +In recent years, the super-resolution for real-world images +has been under the spotlight, which involves upsampling, +denoising, and hole-filling. Early traditional depth enhance- +ment methods (Yang et al. 2014; Liu et al. 2016, 2018) are +based on complex and time-consuming optimization. For +fast CNN-based DSR, AIR (Song et al. 2020) simulates +the real LR depth map by combining the interval degrada- +tion and the bicubic degradation, and proposes a channel at- +tention based network for real DSR. PAC (Su et al. 2019) +and DKN (Kim, Ponce, and Ham 2021) utilize the adap- +tive kernels calculated by the neighborhood pixels in RGB +image for robust DSR. FDSR(He et al. 2021) proposes the +octave convolution for frequency domain separation, which +achieves outstanding performance in real datasets. Although +these methods handle the large modality gap between the +guidance image and depth map, the structure misalignment +between the depth map and the RGB image still leads them + +Cross +Attention +Trilateral +Self- +attention +Grid +Sample +������������������������������������ +������������������������������������ +������������������������������������������������������������������������������������ +Encoder +������������������������������������ +Cross-modality Flow-guided Upsampling + Flow-enhanced Pyramid Edge Attention +Edge +Decoder +������������������������������������������������������������������������������������ +Flow-enhanced +Pyramid Attention +Add +Multi-scale +features +Flow +Decoder +Upsample +Decoder +Flow maps +Figure 2: The pipeline of our structure flow-guided DSR framework. Given the LR depth map and the RGB image, the left +block (blue) first generates the flow maps through a trilateral self-attention module and a cross-attention module, and predicts +the coarse depth map Dcoarse with the flow-based grid-sampling. Then, the right block (yellow) integrates the RGB/depth +features and the flow map (as edge attention) to learn the edge-focused guidance feature for edge refinement (Drefine). +to suffer from serious errors around the edge regions. Dif- +ferent from the general paradigms, we introduce a novel +structure flow-guided framework, which exploits the cross- +modality flow map to guide the RGB-structure information +transferring for real DSR. +Approach +In the following, we introduce our structure flow-guided +DSR framework for robust real-world DSR. As shown in +Fig. 2, our framework consists of two modules: a cross- +modality flow-guided upsampling network (CFUNet) and a +flow-enhanced pyramid edge attention network (PEANet). +Given an LR depth map DLR ∈ RH0×W0 and its cor- +responding HR RGB image I ∈ RH×W ×3 (H/H0 = +W/W0 = s and s is the scale factor), CFUNet first learns +the cross-modality flow to guide the structure alignment be- +tween depth the RGB for coarse HR depth prediction. Then, +PEANet exploits the structure flow as edge attention to learn +the edge-focused guidance feature for edge refinement. +Cross-modality Flow-guided Upsampling Network +As demonstrated in Fig. 1 (c), the structural distortion of the +real LR depth map leads to the significant structure misalign- +ment between the RGB image and the depth map, which po- +tentially damages the structure guidance of RGB images for +depth edge recovery. To handle it, our solution is to learn an +effective cross-modality flow map between the depth and the +RGB to identify their structure relationship. Then, guided by +the learned flow map, we align the structure of the depth map +to the RGB image for the recovery of the distorted depth +edge. Next, we will describe our network in terms of the +feature extraction, the trilateral attention-based flow genera- +tion, and the flow-guided depth upsampling. +Feature extraction. To achieve the consistent input size, +we first upsample the LR depth map DLR to a resolution +map DBic ∈ RH×W with the bicubic interpolation. +Then, we feed the upsampled depth map and the RGB +image into an encoder for their feature extraction: {Fl ∈ +RH×W ×D}L +l=1 and {Gl ∈ RH×W ×D}L +l=1 where the sub- +script l denotes the feature output in l-th layer of the encoder. +Trilateral attention-based flow generation. The key to +generating a reliable cross-modality flow map is to model +a robust relationship between the RGB and the depth map. +Nevertheless, the serious structural distortion caused by the +natural degradation potentially increases the modality gap +between the depth and the RGB. Thereby, it’s difficult to +directly exploit a general attention mechanism to model +such a relationship. To mitigate it, we target at enhanc- +ing the depth feature through a proposed trilateral self- +attention block so that the distorted depth-structure informa- +tion can be largely complemented for relationship modeling. +As shown in Fig. 3, our trilateral self-attention block fuses +the geometric-level correlation and the semantic-level cor- +relation to jointly guide the depth feature enhancement. It’s +noted that we just enhance the depth feature FL in the last +layer (L-th layer): +¯F(i) +L = +� +j +αi,j · (βlow +i,j + βhigh +i,j +) · γi,j · F(j) +L + F(j) +L , +(1) +where F(j) +L (1 ≤ j ≤ H × W) denotes the j-th depth-pixel +feature and ¯F(i) +L +denotes the i-th enhanced depth feature +(1 ≤ i ≤ H × W). The geometric-level correlation con- +tains a spatial kernel α ∈ R(H×W )×(H×W ) and a low-level +color kernel βlow ∈ R(H×W )×(H×W ), while the semantic- +level correlation contains a high-level color semantic ker- +nel βhigh ∈ R(H×W )×(H×W ) and a depth semantic kernel +γ ∈ R(H×W )×(H×W ). In detail, we formulate the spatial +kernel as a coordinate distance-aware Gaussian kernel: +αi,j = Gaussian(∥ Coor(i) − Coor(j)∥2, σs), +(2) +where Gaussian(x, σ) = +1 +σ +√ +2π exp (− x2 +2σ2 ) is the Gaussian +function. Coor(i) ∈ R2 denotes the row-column coordi- +nates of pixel i at the depth map and σs is the kernel vari- +ance. The low-level and high-level color kernels are defined + +K +V +Q +Cross-attention Module +Depth Transformation +Self- +Attention +Add & +Norm +Self- +Attention +Add & +Norm +Color Transformation +V +K +Q +������������������������� +������������������������ +������������������������′ +C +������������������������+������������ +C Concatenation +P Position Embedding +Element-wise Production +P +Trilateral Self-attention Module +{������������������������}������������>������������ +������������ +������������������������������������������������ +������������������������������������������������������������ +������������ +Geometric-level +Semantic-level +������������������������ +������������������������ +Figure 3: The architecture of the trilateral self-attention module and the cross-attention module. +by the Gaussian kernels with the low-level and the semantic- +level RGB feature similarity, whose kernel sum is: +βlow +i,j + βhigh +i,j += +L +� +l=0 +Gaussian(∥G(i) +l +− G(j) +l ∥2, σc). +(3) +The depth semantic kernel is designed based on the depth +feature similarity in the L-th layer: +γi,j = Gaussian(∥F(i) +L − F(j) +L ∥2, σd). +(4) +Guided by the geometric and semantic kernels above, the +correlated depth information can be effectively aggregated +into each depth feature through Eq.1 for depth feature com- +pletion and enhancement. +Then, we feed the enhanced depth feature ¯FL and the +RGB feature GL into the cross-attention block for their ef- +ficient cross-modality feature intersection: +˜F(i) +L = ¯F(i) +L + MLP( +� +j +softmaxj(φq(¯F(i) +L )⊤φk(G(j) +L ))φv(G(j) +L )), +˜G(i) +L = G(i) +L + MLP( +� +j +softmaxj(φq(G(i) +L )⊤φk(¯F(j) +L )φv(¯F(j) +L )), +(5) +where φq, φk and φv are the projection functions of the +query, the key and the value in our nonlocal-style cross- +attention module. With the query-key similarity, the value +can be retrieved for feature enhancement. Then, we concate- +nate the enhanced depth feature ˜FL and RGB feature ˜GL +and pass them into a multi-layer convolutional network to +obtain their correlated feature at each layer {Gl}L′ +l=L+1. Fi- +nally, following (Dosovitskiy et al. 2015), based on the pre- +viously extracted features {Gl}L +l=1 and the correlated fea- +tures {Gl}L′ +l=L+1, we exploit a decoder network to generate +the multi-layer flow maps {∆l}L′ +l=1, where the flow genera- +tion in layer l can be formulated as: +Gflow +l+1 , ∆l+1 = deconv(Cat[Gflow +l +, ∆l, GL′−l−1]), (6) +where Gflow +l +denotes the intermediate flow feature and +deconv consisting of a deconvolution operation and a con- +volutional block (Gflow +1 +, ∆1 = deconv(GL′)). +Flow-guided depth upsampling module. With the +learned flow map ∆L′ in the last layer, we combine it with +the grid-sampling strategy for the HR depth map predic- +tion. In detail, the value of the HR depth map is the bilinear +interpolation of the neighborhood pixels in LR depth map +DLR, where the neighborhoods are defined according to the +learned flow field, which can be formulated as: +Dcoarse = Grid-Sample(DLR, ∆L′), +(7) +where Grid-Sample denotes the upsampling operation com- +puting the output using pixel values from neighborhood pix- +els and pixel locations from the grid (Li et al. 2020). +Flow-enhanced Pyramid Edge Attention Network +In order to further improve our DSR precision in the case of +the edge noise problem, we propose a flow-enhanced pyra- +mid network, where the learned structure flow is served as +the edge attention to hierarchically mine edge-focused guid- +ance feature from the RGB image for the edge refinement +of Dcoarse. Specifically, we first feed the previously pre- +dicted HR depth map Dcoarse and the RGB image into an +encoder network to extract their features: {Fcoarse +t +}T +1 +t=1 and +{Gt}T +t=1, where subscript t indicates the extracted feature at +the t-th layer. Then, we propose the flow-enhanced pyramid +attention module and the edge decoder module as follows +for refined HR depth prediction. +Flow-enhanced pyramid attention module. In this mod- +ule, we target at combining the RGB feature and the flow +map to learn the edge-focused guidance feature {Gguide +t +} +at each layer. In detail, for the t-th layer, with the RGB fea- +ture Gt and its corresponding flow map ∆L′−t, we first fuse +the flow information into the RGB feature to form the flow- +enhanced RGB feature, +Gflow +t += ∆L′−t · Gt + Gt, +(8) + +Decoded Search Feature +Decoder +Add&Ins.Norm +formator +MaskT +Encoded +Add&Ins.Norm +Mul & Iins, Norm +Template Feature +Cross-Attention +Cross-Attention +Encoder +? +Add&ins.Nom +Add&Ins.Norm +Self-Attention +Self-Attention +eight Sharing +TemplateFeature +SearchFeature +Element-wiseProduction +@ Template Feature Mask +Figure 4. An overview of the proposed transformer architectureC +Scale Unify & Concat +CONV +CONV +CONV +CONV +CONV +CONV +∆������������′−������������ +Flow-enhanced Pyramid Attention Module +×K +������������������������ +������������������������ +������������������������������������������������������������������������ +������������������������ +������������������������������������������������������������ +Figure 4: The architecture of the pyramid attention module. +The subscript t denotes the feature output in the t-th layer of +the encoder (1 ≤ t ≤ T). ‘×K’ indicates the iteration times +of the guidance feature updating. +where ∆L′−t ·Gt is expected to exploit the significant flow- +value fluctuations at the edge region in ∆L′−t to better +highlight the structure region of the RGB feature. To fur- +ther smooth the texture feature in Gflow +t +, we concatenate +it with the texture-less depth feature Fcoarse +t +to obtain the +texture-degraded RGB feature ˜Gflow +t +. Then, we feed ˜Gflow +t +into a pyramid network to extract its edge-focused guid- +ance features { ˜Gflow +t,k +}K +k=1 at different scales. The low-level +guidance feature is to filter the texture noise (guided by the +flow map) while the high-level is to exploit the rich context +information for edge-feature capture. After that, we unify +the scales of the hierarchical feature { ˜Gflow +t,k +}K +k=1 using the +bicubic interpolation and pass the concatenated feature into +a convolutional block to generate the flow-enhanced RGB +guidance feature Gguide +t +at the t-th layer. Notably, we de- +sign an iterative architecture to progressively refine the RGB +guidance feature as illustrated in Fig. 4. +Edge decoder. Guided by the flow-based guidance fea- +tures {Gguide +t +}T +t=1 learned at each layer, we progressively +decode the depth feature in an iterative manner: +Fedge +t+1 = FU(Cat(Fedge +t +, Gguide +T −t+1, Fcoarse +T −t+1)), +(9) +where FU function indicates the fusion and upsampling op- +eration following (Guo et al. 2020) and the initial feature +Fedge +1 +is obtained by the convolutional operation on Fcoarse +T +1 . +Finally, we pass Fedge +T +1 into a convolutional block to obtain +the edge-refined HR depth map Drefine. +Loss Function +We train our model by minimizing the smooth-L1 loss be- +tween the ground-truth depth map Dgt and the network out- +put of each sub-network, including the coarse depth predic- +tion Dcoarse and the refined one Drefine: +Ldsr = +H×W +� +i=1 +ℓ +� +Dcoarse +i +− Dgt +i +� ++ ℓ +� +Drefine +i +− Dgt +i +� +, (10) +where the subscript i denote the pixel index and the smooth- +L1 loss function is defined as: +ℓ(u) = +�0.5u2, +if |u| ≤ 1 +(|u| − 0.5) , +otherwise. +(11) +Experiments +Experimental Setting +To evaluate the performance of our method, we perform ex- +tensive experiments on real-world RGB-D-D dataset (He +et al. 2021), ToFMark dataset (Ferstl et al. 2013) and syn- +thetic NYU-v2 dataset (Silberman et al. 2012). We imple- +ment our model with PyTorch and conduct all experiments +on a server containing an Intel i5 2.2 GHz CPU and a TITAN +RTX GPU with almost 24 GB. During training, we randomly +crop patches of resolution 256 × 256 as groundtruth and the +training and testing data are normalized to the range [0, 1]. +In order to balance the training time and network perfor- +mance, the parameters L, L′, K, T are set to 3, 6, 3, 2 in this +paper. We quantitatively and visually compare our method +with 13 state-of-the-art (SOTA) methods: TGV (Ferstl et al. +2013), FBS (Barron and Poole 2016), MSG (Tak-Wai, Loy, +and Tang 2016), DJF (Li et al. 2016), DJFR (Li et al. 2019), +GbFT (AlBahar and Huang 2019), PAC (Su et al. 2019), +CUNet (Deng and Dragotti 2020), FDKN (Kim, Ponce, and +Ham 2021), DKN (Kim, Ponce, and Ham 2021), FDSR (He +et al. 2021), CTKT (Sun et al. 2021) and DCTNet (Zhao +et al. 2022). For simplicity, we name our Structure Flow- +Guided method as SFG. +Experiments on Real Datasets +Depth maps captured by cheap depth sensors usually suf- +fer from structural distortion and edge noise. To verify the +efficiency and robustness of our proposed method, we em- +ploy our method on two challenging benchmarks: RGB-D-D +dataset and ToFMark dataset. +Evaluation on hand-filled RGB-D-D. To evaluate the per- +formance of our method on real LR depth maps, we conduct +experiments on RGB-D-D datasets captured by two RGB- +D sensors: Huawei P30 Pro (captures RGB images and LR +depth maps) and Helios TOF camera (captures HR depth +maps). The LR inputs are shown in Fig. 5, which suffer from +the low resolution (LR size is 192 × 144 and target size is +512 × 384) and random structural missing in the edge re- +gion. Following FDSR (He et al. 2021), we first use 2215 +hand-filled RGB/D pairs for training and 405 RGB/D pairs +for testing. As listed in the first row of Table 1, the proposed +model outperforms SOTA methods by a significant margin. +The first two rows in Fig. 5 show the visual DSR com- +parisons on hand-filled RGB-D-D dataset. We can see that +edges in the results of DKN (Kim, Ponce, and Ham 2021) +and DCTNet (Zhao et al. 2022) are over-smoothed and the +artifacts are visible in the FDSR results. In contrast, our re- +sults show more accurate structures without texture copying. +Evaluation on incomplete RGB-D-D. To further verify the +DSR performance of our method in the case of edge noise +(e.g., edge holes), instead of the hole completion above, we +directly test SFG on unfilled RGB-D dataset and achieve the + +(a) LR depth +(f) Groundtruth +(d) DCTNet +(c) FDSR +(e) SFG (ours) +(b) DKN +Hand-filled +Incomplete +Figure 5: Visual comparison on RGB-D-D dataset. The first (last) two rows show DSR results of hand-filled (incomplete) LR. +RMSE +Bicubic +MSG +DJF +DJFR +CUNet +DKN +FDKN +FDSR +DCTNet +SFG (ours) +Hand-filled +7.17 +5.50 +5.54 +5.52 +5.84 +5.08 +5.37 +5.34 +5.28 +3.88 +Incomplete +- +7.90 +5.70 +5.52 +6.54 +5.43 +5.87 +5.59 +5.49 +4.79 +Noisy +11.57 +10.36 +5.62 +5.71 +6.13 +5.16 +5.54 +5.63 +5.16 +4.45 +Table 1: Quantitative comparison on RGB-D-D dataset. Best and second best results are in bold and underline, respectively. +(b) Bicubic +(d) DCTNet +(c) DKN +(e) SFG (ours) +(a) Groundtruth +Figure 6: Visual comparison on ToFMark dataset. +DJFR +DKN +FDKN +FDSR +DCTNet +SFG (ours) +RMSE +0.27 +0.26 +0.28 +0.28 +0.27 +0.25 +Table 2: Quantitative comparison on ToFMark dataset. +lowest RMSE as shown in the second row of Table 1. More- +over, as shown in the last two rows in Fig. 5, the edges recov- +ered by our method are sharper with fewer artifacts and vi- +sually closest to the ground-truth map. It’s mainly attributed +to the edge-focused guidance feature learning with our flow- +enhanced pyramid edge attention network. +Evaluation on noisy RGB-D-D and ToFMark. We evalu- +ate the denoising and generalization ability of our method on +ToFMark dataset consisting of three RGB-D pairs. The LR +inputs have irregular noise and limited resolution (LR depth +is 120 × 160 and target size is 610 × 810). To simulate the +similar degradation for training, we add the Gaussian noise +(mean 0 and standard deviation 0.07) and the Gaussian blur +(kernel size 5) on the 2215 RGB-D pairs from RGB-D-D +dataset to generate the noisy training dataset. Testing dataset +consists of 405 RGB-D pairs from noisy RGB-D-D dataset +and 3 RGB-D pairs from ToFMark dataset. As shown in the +last row of Table 1 and Table 2, our method achieves the low- +est RMSE in noisy RGB-D-D dataset and the lowest RMSE +in ToFMark dataset, which proves its ability for noise re- +moving. As shown in Fig. 6, it is observed that DKN (Kim, +Ponce, and Ham 2021) and DCTNet (Zhao et al. 2022) intro- +duce some texture artifacts and noise in the low-frequency +region, while SFG recovers clean surface owing to PEA with +effective texture removing. +Experiments on Synthetic Datasets +Since most popular methods are designed for synthetic +datasets, we further evaluate our method on NYU-v2 +datasets for a more comprehensive comparison. Following +the widely used data splitting criterion, we sample 1000 + +T(a) LR depth +(b) DJFR +(c) DKN +(d) FDSR +(e) CFUNet +(f) SFG (ours) +(g) Groundtruth +× 8 +× 16 +Figure 7: Visual comparison of ×8 and ×16 DSR results on NYU-v2 dataset. +RMSE +TGV +FBS +DJFR +GbFT +PAC +CUNet +FDKN +DKN +FDSR +DCTNet +CTKT +SFG (ours) +×4 +4.98 +4.29 +2.38 +3.35 +2.39 +1.89 +1.86 +1.62 +1.61 +1.59 +1.49 +1.45 +×8 +11.23 +8.94 +4.94 +5.73 +4.59 +3.58 +3.33 +3.26 +3.18 +3.16 +2.73 +2.84 +×16 +28.13 +14.59 +9.18 +9.01 +8.09 +6.96 +6.78 +6.51 +5.86 +5.84 +5.11 +5.56 +Table 3: Quantitative comparison on NYU-v2 dataset in terms of average RMSE (cm). +Model +RMSE +CFUNet +4.22 +CFUNet w/o TriSA +4.34 +CFUNet w/o cross-attention +4.57 +Table 4: Ablation study of CFUNet on RGB-D-D dataset. +Datasets +SFG +SFG w/o PEANet +RGB-D-D +3.88 +4.22 +NYU-v2 (×4) +1.45 +1.82 +NYU-v2 (×8) +2.84 +3.76 +NYU-v2 (×16) +5.55 +5.90 +Table 5: Ablation study (in RMSE) of PEANet. +RGB-D pairs for training and the rest 449 RGB-D pairs +for testing. As shown in the Table 3, the proposed method +still achieves comparable results with the SOTA methods +on all upsampling cases (×4, ×8, ×16). In addition, Fig. 7 +presents that our ×8 and ×16 upsampled depth maps own +higher accuracy and more convincing results. It verifies that +our method not only performs DSR well in low-quality maps +with noise and missing structure, but also achieves high- +quality precision in the case of large-scale upsampling. +Ablation Analysis +Ablation study on CFUNet. As shown in the first row of the +Table 4, we still achieve the lowest RMSE criterion just with +the single CFUNet (SFG w/o PEANet) on RGB-D-D dataset +when compare with SOTA methods. It proves the effective- +ness of the learned structure flow map for real DSR. The +Table 4 also shows that removing the trilateral self-attention +(TriSA) and cross-attention module in CFUNet causes per- +formance degradation on RGB-D-D datasets, which verifies +the necessary of the depth feature enhancement for reliable +flow map generation. +K=0 (w/o FPA) +K=1 +K=2 +K=3 +Figure 8: Visual comparison of guidance features using FPA +with different iteration times K, i.e., from 0 (w/o FPA) to 3. +Ablation study on PEANet. To analyze the effectiveness of +PEANet, we train the network with and without PEANet on +the synthetic dataset (NYU-v2) and the real-world dataset +(RGB-D-D). As shown in the Table 5, PEANet consistently +brings the RMSE gain under both real and synthetic dataset +settings. It’s mainly due to our edge-focused guidance fea- +ture learning for robust edge refinement. In addition, Fig. 8 +shows the guidance features under varying iteration times +in FPA (Flow-enhanced Pyramid Attention) module from +0 (w/o FPA) to 3. Visually, as the number of iterations in- +creases, the edge regions tend to receive more attention. +Conclusion +In this paper, we proposed a novel structure flow-guided +DSR framework for real-world depth super-resolution, +which deals with issues of structural distortion and edge +noise. For the structural distortion, a cross-modality flow- +guided upsampling network was presented to learn a reli- +able cross-modality flow between depth and the correspond- +ing RGB guidance for the reconstruction of the distorted +depth edge, where a trilateral self-attention combines the ge- +ometric and semantic correlations for structure flow learn- +ing. For the edge noise, a flow-enhanced pyramid edge at- +tention network was introduced to produce edge attention +based on the learned flow map and learn the edge-focused +guidance feature for depth edge refinement with a pyramid +network. 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IEEE +Transactions on Circuits and Systems for Video Technology, +30(2): 297–306. + diff --git a/0tFQT4oBgHgl3EQf0TbP/content/tmp_files/load_file.txt b/0tFQT4oBgHgl3EQf0TbP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6201e2ede05d976297f418db1be7da334ce85751 --- /dev/null +++ b/0tFQT4oBgHgl3EQf0TbP/content/tmp_files/load_file.txt @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf,len=881 +page_content='Structure Flow-Guided Network for Real Depth Super-Resolution Jiayi Yuan*, Haobo Jiang*, Xiang Li, Jianjun Qian, Jun Li†, Jian Yang† PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education Jiangsu Key Lab of Image and Video Understanding for Social Security School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China {jiayiyuan, jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='hao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='bo, xiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='implus, csjqian, junli, csjyang}@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='cn Real LR Groundtruth RGB image FDSR Ours (c) (e) (d) (f) (g) (h) (i) (j) (k) (l) (a) (b) Synthetic LR Figure 1: In this paper, we propose a novel structure flow-guided method for real-world DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Our method obtains better depth edge recovery (g-h), compared to (e) and (f) using the SOTA method, FDSR (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (a-b) Synthetic LR depth maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (c) Real LR depth map with the structural distortion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (d) Real LR depth map with the edge noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', holes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (i-j) Ground-truth HR depth maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (k-l) RGB image guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Abstract Real depth super-resolution (DSR), unlike synthetic settings, is a challenging task due to the structural distortion and the edge noise caused by the natural degradation in real-world low-resolution (LR) depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' These defeats result in sig- nificant structure inconsistency between the depth map and the RGB guidance, which potentially confuses the RGB- structure guidance and thereby degrades the DSR quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In this paper, we propose a novel structure flow-guided DSR framework, where a cross-modality flow map is learned to guide the RGB-structure information transferring for pre- cise depth upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Specifically, our framework consists of a cross-modality flow-guided upsampling network (CFU- Net) and a flow-enhanced pyramid edge attention network (PEANet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' CFUNet contains a trilateral self-attention module combining both the geometric and semantic correlations for reliable cross-modality flow learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, the learned flow maps are combined with the grid-sampling mechanism for coarse high-resolution (HR) depth prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' PEANet targets These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' †corresponding authors Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' at integrating the learned flow map as the edge attention into a pyramid network to hierarchically learn the edge-focused guidance feature for depth edge refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Extensive exper- iments on real and synthetic DSR datasets verify that our ap- proach achieves excellent performance compared to state-of- the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Introduction With the fast development of cheap RGB-D sensors, depth maps have played a much more important role in a variety of computer vision applications, such as object recognition (Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Eitel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2015), 3D reconstruction (Hou, Dai, and Nießner 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Newcombe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2011), and virtual reality (Meuleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' However, the defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', low resolution and structural distortion) lying in the cheap RGB-D sensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', Microsoft Kinect and HuaweiP30Pro), still hinder their more extensive applications in real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Also, although the popular DSR methods (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Kim, Ponce, and Ham 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021) have achieved excellent DSR accuracy on synthetic LR depth maps, the significant domain gap between the real and the synthetic data largely degrades their DSR precision on the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='13416v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='CV] 31 Jan 2023 This domain gap is mainly caused by different genera- tion mechanisms of the LR depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The synthetic LR depth map is usually generated via artificial degradation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', down-sampling operation), while the real one is from natural degradation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', noise, blur, and distortion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Differ- ent from the synthetic data, there are two challenges of the real-data DSR as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The first one is the severe structural distortion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 1 (c)), especially for the low-reflection glass surface or the infrared-absorbing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The second one is the edge noise even the holes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 1 (d)), caused by the physical limitations or the low processing power of the depth sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Both of the challenges above present a significant difference between the real and the synthetic data, which inherently degrades the generalization precision of the synthetic DSR methods to the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In this paper, we develop a novel structure flow-guided DSR framework to handle the above challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For the structural distortion, we propose a cross-modality flow- guided upsampling network (CFUNet) that learns a struc- tured flow between the depth map and the RGB image to guide their structure alignment for the recovery of the dis- torted depth structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It includes two key components: a trilateral self-attention module and a cross-modality cross- attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In detail, the former leverages the geomet- ric and semantic correlations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', coordinate distance, pixel difference, feature difference) to guide the relevant depth- feature aggregation into each depth feature to supplement the missing depth-structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The latter utilizes the enhanced depth feature and the RGB feature as the in- put for their sufficient message passing and flow-map gen- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Finally, we combine the flow map with the grid- sampling mechanism for the coarse HR depth prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For the edge noise, we present a flow-enhanced pyramid edge attention network (PEANet) that integrates the learned structure flow map as the edge attention into a pyramid net- work to learn the edge-focused guidance feature for the edge refinement of the coarse HR depth map predicted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Considering the structure clue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', edge region tends to own significant flow-value fluctuations) lying in the learned flow map, we combine the flow map with the RGB fea- ture to form the flow-enhanced RGB feature for highlighting the RGB-structure region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we feed the flow-enhanced RGB feature into an iterative pyramid network for its edge- focused guidance feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The low-level guidance features effectively filter the RGB-texture noise (guided by the flow map), while the high-level guidance features exploit the rich context information for more precise edge-feature capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Finally, we pass the learned guidance feature and the depth feature into a decoder network to predict the edge- refined HR depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Extensive experiments on challeng- ing real-world datasets verify the effectiveness of our pro- posed method (see examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 1(g-h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In summary, our contributions are as follows: We propose an effective cross-modality flow-guided up- sampling network (CFUNet), where a structure flow map is learned to guide the structure alignment between the depth map and the RGB image for the recovery of the distorted depth edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' We present a flow-enhanced pyramid edge attention net- work (PEANet) that integrates the flow map as edge at- tention into a pyramid network to hierarchically learn the edge-focused guidance feature for edge refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Extensive experiments on the real and synthetic datasets verify the effectiveness of the proposed framework, and we achieve state-of-the-art restoration performance on multiple DSR dataset benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Related Work Synthetic Depth Super-Resolution The synthetic depth super-resolution (DSR) architectures can be divided into the pre-upsampling methods and the progressive upsampling methods (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The pre-upsampling DSR methods first upsample the input depth with interpolation algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', bicubic) from LR to HR, and then feed it into depth recovery network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2016) introduce the first pre-upsampling network ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As this method handles arbitrary scaling factor depth, more and more similar approaches have been pre- sented to further facilitate DSR task (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Lutio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Chen and Jung 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' However, upsampling in one step is not suitable for large scaling factors simply because it usu- ally leads to losing much detailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To tackle these issues, a progressive upsampling structure is designed in MSG-net(Tak-Wai, Loy, and Tang 2016), which gradually upsamples the LR depth map by transposed convolution lay- ers at different scale levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Since then, various progressive upsample-based methods have been proposed that greatly promote the development of this domain(Tak-Wai, Loy, and Tang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Recently, the joint-task learning framework achieves im- pressive performance, such as DSR & completion (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2022), depth estimation & enhancement (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021) and DSR & depth estimation (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Inspired by these joint-task methods, we combine the alignment task with the super-resolution task to distill the cross-modality knowledge for robust depth upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Real-world Depth Super-Resolution In recent years, the super-resolution for real-world images has been under the spotlight, which involves upsampling, denoising, and hole-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Early traditional depth enhance- ment methods (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2016, 2018) are based on complex and time-consuming optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For fast CNN-based DSR, AIR (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020) simulates the real LR depth map by combining the interval degrada- tion and the bicubic degradation, and proposes a channel at- tention based network for real DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' PAC (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019) and DKN (Kim, Ponce, and Ham 2021) utilize the adap- tive kernels calculated by the neighborhood pixels in RGB image for robust DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' FDSR(He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021) proposes the octave convolution for frequency domain separation, which achieves outstanding performance in real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Although these methods handle the large modality gap between the guidance image and depth map,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' the structure misalignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='between the depth map and the RGB image still leads them ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Trilateral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Cross-modality Flow-guided Upsampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Flow-enhanced Pyramid Edge Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Flow-enhanced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Pyramid Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Add ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Multi-scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Upsample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Flow maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Figure 2: The pipeline of our structure flow-guided DSR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Given the LR depth map and the RGB image, the left block (blue) first generates the flow maps through a trilateral self-attention module and a cross-attention module, and predicts the coarse depth map Dcoarse with the flow-based grid-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, the right block (yellow) integrates the RGB/depth features and the flow map (as edge attention) to learn the edge-focused guidance feature for edge refinement (Drefine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' to suffer from serious errors around the edge regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Dif- ferent from the general paradigms, we introduce a novel structure flow-guided framework, which exploits the cross- modality flow map to guide the RGB-structure information transferring for real DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Approach In the following, we introduce our structure flow-guided DSR framework for robust real-world DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2, our framework consists of two modules: a cross- modality flow-guided upsampling network (CFUNet) and a flow-enhanced pyramid edge attention network (PEANet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Given an LR depth map DLR ∈ RH0×W0 and its cor- responding HR RGB image I ∈ RH×W ×3 (H/H0 = W/W0 = s and s is the scale factor), CFUNet first learns the cross-modality flow to guide the structure alignment be- tween depth the RGB for coarse HR depth prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, PEANet exploits the structure flow as edge attention to learn the edge-focused guidance feature for edge refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Cross-modality Flow-guided Upsampling Network As demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 1 (c), the structural distortion of the real LR depth map leads to the significant structure misalign- ment between the RGB image and the depth map, which po- tentially damages the structure guidance of RGB images for depth edge recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To handle it, our solution is to learn an effective cross-modality flow map between the depth and the RGB to identify their structure relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, guided by the learned flow map, we align the structure of the depth map to the RGB image for the recovery of the distorted depth edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Next, we will describe our network in terms of the feature extraction, the trilateral attention-based flow genera- tion, and the flow-guided depth upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To achieve the consistent input size, we first upsample the LR depth map DLR to a resolution map DBic ∈ RH×W with the bicubic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we feed the upsampled depth map and the RGB image into an encoder for their feature extraction: {Fl ∈ RH×W ×D}L l=1 and {Gl ∈ RH×W ×D}L l=1 where the sub- script l denotes the feature output in l-th layer of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Trilateral attention-based flow generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The key to generating a reliable cross-modality flow map is to model a robust relationship between the RGB and the depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Nevertheless, the serious structural distortion caused by the natural degradation potentially increases the modality gap between the depth and the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Thereby, it’s difficult to directly exploit a general attention mechanism to model such a relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To mitigate it, we target at enhanc- ing the depth feature through a proposed trilateral self- attention block so that the distorted depth-structure informa- tion can be largely complemented for relationship modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 3, our trilateral self-attention block fuses the geometric-level correlation and the semantic-level cor- relation to jointly guide the depth feature enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It’s noted that we just enhance the depth feature FL in the last layer (L-th layer): ¯F(i) L = � j αi,j · (βlow i,j + βhigh i,j ) · γi,j · F(j) L + F(j) L , (1) where F(j) L (1 ≤ j ≤ H × W) denotes the j-th depth-pixel feature and ¯F(i) L denotes the i-th enhanced depth feature (1 ≤ i ≤ H × W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The geometric-level correlation con- tains a spatial kernel α ∈ R(H×W )×(H×W ) and a low-level color kernel βlow ∈ R(H×W )×(H×W ), while the semantic- level correlation contains a high-level color semantic ker- nel βhigh ∈ R(H×W )×(H×W ) and a depth semantic kernel γ ∈ R(H×W )×(H×W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In detail, we formulate the spatial kernel as a coordinate distance-aware Gaussian kernel: αi,j = Gaussian(∥ Coor(i) − Coor(j)∥2, σs), (2) where Gaussian(x, σ) = 1 σ √ 2π exp (− x2 2σ2 ) is the Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Coor(i) ∈ R2 denotes the row-column coordi- nates of pixel i at the depth map and σs is the kernel vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The low-level and high-level color kernels are defined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Cross-attention Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Depth Transformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Add & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Add & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Color Transformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������+������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='C Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='P Position Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Element-wise Production ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Trilateral Self-attention Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='{������������������������}������������>������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Geometric-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Semantic-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Figure 3: The architecture of the trilateral self-attention module and the cross-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' by the Gaussian kernels with the low-level and the semantic- level RGB feature similarity, whose kernel sum is: βlow i,j + βhigh i,j = L � l=0 Gaussian(∥G(i) l − G(j) l ∥2, σc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (3) The depth semantic kernel is designed based on the depth feature similarity in the L-th layer: γi,j = Gaussian(∥F(i) L − F(j) L ∥2, σd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (4) Guided by the geometric and semantic kernels above, the correlated depth information can be effectively aggregated into each depth feature through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='1 for depth feature com- pletion and enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we feed the enhanced depth feature ¯FL and the RGB feature GL into the cross-attention block for their ef- ficient cross-modality feature intersection: ˜F(i) L = ¯F(i) L + MLP( � j softmaxj(φq(¯F(i) L )⊤φk(G(j) L ))φv(G(j) L )), ˜G(i) L = G(i) L + MLP( � j softmaxj(φq(G(i) L )⊤φk(¯F(j) L )φv(¯F(j) L )), (5) where φq, φk and φv are the projection functions of the query, the key and the value in our nonlocal-style cross- attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' With the query-key similarity, the value can be retrieved for feature enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we concate- nate the enhanced depth feature ˜FL and RGB feature ˜GL and pass them into a multi-layer convolutional network to obtain their correlated feature at each layer {Gl}L′ l=L+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Fi- nally, following (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2015), based on the pre- viously extracted features {Gl}L l=1 and the correlated fea- tures {Gl}L′ l=L+1, we exploit a decoder network to generate the multi-layer flow maps {∆l}L′ l=1, where the flow genera- tion in layer l can be formulated as: Gflow l+1 , ∆l+1 = deconv(Cat[Gflow l , ∆l, GL′−l−1]), (6) where Gflow l denotes the intermediate flow feature and deconv consisting of a deconvolution operation and a con- volutional block (Gflow 1 , ∆1 = deconv(GL′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Flow-guided depth upsampling module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' With the learned flow map ∆L′ in the last layer, we combine it with the grid-sampling strategy for the HR depth map predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In detail, the value of the HR depth map is the bilinear interpolation of the neighborhood pixels in LR depth map DLR, where the neighborhoods are defined according to the learned flow field, which can be formulated as: Dcoarse = Grid-Sample(DLR, ∆L′), (7) where Grid-Sample denotes the upsampling operation com- puting the output using pixel values from neighborhood pix- els and pixel locations from the grid (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Flow-enhanced Pyramid Edge Attention Network In order to further improve our DSR precision in the case of the edge noise problem, we propose a flow-enhanced pyra- mid network, where the learned structure flow is served as the edge attention to hierarchically mine edge-focused guid- ance feature from the RGB image for the edge refinement of Dcoarse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Specifically, we first feed the previously pre- dicted HR depth map Dcoarse and the RGB image into an encoder network to extract their features: {Fcoarse t }T +1 t=1 and {Gt}T t=1, where subscript t indicates the extracted feature at the t-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we propose the flow-enhanced pyramid attention module and the edge decoder module as follows for refined HR depth prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Flow-enhanced pyramid attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In this mod- ule, we target at combining the RGB feature and the flow map to learn the edge-focused guidance feature {Gguide t } at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In detail, for the t-th layer, with the RGB fea- ture Gt and its corresponding flow map ∆L′−t, we first fuse the flow information into the RGB feature to form the flow- enhanced RGB feature, Gflow t = ∆L′−t · Gt + Gt, (8) Decoded Search Feature Decoder Add&Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Norm formator MaskT Encoded Add&Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Norm Mul & Iins, Norm Template Feature Cross-Attention Cross-Attention Encoder ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Add&ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Nom Add&Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='Norm Self-Attention Self-Attention eight Sharing TemplateFeature SearchFeature Element-wiseProduction @ Template Feature Mask Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' An overview of the proposed transformer architectureC Scale Unify & Concat CONV CONV CONV CONV CONV CONV ∆������������′−������������ Flow-enhanced Pyramid Attention Module ×K ������������������������ ������������������������ ������������������������������������������������������������������������ ������������������������ ������������������������������������������������������������ Figure 4: The architecture of the pyramid attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The subscript t denotes the feature output in the t-th layer of the encoder (1 ≤ t ≤ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' ‘×K’ indicates the iteration times of the guidance feature updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' where ∆L′−t ·Gt is expected to exploit the significant flow- value fluctuations at the edge region in ∆L′−t to better highlight the structure region of the RGB feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To fur- ther smooth the texture feature in Gflow t , we concatenate it with the texture-less depth feature Fcoarse t to obtain the texture-degraded RGB feature ˜Gflow t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Then, we feed ˜Gflow t into a pyramid network to extract its edge-focused guid- ance features { ˜Gflow t,k }K k=1 at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The low-level guidance feature is to filter the texture noise (guided by the flow map) while the high-level is to exploit the rich context information for edge-feature capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' After that, we unify the scales of the hierarchical feature { ˜Gflow t,k }K k=1 using the bicubic interpolation and pass the concatenated feature into a convolutional block to generate the flow-enhanced RGB guidance feature Gguide t at the t-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Notably, we de- sign an iterative architecture to progressively refine the RGB guidance feature as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Edge decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Guided by the flow-based guidance fea- tures {Gguide t }T t=1 learned at each layer, we progressively decode the depth feature in an iterative manner: Fedge t+1 = FU(Cat(Fedge t , Gguide T −t+1, Fcoarse T −t+1)), (9) where FU function indicates the fusion and upsampling op- eration following (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2020) and the initial feature Fedge 1 is obtained by the convolutional operation on Fcoarse T +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Finally, we pass Fedge T +1 into a convolutional block to obtain the edge-refined HR depth map Drefine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Loss Function We train our model by minimizing the smooth-L1 loss be- tween the ground-truth depth map Dgt and the network out- put of each sub-network, including the coarse depth predic- tion Dcoarse and the refined one Drefine: Ldsr = H×W � i=1 ℓ � Dcoarse i − Dgt i � + ℓ � Drefine i − Dgt i � , (10) where the subscript i denote the pixel index and the smooth- L1 loss function is defined as: ℓ(u) = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='5u2, if |u| ≤ 1 (|u| − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='5) , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (11) Experiments Experimental Setting To evaluate the performance of our method, we perform ex- tensive experiments on real-world RGB-D-D dataset (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021), ToFMark dataset (Ferstl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2013) and syn- thetic NYU-v2 dataset (Silberman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' We imple- ment our model with PyTorch and conduct all experiments on a server containing an Intel i5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='2 GHz CPU and a TITAN RTX GPU with almost 24 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' During training, we randomly crop patches of resolution 256 × 256 as groundtruth and the training and testing data are normalized to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In order to balance the training time and network perfor- mance, the parameters L, L′, K, T are set to 3, 6, 3, 2 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' We quantitatively and visually compare our method with 13 state-of-the-art (SOTA) methods: TGV (Ferstl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2013), FBS (Barron and Poole 2016), MSG (Tak-Wai, Loy, and Tang 2016), DJF (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2016), DJFR (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019), GbFT (AlBahar and Huang 2019), PAC (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2019), CUNet (Deng and Dragotti 2020), FDKN (Kim, Ponce, and Ham 2021), DKN (Kim, Ponce, and Ham 2021), FDSR (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021), CTKT (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021) and DCTNet (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For simplicity, we name our Structure Flow- Guided method as SFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Experiments on Real Datasets Depth maps captured by cheap depth sensors usually suf- fer from structural distortion and edge noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To verify the efficiency and robustness of our proposed method, we em- ploy our method on two challenging benchmarks: RGB-D-D dataset and ToFMark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Evaluation on hand-filled RGB-D-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To evaluate the per- formance of our method on real LR depth maps, we conduct experiments on RGB-D-D datasets captured by two RGB- D sensors: Huawei P30 Pro (captures RGB images and LR depth maps) and Helios TOF camera (captures HR depth maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The LR inputs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 5, which suffer from the low resolution (LR size is 192 × 144 and target size is 512 × 384) and random structural missing in the edge re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Following FDSR (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2021), we first use 2215 hand-filled RGB/D pairs for training and 405 RGB/D pairs for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As listed in the first row of Table 1, the proposed model outperforms SOTA methods by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The first two rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 5 show the visual DSR com- parisons on hand-filled RGB-D-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' We can see that edges in the results of DKN (Kim, Ponce, and Ham 2021) and DCTNet (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2022) are over-smoothed and the artifacts are visible in the FDSR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In contrast, our re- sults show more accurate structures without texture copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Evaluation on incomplete RGB-D-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To further verify the DSR performance of our method in the case of edge noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', edge holes), instead of the hole completion above, we directly test SFG on unfilled RGB-D dataset and achieve the (a) LR depth (f) Groundtruth (d) DCTNet (c) FDSR (e) SFG (ours) (b) DKN Hand-filled Incomplete Figure 5: Visual comparison on RGB-D-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The first (last) two rows show DSR results of hand-filled (incomplete) LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' RMSE Bicubic MSG DJF DJFR CUNet DKN FDKN FDSR DCTNet SFG (ours) Hand-filled 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='88 Incomplete 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='70 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='59 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='79 Noisy 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='57 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='71 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='63 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='45 Table 1: Quantitative comparison on RGB-D-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Best and second best results are in bold and underline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' (b) Bicubic (d) DCTNet (c) DKN (e) SFG (ours) (a) Groundtruth Figure 6: Visual comparison on ToFMark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' DJFR DKN FDKN FDSR DCTNet SFG (ours) RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='25 Table 2: Quantitative comparison on ToFMark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' lowest RMSE as shown in the second row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' More- over, as shown in the last two rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 5, the edges recov- ered by our method are sharper with fewer artifacts and vi- sually closest to the ground-truth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It’s mainly attributed to the edge-focused guidance feature learning with our flow- enhanced pyramid edge attention network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Evaluation on noisy RGB-D-D and ToFMark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' We evalu- ate the denoising and generalization ability of our method on ToFMark dataset consisting of three RGB-D pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The LR inputs have irregular noise and limited resolution (LR depth is 120 × 160 and target size is 610 × 810).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To simulate the similar degradation for training, we add the Gaussian noise (mean 0 and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='07) and the Gaussian blur (kernel size 5) on the 2215 RGB-D pairs from RGB-D-D dataset to generate the noisy training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Testing dataset consists of 405 RGB-D pairs from noisy RGB-D-D dataset and 3 RGB-D pairs from ToFMark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in the last row of Table 1 and Table 2, our method achieves the low- est RMSE in noisy RGB-D-D dataset and the lowest RMSE in ToFMark dataset, which proves its ability for noise re- moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 6, it is observed that DKN (Kim, Ponce, and Ham 2021) and DCTNet (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 2022) intro- duce some texture artifacts and noise in the low-frequency region, while SFG recovers clean surface owing to PEA with effective texture removing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Experiments on Synthetic Datasets Since most popular methods are designed for synthetic datasets, we further evaluate our method on NYU-v2 datasets for a more comprehensive comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Following the widely used data splitting criterion, we sample 1000 T(a) LR depth (b) DJFR (c) DKN (d) FDSR (e) CFUNet (f) SFG (ours) (g) Groundtruth × 8 × 16 Figure 7: Visual comparison of ×8 and ×16 DSR results on NYU-v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' RMSE TGV FBS DJFR GbFT PAC CUNet FDKN DKN FDSR DCTNet CTKT SFG (ours) ×4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='45 ×8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='23 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='59 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='84 ×16 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='13 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='59 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='51 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='56 Table 3: Quantitative comparison on NYU-v2 dataset in terms of average RMSE (cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Model RMSE CFUNet 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='22 CFUNet w/o TriSA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='34 CFUNet w/o cross-attention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='57 Table 4: Ablation study of CFUNet on RGB-D-D dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Datasets SFG SFG w/o PEANet RGB-D-D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='22 NYU-v2 (×4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='82 NYU-v2 (×8) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='76 NYU-v2 (×16) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='90 Table 5: Ablation study (in RMSE) of PEANet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' RGB-D pairs for training and the rest 449 RGB-D pairs for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in the Table 3, the proposed method still achieves comparable results with the SOTA methods on all upsampling cases (×4, ×8, ×16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 7 presents that our ×8 and ×16 upsampled depth maps own higher accuracy and more convincing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It verifies that our method not only performs DSR well in low-quality maps with noise and missing structure, but also achieves high- quality precision in the case of large-scale upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Ablation Analysis Ablation study on CFUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in the first row of the Table 4, we still achieve the lowest RMSE criterion just with the single CFUNet (SFG w/o PEANet) on RGB-D-D dataset when compare with SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It proves the effective- ness of the learned structure flow map for real DSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' The Table 4 also shows that removing the trilateral self-attention (TriSA) and cross-attention module in CFUNet causes per- formance degradation on RGB-D-D datasets, which verifies the necessary of the depth feature enhancement for reliable flow map generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' K=0 (w/o FPA) K=1 K=2 K=3 Figure 8: Visual comparison of guidance features using FPA with different iteration times K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=', from 0 (w/o FPA) to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Ablation study on PEANet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' To analyze the effectiveness of PEANet, we train the network with and without PEANet on the synthetic dataset (NYU-v2) and the real-world dataset (RGB-D-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' As shown in the Table 5, PEANet consistently brings the RMSE gain under both real and synthetic dataset settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' It’s mainly due to our edge-focused guidance fea- ture learning for robust edge refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' 8 shows the guidance features under varying iteration times in FPA (Flow-enhanced Pyramid Attention) module from 0 (w/o FPA) to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Visually, as the number of iterations in- creases, the edge regions tend to receive more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Conclusion In this paper, we proposed a novel structure flow-guided DSR framework for real-world depth super-resolution, which deals with issues of structural distortion and edge noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For the structural distortion, a cross-modality flow- guided upsampling network was presented to learn a reli- able cross-modality flow between depth and the correspond- ing RGB guidance for the reconstruction of the distorted depth edge, where a trilateral self-attention combines the ge- ometric and semantic correlations for structure flow learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' For the edge noise, a flow-enhanced pyramid edge at- tention network was introduced to produce edge attention based on the learned flow map and learn the edge-focused guidance feature for depth edge refinement with a pyramid network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Extensive experiments on both real-world and syn- thetic datasets demonstrated the superiority of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Science Fund of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' U1713208 and 62072242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=' References AlBahar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQf0TbP/content/2301.13416v1.pdf'} diff --git a/29AyT4oBgHgl3EQf1vl1/content/tmp_files/2301.00740v1.pdf.txt b/29AyT4oBgHgl3EQf1vl1/content/tmp_files/2301.00740v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c81e1f5ef176f5768b20764b4fce6a99ced05e3 --- /dev/null +++ b/29AyT4oBgHgl3EQf1vl1/content/tmp_files/2301.00740v1.pdf.txt @@ -0,0 +1,1514 @@ +P3DC-Shot: Prior-Driven Discrete Data Calibration for +Nearest-Neighbor Few-Shot Classification +Shuangmei Wanga,∗, Rui Maa,b,∗, Tieru Wua,b,∗∗, Yang Caoa,∗∗ +aJilin University, No. 2699 Qianjin Street, Changchun, 130012, China +bEngineering Research Center of Knowledge-Driven Human-Machine Intelligence, MOE, No. 2699 Qianjin Street, Changchun, 130012, China +Abstract +Nearest-Neighbor (NN) classification has been proven as a simple and effective approach for few-shot learning. The +query data can be classified efficiently by finding the nearest support class based on features extracted by pretrained +deep models. However, NN-based methods are sensitive to the data distribution and may produce false prediction if +the samples in the support set happen to lie around the distribution boundary of different classes. To solve this issue, +we present P3DC-Shot, an improved nearest-neighbor based few-shot classification method empowered by prior- +driven data calibration. Inspired by the distribution calibration technique which utilizes the distribution or statistics of +the base classes to calibrate the data for few-shot tasks, we propose a novel discrete data calibration operation which is +more suitable for NN-based few-shot classification. Specifically, we treat the prototypes representing each base class +as priors and calibrate each support data based on its similarity to different base prototypes. Then, we perform NN +classification using these discretely calibrated support data. Results from extensive experiments on various datasets +show our efficient non-learning based method can outperform or at least comparable to SOTA methods which need +additional learning steps. +Keywords: +Few-Shot Learning, Image Classification, Prototype, Calibration +1. Introduction +Deep learning has triggered significant breakthroughs +in many computer vision tasks, such as image classifi- +cation [1, 2, 3], object detection [4, 5, 6], and seman- +tic segmentation [7, 8, 9] etc. One key factor for the +success of deep learning is the emergence of large-scale +datasets, e.g., ImageNet [2], MSCOCO [10], Cityscapes +[11], just to name a few. However, it is difficult and +expensive to collect and annotate sufficient data sam- +ples to train a deep model with numerous weights. The +data limitation has become a main bottleneck for more +broader application of deep leaning, especially for the +tasks involving rarely seen samples. On the other hand, +human can learn to recognize novel visual concepts +from only a few samples. There is still a notable gap +• This work is supported in part by the National Key Research +and Development Program of China (Grant No. 2020YFA0714103) +and the National Natural Science Foundation of China (Grant No. +61872162 and 62202199). +∗Co-first authors. +∗∗Corresponding authors. +between human intelligence and the deep learning based +artificial intelligence. Few-shot learning (FSL) aims to +learn neural models for novel classes with only a few +samples. Due to its ability for generalization, FSL has +attracted extensive interests in recent years [12, 13, 14]. +Few-shot classification is the most widely studied +FSL task which attempts to recognize new classes or +classify data in an unseen query set. Usually, few-shot +classification is formulated in a meta-learning frame- +work [15, 16, 17, 18, 19, 20, 21, 22, 23]. +In the +meta-training stage, the N-way K-shot episodic training +paradigm is often employed to learn generalizable clas- +sifiers or feature extractors for data of the base classes. +Then, in the meta-testing stage, the meta-learned clas- +sifiers can quickly adapt to a few annotated but unseen +data in a support set and attain the ability to classify the +novel query data. Although meta-learning has shown +the effectiveness for few-shot classification, it is unclear +how to set the optimal class number (N) and per-class +sample number (K) when learning the classifiers. Also, +the learned classifier may not perform well when the +sample number K used in meta-testing does not match +Preprint submitted to Elsevier +January 3, 2023 +arXiv:2301.00740v1 [cs.CV] 2 Jan 2023 + +the one used in the meta-training [24]. +On the other hand, nearest-neighbor (NN) based clas- +sification has been proven as a simple and effective ap- +proach for FSL. Based on features obtained from the +meta-learned feature extractor [15, 16] or the pretrained +deep image models [25], the query data can be effi- +ciently classified by finding the nearest support class. +Specifically, the prediction is determined by measuring +the similarity or distance between the query feature and +the prototypes (i.e., average or centroid) of the support +features. From the geometric view, NN-based classi- +fication can be solved using a Voronoi Diagram (VD) +which is a partition of the space formed by the support +features [26, 27]. Given a query feature, its class can +be predicted by computing the closest Voronoi cell that +corresponds to a certain support class. With proper VD +construction and feature distance metrics, the state-of- +the-art performance can be achieved for few-shot clas- +sification [28]. +However, due to the limited number +of support samples, NN-based few-shot classification is +sensitive to the distribution of the sampled data and may +produce false prediction if the samples in the support set +happen to lie around the distribution boundary of differ- +ent classes (see Figure 1 left). +To solve above issues, various efforts have been paid +to more effectively utilize the knowledge or priors from +the base classes for few-shot classification. One natural +way is to learn pretrained classifiers or image encoders +with the abundant labeled samples of base classes and +then adapt them the novel classes via transfer learning +[29, 30, 31, 23]. Meanwhile, it has been shown that +variations in selecting the base classes can lead to dif- +ferent performance on the novel classes [32, 33, 34] and +how to select the base classes for better feature repre- +sentation learning still needs more investigation. +On +the other hand, a series of works [35, 36, 37, 38] per- +form data calibration to the novel classes so that the re- +sults are less affected by the limited number of support +samples. One representative is Distribution Calibration +(DC) [38] which assumes the features of the data fol- +low the Gaussian distribution and transfers the statis- +tics from the similar base classes to the novel classes. +Then, DC trains a simple logistic regression classifier +to classify the query features using features sampled +from the calibrated distributions of the novel classes. +Although DC has achieved superior performance than +previous meta-learning [19, 21, 22] or transfer-learning +[29, 30, 31, 23] based methods, it relies on the strong as- +sumption for Gaussian-like data distribution and it can- +not be directly used for NN-based few-shot classifica- +tion. +In this paper, we propose P3DC-Shot, an improved +Support sample Query sample Calibrated support sample +Figure 1: When samples in the support set lie around the distribution +boundary of different classes, the NN classifier may produce false pre- +diction. By performing discrete calibration for each support sample +using priors from the base classes, the calibrated support data is trans- +formed closer to the actual class centroid and can lead to less-biased +NN classification. The colored regions represent the underlying data +distribution of different classes. The gray lines are the predicted deci- +sion boundaries by the NN classifier. +NN-based few-shot classification method that employs +prior information from base classes to discretely cali- +brate or adjust the support samples so that the calibrated +data is more representative for the underlying data dis- +tribution (Figure 1 right). Our main insight is even the +novel classes have not been seen before, they still share +similar features to some base classes, and the prior in- +formation from the base classes can serve as the context +data for the novel classes. When only a few support +samples are available for the novel classes, performing +prior-driven calibration can alleviate the possible bias +introduced by the few-shot support samples. With the +calibrated support samples, the query data can be more +accurately classified by a NN-based classifier. +Specifically, for the prior information, we compute +the prototype, i.e., the average of features, for each base +class. Then, we propose three different schemes for se- +lecting the similar prototypes to calibrate the support +data. Firstly, we propose the sample-level calibration +which selects the top M most similar base prototypes for +each support sample and then apply weighted averaging +between each support sample and selected prototypes to +obtain the calibrated support sample. Secondly, to uti- +lize more context from the base classes, we propose the +task-level calibration which combines the most similar +base prototypes for each support sample into a union +and performs the calibration for the support samples us- +ing each prototype in the union. In addition, we pro- +pose a unified calibration scheme that combines the two +above schemes so that the calibration can exploit dif- +ferent levels of prior information from the base classes. +To utilize the calibrated support samples for the NN- +based classification, we further obtain the prototypes of +2 + +the support class using an attention-weighted averaging, +while the attention weights are computed between the +query sample and each calibrated support sample. Fi- +nally, the classification of a query sample is simply de- +termined by finding its nearest support prototype mea- +sured by the cosine similarity. +Comparing to DC, our P3DC-Shot adopts the simi- +lar idea of transferring the information or statistics from +the base classes to the novel classes. The key differ- +ence is our data calibration is performed on each indi- +vidual support sample rather than the distribution pa- +rameters and we employ the NN-based classification in- +stead of the learned classifier as in DC. Comparing to +other NN-based few-shot classification methods such as +SimpleShot [25], since our support data is calibrated, +the NN classification is less affected by the sampling +bias for the support data, e.g, the calibrated data is more +likely to be close to the center of the corresponding +novel class. We conduct extensive comparisons with re- +cent state-of-the-art few-shot classificaiton methods on +miniImageNet [2], tiredImageNet [39] and CUB [40] +and the results demonstrate the superiority and general- +izability of our P3DC-Shot. Ablation studies on differ- +ent calibration schemes, i.e., different weights between +the sample-level and task-level calibration also show the +necessity of combining two schemes for better results. +In summary, our contributions are as follows: +1. We +propose +P3DC-Shot, +a +prior-driven +dis- +crete data calibration strategy for nearest-neighbor +based few-shot classification to enhance the +model’s robustness to the distribution of the sup- +port samples. +2. Without additional training and expensive compu- +tation, the proposed method can efficiently cali- +brate each support sample using information from +the prototypes of the similar base classes. +3. We conduct extensive evaluations on three discrete +calibration schemes on various datasets and the re- +sults show our efficient non-learning based method +can outperform or at least comparable to SOTA +few-shot classification methods. +2. Related Work +In this section, we first review the representative +meta-learning and transfer learning based few-shot clas- +sification techniques. Then, we summarize the nearest- +neighbor and data calibration based approaches which +are most relevant to our P3DC-Shot. +Meta-learning based few-shot classification. Meta- +learning [41] has been widely adopted for few-shot clas- +sification. +The core idea is to leverage the episodic +training paradigm to learn generalizable classifiers or +feature extractors using the data from the base classes +in an optimization-based framework [18, 19, 20, 21, +22], as well as learn a distance function to measure +the similarity between the support and query samples +through metric-learning [42, 15, 17, 43, 44, 37]. For +example, MAML [19] is one of the most representa- +tive optimization-based meta-learning method for few- +shot classification and its goal is to learn good net- +work initialization parameters so that the model can +quickly adapt to new tasks with only a small amount +of new training data from the novel classes. For metric- +learning based methods such as the Matching Networks +[15], Prototypical Networks [16] and Relation Net- +works [17], the network is trained to either learn an +embedding function with a given distance function or +learn both the embedding and the distance function in +a meta-learning architecture. Unlike the optimization +and metric-learning based methods which require so- +phisticated meta-learning steps, our method can directly +utilize the features extracted by the pretrained models +and perform the prior-driven calibration to obtain less- +biased support features for classification. +Transfer learning based few-shot classification. +Transfer learning [45, 46, 47] is a classic machine learn- +ing or deep learning technique that aims to improve +the the learning of a new task through the transfer of +knowledge from one or more related tasks that have al- +ready been learned. Pretraining a deep network on the +base dataset and transferring knowledge to the novel +classes via fine-tuning [31, 48, 30] has been shown as +the strong baseline for the few-shot classification. To +learn better feature representations which can lead to +improved few-shot fine-tuning performance, Mangla et +al. [29] propose S2M2, the Self-Supervised Manifold +Mixup, to apply regularization over the feature mani- +fold enriched via the self-supervised tasks. In addition +to training new linear classifiers based on the pretrained +weights learned from the base classes, Meta-Baseline +[23] performs meta-learning to further optimize the pre- +trained weights for few-shot classification. On the other +hand, it has been shown the results of the transfer learn- +ing based methods depend on different selections of the +base classes for pretraining [32, 33], while how to se- +lect the base classes to achieve better performance is +still challenging [34]. In comparison, our P3DC-shot +does not need the additional cost for feature represen- +tation learning and can more effectively utilize the base +classes in a NN-based classification framework. +Nearest neighbor based few-shot classification. +NN-based classification has also been investigated for +few-shot classification. The main idea is to compute the +3 + +prototypes of the support samples, i.e., the mean or cen- +troid of the support features, and classify the query sam- +ple using metrics such as L2 distance, cosine similarity +or a learned distance function. In SimpleShot [25], it +shows nearest neighbor classification with features sim- +ply normalized by L2 norm and measured by Euclidean +distance can achieve competitive few-shot classification +results. Instead of performing nearest neighbor classifi- +cation on the image-level features, Li et al. [49] intro- +duces a Deep Nearest Neighbor Neural Network which +performs nearest neighbor search over the deep local +descriptors and defines an image-to-class measure for +few-shot classification. From a geometric view, Ma et +al. [50] utilize the Cluster-induced Voronoi Diagram +(CIVD) to incorporate cluster-to-point and cluster-to- +cluster relationships to the nearest neighbor based clas- +sification. +Similar to above methods, our method is +based on the nearest prototype classification, while +we perform the prior-driven data calibration to obtain +less-biased support data for the prototype computation. +Meanwhile, computing the attentive or reweighted pro- +totypes [51, 52, 53] that are guided by the base classes +or query samples has also been investigated recently. +We follow the similar idea and compute the attention- +weighted prototypes for NN-based classification. +Data calibration for few-shot classification. Due to +the limited number of samples, the prototypes or cen- +troids computed from the few-shot support data may be +biased and cannot represent the underlying data distri- +bution. Simply performing NN-based classification on +these biased prototypes will lead to inaccurate classi- +fication. Several methods have been proposed to cali- +brate or rectify the data to obtain better samples or pro- +totypes of the support class [35, 36, 37, 54, 38]. Using +the images in the base classes, RestoreNet [35] learns +a class agnostic transformation on the feature of each +image to move it closer to the class center in the fea- +ture space. To reduce the bias caused by the scarcity +of the support data, Liu et al., [36] employ the pseudo- +labeling to add unlabelled samples with high prediction +confidence into the support set for prototype rectifica- +tion. In [37], Guo et al. propose a Pair-wise Similar- +ity Module to generate calibrated class centers that are +adapted to the query sample. Instead of calibrating in- +dividual support samples, Distribution Calibration (DC) +[38] aims to calibrate the underlying distribution of the +support classes by transferring the Gaussian statistics +from the base classes. With sufficient new support data +sampled from the calibrated distribution, an additional +classifier is trained in [38] to classify the query sam- +ple. In contrast to these methods, we do not require +additional training or assumption of the underlying dis- +tribution. Instead, we directly use the prototypes of the +base classes to calibrate each support sample individ- +ually and we adopt the NN-based classification which +makes the whole pipeline discrete and efficient. One +recent work that is similar to ours is Xu et al. +[54] +which proposes the Task Centroid Projection Removing +(TCPR) module and transforms all support and query +features in a given task to alleviate the sample selection +bias problem. Comparing to [54], we only calibrate the +support samples using the priors from the base classes +and keep the query samples unchanged. +3. Method +To effectively utilize the prior knowledge from the +base classes, we first propose two independent calibra- +tion strategies, i.e., sample-level calibration and task- +level calibration, which exploit different levels of infor- +mation from the base classes. Then, we combine the +sample-level and task-level calibration together to ob- +tain the final calibrated support samples which will be +used for the nearest neighbor classification. +Figure 2 shows an illustration of the P3DC-Shot +pipeline. Given a pretrained feature extractor F and a +set of prototypes of base classes, we perform the prior- +driven discrete calibration to the normalized features of +the support data. Initially, the query sample in green +is closer to the support sample in yellow. +After the +proposed calibration using the related base class proto- +types, the query sample becomes closer to the calibrated +support sample in blue. In the following, we provide +technical details of the P3DC-Shot for few-shot classi- +fication. +3.1. Problem Statement +In this paper, we focus on the few-shot image clas- +sification which aims to classify the new image sam- +ples from the novel classes with just a few labeled im- +age samples. Normally, the new data sample is called +a query sample and the labelled samples are called sup- +port samples. With the aid of a set of base classes rep- +resented by their prototypes Pb = {pb +i }nb +i=1, our goal is to +calibrate the support samples from novel-class so that +they can be better matched with the query samples by +a nearest neighbor classifier. Here, all data samples are +represented by the features computed from a pretrained +feature extractor F(·) : X → Rd, while X is the domain +of the image space and d is the dimension of the feature +space; pb +i is the prototype of a base class, which is com- +puted as the average feature of the samples within the +4 + +L2 norm +Calibration +������������ +Feature +extraction +������������1 +������������2 +������������ +Support data +Query data +Final calibrated support features +Endpoint of sample-level calibration +Endpoint of task-level calibration +All base class prototypes +or +Selected prototypes for a sample ++ +Selected prototypes for a task +̅������������1 +̅������������1 +̅������������2 +̅������������2 +������������1 +������������ +������������2 +������������ +̅������������1 +������������� +̅������������2 +Figure 2: An illustration of the P3DC-Shot pipeline for the 2-way 1-shot scenario. Note that the direct interpolation of the three triangle vertices +return a feature on the triangle plane. After normalization, the final calibrated features ¯xu +1 and ¯xu +2 are on the hypersphere of the normalized space. +class; nb is the number of all base classes. For simplic- +ity, we directly use xi to represent the feature F(xi) of +an image xi. +We follow the conventional few-shot learning setting, +i.e., build a series of N-way K-shot tasks where N is the +number of novel classes and K is the number of sup- +port samples in each task. +Formally, each task con- +sists of a support set S = {(xi, yi)}N×K +i=1 +and a query set +Q = {qi}N×K+N×Q +i=N×K+1 . Here, yi is the label of the corre- +sponding sample, which is known for the support set +and unknown for the query set; Q is the number of query +sample for each novel class in the current task. Given a +support feature xi, we perform our prior-driven calibra- +tion to obtain the calibrated support feature xc +i = C(xi), +where C(·) : Rd → Rd conducts feature transformation +based on the information from the base classes. Then, +we predict the label of a query feature by performing +nearest neighbor classification w.r.t the novel class pro- +totypes computed from the calibrated support feature(s). +3.2. Prior-Driven Discrete Data Calibration +Before we perform calibration to the support data, we +first apply L2 normalization to the support and query +features. +It is shown in SimpleShot [25] that using +L2-normalized feature with a NN-based classifier can +lead to competitive results for few-shot classification. +Hence, we obtain ¯xi for a support feature xi by: +¯xi = normalize(xi) = +xi +∥xi∥2 +. +(1) +Similarly, the normalization of the query features are +also computed: ¯qi = normalize(qi). By working with +the normalized features, we can obviate the absolute +scales of the features and focus on the similarities and +differences on their directions. Note that, the normal- +ized features are used in the feature combination step +(Eq. 7, 10 and 11) for obtaining the interpolation be- +tween the normalized features and in the NN-based clas- +sification step (Eq. 12) for performance improvement. +Next, we propose the sample-level and task-level cal- +ibration, and their combination to utilize the priors from +the base classes for obtaining the less-biased support +features. +3.2.1. Sample-Level Calibration +According to previous works [55, 38] which also use +the information from base classes for classifying the +new classes, the base classes with higher similarities +to the query classes are more important than other base +classes. Hence, we first propose to perform calibration +based on the top similar base classes for each support +sample. Moreover, following DC [38], we apply the +Tukeys’s Ladder of Powers transformation [56] to the +features of the support samples before the calibration: +˜xi = +� xλ +i +if λ � 0 +log(xi) +if λ = 0 +(2) +Here, λ is a hyperparameter which controls the distri- +bution of the transformed feature, with a smaller λ can +lead to a less skewed feature distribution. We set λ = 0.5 +and obtain the transformed support feature ˜xi from the +original feature xi. +Then, we select the top M base classes with higher +similarities to a transformed support feature ˜xi: +ΛM +i = {pb +j| j ∈ topM(Si)}, +(3) +where Si = {< ˜xi, pb +j > | j ∈ {1, . . . nb}}. +(4) +Here, ΛM +i stores the M nearest base prototypes with re- +spect to a transformed support feature vector ˜xi; topM(·) +is an operator that returns the index of top M elements +from Si, the similarity set of ˜xi, while the similarity be- +tween ˜xi and a base prototype pb +j is computed by the +5 + +inner product < ·, · >. In DC [38], the distributions +of the base and novel classes are assumed as Gaussian +distribution and the statistics (mean and co-variance) of +the base classes are used to calibrate the distribution of +the novel classes. In contrast, we directly use the sim- +ilar base prototypes to calibrate each support feature. +Specifically, the calibration for ˜xi driven by base proto- +types pb +j ∈ ΛM +i is computed as: +si = ˜xi + +� +j∈ΛM +i +wijpb +j, +(5) +where the weights of the M nearest base classes proto- +types in ΛM +i +are obtained by applying Softmax to the +similarities between ˜xi and these prototypes: +wij = +e<˜xi,pb +j> +� +k∈ΛM +i e<˜xi,pb +k> , j ∈ ΛM +i . +(6) +It should be noted that, in Eq. 5, the support feature ˜xi +is a transformed feature, while the base prototypes are +in the original feature space. This setting is the same +as DC does for calibrating the distribution of the novel +classes and it can be understood as follows: 1) the trans- +formation can initially reduce the skewness of the few- +shot-sampled support features; 2) the term wijpb +j can be +regarded as the projection of ˜xi w.r.t prototype pb +j; 3) +˜xi is calibrated based on its projects to all of its similar +base prototypes in ΛM +i . +Finally, the sample-level calibration for a normalized +support sample ¯xi is defined as: +¯xs +i = normalize((1 − α)¯xi + α¯si), +(7) +where α ∈ [0, 1] is a parameter to linearly combine +the normalized support feature ¯xi and normalized base- +prototypes-driven calibration ¯si = norm(si). As shown +in Figure 2, ¯xi and ¯si form a line in the normalized fea- +ture space and ¯xs +i is the normalization of a in-between +point on this line. In general, the sample-level calibra- +tion can rectify each support sample based on its own +top M most similar base classes. +3.2.2. Task-Level Calibration +By performing the sample-level calibration, the bias +induced by the few-shot support samples can be reduced +to a certain degree. However, when the sampling bias +is too large, e.g., the support sample is lying near the +boundary of a class, the set of similar base classes ΛM +i +obtained by Eq. 3 may also be biased. To alleviate such +bias, we propose the task-level calibration which utilizes +the base prototypes related to all support samples when +calibrating each individual support feature. Concretely, +for a support set S = {(xi, yi)}N×K +i=1 w.r.t a task T , we col- +lect the top M similar base prototypes for each support +sample and form a union of related base prototypes for +T : +ΛT = +N×K +� +i=1 +ΛM +i . +(8) +Then, for a transformed support sample ˜xi obtained +by Eq. 2, the calibration using all of the task-related +base prototypes is computed by: +ti = ˜xi + +� +j∈ΛT +wi jpb +j, +(9) +where wi j is calculated in the similar way as Eq. 6, but +the similarities are computed using the prototypes from +ΛT instead of ΛM +i . By involving more prototypes to cal- +ibrate the support samples, the bias caused by only using +nearby prototypes for a near-boundary support sample +can be reduced. +Then, we define the task-level calibration for a nor- +malized support sample ¯xi as: +¯xt +i = normalize((1 − β)¯xi + β¯ti), +(10) +where ¯ti is the normalization of ti. Similar to the sample- +level calibration, ¯xi and ¯ti also form a line in the normal- +ized feature space, while the calibration for each support +sample is based on the union of all related base proto- +types ΛT . +3.2.3. Unified Model +The sample-level and task-level calibration utilize +different levels of information from the base classes to +rectify the support samples in a discrete manner. To fur- +ther attain the merits of both calibration schemes, we +propose a unified model which linearly combines the +sample-level and task-level calibration: +xc +i = ¯xu +i = normalize((1 − α − β)¯xi + α¯si + β¯ti). +(11) +Here, ¯xu +i which is also denoted as xc +i , is the final calibra- +tion for a normalized support sample ¯xi . Geometrically, +xc +i can be understood as the normalization of an interpo- +lated feature point xu +i locating in the triangle formulated +by the three vertices ¯xi, ¯si and ¯ti, while 1 − α − β, α and +β are the barycentric coordinates of xu +i . Different α and +β values can lead to different calibration effects. When +β = 0, the unified model degenerates to the sample- +level calibration, while when α = 0, the model becomes +to the task-level calibration. We quantitatively evaluate +the effects of different α and β values in Section 4.4. +6 + +3.3. Nearest Prototype Classifier +With the calibrated support set Sc = {(xc +i , yi)}N×K +i=1 , we +compute the prototypes {pn}N +n=1 for the novel classes and +perform cosine similarity based nearest classification +for a query feature q. To simplify the notation, we fur- +ther represent Sc = {Sc +n}N +n=1, while Sc +n = {(xc +k, yk = n)}K +k=1 +is the support set for a novel class CLS n. +For the 1-shot case, each calibrated support sample +becomes one prototype and the class of the query fea- +ture is predicted by the nearest prototype classifier: +y∗ = max +pn cos(¯q, pn), +(12) +where pn = xc +n is the calibrated prototype for novel class +CLS n and ¯q is the normalization of query q. +For the multi-shot case, one way to obtain the pro- +totype for a novel class is simply to compute the av- +erage of all support features for the given class as in +Prototypical Networks [16]. However, merely using the +unweighted average of the support features as prototype +does not consider the importance of the support samples +w.r.t the query. Therefore, we adopt the idea of attentive +prototype which is proposed in recent works [51, 53] for +query-guided prototype computation. In our implemen- +tation, we define the attention-weighted prototype as: +pq +n = +� +xc +k∈Scn +akxc +k, +(13) +where ak = +e +� +xcm∈Scn e . +(14) +Here, xc +k and xc +m are the calibrated support samples be- +longing to the CLS n’s support set Sc +n and ak is the atten- +tion weight computed by applying Softmax to the sim- +ilarities between query q and these calibrated support +samples; pq +n is the CLS n’s prototype guided by query +q. Similar to Eq. 12, the prediction for a query q is +obtained by finding the novel class with the nearest pro- +totype pq +n. +4. Experiments +In this section, we perform quantitative compar- +isons between our P3DC-Shot and state-of-the-art +few-shot classification methods on three represen- +tative datasets. +We also conduct ablation studies +on evaluating different hyperparameters and design +choices for our methods. +Our code is available at: +https://github.com/breakaway7/P3DC-Shot. +4.1. Datasets +We evaluate our prior-driven data calibration strate- +gies on three popular datasets for benchmarking few +shot classificaiton: miniImageNet [2], tieredImageNet +[39] and CUB [40]. miniImageNet and tieredImageNet +contain a broad range of classes including various an- +imals and objects, while CUB is a more fine-grained +dataset that focuses on various species of birds. +Specifically, the miniImageNet [2] is derived from +the ILSVRC-2012 [58] and it contains a subset of 100 +classes, each of which consisting of 600 images. We +follow the split used in [18] and obtain 64 base, 16 val- +idation and 20 novel classes for miniImageNet. Comar- +ing to miniImageNet, the tieredImageNet [39] is a larger +subset of [58] which contains 608 classes and therefore +more challenging. We follow [39] and split the tiered- +ImageNet into 351, 97, and 160 classes for base, vali- +dation, and novel classes, respectively. For CUB [40], it +is the short name for Caltech-UCSD Birds 200 dataset, +which contains a total of 11,788 images covering 200 +categories of different bird species. We split the CUB +dataset into 100 base, 50 validation and 50 novel classes +following [31]. Note that the set formed by the base +classes can also be regarded as the train set and the novel +classes correspond to the test set. +4.2. Implementation Details +For each image in the dataset, we represent it as a +640-dimensional feature vector which is extracted us- +ing the WideResNet [59] pretrained by the S2M2 [29] +work. Our calibration pipeline can efficiently proceed +in four steps: 1) find the M = 5 nearby base prototypes +for each support sample xi; 2) compute the endpoint +of the sample-level calibration for xi, i.e., si; 3) col- +lect all nearby base prototypes for all support samples +in the task and compute the endpoint of the task-level +calibration for xi, i.e., ti; 4) combine the sample-level +and task-level calibration and obtain the final calibrated +support sample xc +i . The parameter α and β for weighting +the sample-level and task-level calibration are selected +based on the best results obtained on the validation set +for each dataset. All experiments are conducted on a +PC with a 2.70GHz CPU and 16G memory. No GPU +is needed during the calibration. On average, for a 5- +way 5-shot task, it takes 0.027 seconds to calibrate the +support samples and 0.002 seconds for performing the +nearest prototype classification. +4.3. Comparison and Evaluation +To evaluate the performance of our P3DC-Shot, we +first conduct quantitative comparisons with some rep- +resentative and state-of-the-art few-short classification +7 + +Table 1: +Quantitative comparison on the test set of miniImageNet, tieredImageNet and CUB. The 5-way 1-shot and 5-way 5-shot classification +accuracy (%) with 95% confidence intervals are measured. Best results are highlighted in bold and second best are in italic. The last line shows the +α and β selected based on the valiation set for each dataset. * 8 and 20 are the number of ensembles in DeepVoro and DeepVoro++. † The results +of [54] on tieredImageNet are obtained using its released code. +Methods +miniImageNet +tieredImageNet +CUB +5-way 1-shot +5-way 5-shot +5-way 1-shot +5-way 5-shot +5-way 1-shot +5-way 5-shot +Meta-learning (metric-learning) +MatchingNet [15] (2016) +64.03 ± 0.20 +76.32 ± 0.16 +68.50 ± 0.92 +80.60 ± 0.71 +73.49 ± 0.89 +84.45 ± 0.58 +ProtoNet [16] (2017) +54.16 ± 0.82 +73.68 ± 0.65 +65.65 ± 0.92 +83.40 ± 0.65 +72.99 ± 0.88 +86.64 ± 0.51 +RelationNet [17] (2018) +52.19 ± 0.83 +70.20 ± 0.66 +54.48 ± 0.93 +71.32 ± 0.78 +68.65 ± 0.91 +81.12 ± 0.63 +Meta-learning (optimization) +MAML [19] (2017) +48.70 ± 1.84 +63.10 ± 0.92 +51.67 ± 1.81 +70.30 ± 0.08 +50.45 ± 0.97 +59.60 ± 0.84 +LEO [21] (2019) +61.76 ± 0.08 +77.59 ± 0.12 +66.33 ± 0.15 +81.44 ± 0.09 +68.22 ± 0.22 +78.27 ± 0.16 +DCO [22] (2019) +62.64 ± 0.61 +78.63 ± 0.46 +65.99 ± 0.72 +81.56 ± 0.53 +- +- +Transfer learning +Baseline++ [31] (2019) +57.53 ± 0.10 +72.99 ± 0.43 +60.98 ± 0.21 +75.93 ± 0.17 +70.40 ± 0.81 +82.92 ± 0.78 +Negative-Cosine [57] (2020) +62.33 ± 0.82 +80.94 ± 0.59 +- +- +72.66 ± 0.85 +89.40 ± 0.43 +S2M2R [29] (2020) +64.65 ± 0.45 +83.20 ± 0.30 +68.12 ± 0.52 +86.71 ± 0.34 +80.14 ± 0.45 +90.99 ± 0.23 +Nearest neighbor +SimpleShot [25] (2019) +64.29 ± 0.20 +81.50 ± 0.14 +71.32 ± 0.22 +86.66 ± 0.15 +- +- +DeepVoro(8)∗ [50] (2022) +66.45 ± 0.44 +84.55 ± 0.29 +74.02 ± 0.49 +88.90 ± 0.29 +80.98 ± 0.44 +91.47 ± 0.22 +DeepVoro++(20)∗ [50] (2022) +68.38 ± 0.46 +83.27 ± 0.31 +74.48 ± 0.50 +- +80.70 ± 0.45 +- +Data calibration +RestoreNet [35] (2020) +59.28 ± 0.20 +- +- +- +74.32 ± 0.91 +- +DC [38] (2021) +67.79 ± 0.45 +83.69 ± 0.31 +74.24 ± 0.50 +88.38 ± 0.31 +79.93 ± 0.46 +90.77 ± 0.24 +MCL-Katz+PSM [37] (2022) +67.03 +84.03 +69.90 +85.08 +85.89 +93.08 +S2M2+TCPR† [54] (2022) +68.05 ± 0.41 +84.51 ± 0.27 +72.67 ± 0.48 +87.96 ± 0.31 +- +- +P3DC-Shot (α = 0, β = 0) +65.93 ± 0.45 +84.06 ± 0.30 +73.56 ± 0.49 +88.50 ± 0.32 +81.61 ± 0.43 +91.36 ± 0.22 +P3DC-Shot (α = 1, β = 0) +68.41 ± 0.44 +83.06 ± 0.32 +74.84 ± 0.49 +88.01 ± 0.33 +81.51 ± 0.44 +90.83 ± 0.24 +P3DC-Shot (α = 0, β = 1) +68.67 ± 0.44 +83.64 ± 0.31 +75.20 ± 0.48 +88.29 ± 0.33 +81.58 ± 0.44 +91.02 ± 0.23 +P3DC-Shot (α = 1 +3, β = 1 +3) +68.33 ± 0.44 +84.19 ± 0.30 +74.91 ± 0.49 +88.54 ± 0.32 +81.75 ± 0.43 +91.21 ± 0.23 +P3DC-Shot (selected α, β) +68.68 ± 0.44 +84.37 ± 0.30 +75.20 ± 0.48 +88.67 ± 0.32 +81.86 ± 0.43 +91.36 ± 0.23 +(0.0, 0.9) +(0.0, 0.4) +(0.0, 1.0) +(0.0, 0.3) +(0.2, 0.4) +(0.0, 0.4) +methods. Then, we compare with different data trans- +formation or calibration schemes and provide qualita- +tive visualization for showing the difference of our cali- +bration results w.r.t existing works. In addition, we eval- +uate the generalizability of our method by performing +classification tasks with different difficulties. +Quantitative comparisons. As there are numerous +efforts have been paid to the few-shot classification, +we mainly compare our P3DC-Shot with representative +and SOTA works which cover different types of few- +shot learning schemes. The compared methods include +the metric-learning based meta-learning [15, 16, 17], +optimization-based meta-learning [19, 21, 22], transfer +learning [31, 57, 29], nearest neighbor [25, 50] and cal- +ibration [35, 38, 37, 54] based methods. For certain +methods such as [29, 28], we only compare with their +basic versions and do not consider their model trained +with data augmentation. Note that as not every method +has conducted experiments on all three datasets, we +mainly compare with their reported results. One excep- +tion is for [54], we compare with its results generated +using its released code. +For our method, we report the results of our model +with different hyperparameters α and β. In particular, +we consider the case when α and β are both zero, which +makes our method a simple NN-based method with no +data calibration and only shows the effect for using the +query-guided prototype computation (Eq. 13). We also +compare with the results of α or β is 1, or both of them +are equal to 1 +3, which correspond to the cases that the +endpoint of the sample-level or task-level calibration or +the barycenter of the calibration triangle (Figure 2). In +the end, we provide our best results with the α or β se- +lected based on the validation set. +For each dataset, we evaluate on the 5-way 1-shot +and 5-way 5-shot classification setting. For each set- +ting, 2,000 testing tasks, each of which contains 5 × K +(K = 1 or 5) samples for the support set and 5 × 15 +8 + +•• +． +口� +本 +·.. -­ +． ． +. +，，， ＂ +•• ·护h.． +心 +..了 '. .:. -� +～ ．沁：．．“心.. +°'“)．一..,... +·- +--■一一- . 一护 +． ＿ +． +I I ． . .. +． ． ． +•. +lJ +. +． ． +• +． + +． + .1 + +� + +-.. + +． + +．． + +． + +． ． ． +． + +． + +•• + +气 +． +它v +二, +□ +女 +．． + +． + +． + +炉 +女 +． + +． + +女 +0 +v + +． +·`i.．． ． +． +． ． +． ． +觅 +． +． +． +炉 +•• +．． +＇， ． +． ． ． ． +． .,. ． +•• +(a) +(b) +(c) +Figure 3: T-SNE visualization of the calibration on example support samples from the test set of miniImageNet (a), tieredImageNet (b), and CUB +(c). The colored dots are data from the same underlying classes as the selected sample and the star is the center of each class. Given a support +sample (represented in square), the upside down triangle is our calibration result and the lozenge is the calibration result of DC [38]. +samples for the query set, are randomly generated from +the test split of the corresponding dataset. Table 1 shows +the quantitative comparison results on three datasets. It +can be seen that our best results outperform most meth- +ods in the 5-way 1-shot setting and are comparable to +the SOTA methods [28, 38] for the 5-way 5-shot set- +ting. Note that although [37] achieves best results on the +CUB dataset, it is inferior on miniImageNet and tiered- +ImageNet. Moreover, since [37] follows a metric-based +few-shot learning pipeline, it still requires to train the +feature extractor and the metric module for each dataset. +For [28], it performs generally well on all three datasets, +but as an ensemble-based method, its computation time +is much longer than our method, especially when the +ensemble number is large. In contrast, our method does +not require any training and only needs to perform an +efficient calibration step for each testing task. +Also, from results of our method with different α and +β values in Table 1, it can be found when α and β is +zero, the query-guided prototype computation can lead +to better performance than the simple NN-based Sim- +pleShot [25]. When either the sample-level or task-level +calibration is applied, i.e., α or β is not zero, the results +are better than the non-calibrated version, showing the +calibration can indeed reduce the bias for the support +samples. Meanwhile, which calibration type is more +suitable is depending on the underlying data distribu- +tion of the dataset. By selecting the α and β based on +the validation set of each dataset, the results are further +improved. In the ablation study, we perform more ex- +periments and analysis of different α and β values. +Comparison with different data transformation or +calibration schemes. To further verify the effectiveness +Table 2: Comparison with different data transformation or calibration +schemes. Accuracy (%) for 5-way 1-shot task on the test set of mini- +ImageNet are measured. +Model +miniImageNet +CUB +5-way 1-shot +5-way 1-shot +NN +47.50 +76.40 +L2N+NN +65.93 +81.61 +CL2N+NN +65.96 +81.54 +DC+L2N+NN +66.23 +79.49 +P3DC-Shot +68.68 +81.86 +(selected α, β) +(0.0,0.9) +(0.2,0.4) +of our prior-driven data calibration, we compare with +several NN-based baseline methods which perform dif- +ferent data transformation or calibration schemes and +the results are shown in Table 2. In this experiment, all +methods are based on the pretrained WideResNet fea- +tures. +Also, only the 5-way 1-shot classification ac- +curacy is measured so that the comparison is focused +on feature transformation instead of the prototype com- +putation schemes. The first baseline is NN, which is +a naive inner product based nearest neighbor classifier. +Then, L2N and CL2N represent L2 normalization and +centered L2 normalization which have been shown as +effective in SimpleShot [25]. In addition, another base- +line that follows the data calibration scheme in DC [38] +is compared. Comparing to the original DC, this base- +line directly takes the calibrated and then normalized +features and employs NN for classification instead of +training new classifiers using the sampled data. From +Table 2, it can be observed the data normalization or cal- +ibration can significantly improve the NN-based classi- +9 + +★Table 3: Generalizability test on different N in N-way 1-shot tasks. Accuracy (%) on the test set of miniImageNet are measured. For our P3DC-Shot, +the same α = 0 and β = 0.9 selected based on the validation set for the 5-way 1-shot case are used for all experiments. +Models +5-way +7-way +9-way +11-way +13-way +15-way +20-way +RestroreNet [35] +59.56 +50.55 +44.54 +39.98 +36.34 +33.52 +28.48 +L2N+NN +65.93 +57.86 +52.45 +48.25 +44.80 +42.12 +37.06 +CL2N+NN +65.96 +57.69 +52.23 +47.93 +44.36 +41.85 +36.65 +P3DC-Shot +68.68 +60.58 +55.03 +50.75 +47.21 +44.43 +39.33 +fication. In addition, our data calibration achieves the +best results comparing to other baselines. The main rea- +son is the L2N and CL2N only perform transformation +rather than calibration using the base priors, while the +modified DC does not consider the attentive similarity +between the support samples and the base classes when +performing the calibration. +Visualization of the calibration. +To qualitatively +verify the effectiveness of our calibration, we show the +T-SNE [60] visualization of the calibration results for +some example support samples in Figure 3. The results +of calibrating the same sample using DC [38] are also +compared. It can be seen from Figure 3 that our calibra- +tion can more effectively transform the support samples +closer to the center of the underlying classes. For DC, +the calibration may be minor or even be far away from +the center. The reason is still due to it treats the nearby +base classes with the same weights. In contrast, our cal- +ibration pays more attention to the similar base classes +when determining the weights for combining the base +prototypes (Eq. 5 and 9). +Generalizability test on different N in N-way clas- +sification. Following [35], we conduct a series of N- +way 1-shot experiments on miniImageNet to test the +generalizability of the proposed calibration for differ- +ent classification tasks. Table 3 shows the results of the +baseline methods [35], L2N and CL2N and ours. Note +that with the N increases, there are more data samples in +a test task and the classification becomes more difficult. +It can be observed that our P3DC-Shot achieves con- +sistent best results comparing to the baseline methods, +verifying our method is generalizable to classification +tasks with different difficulties. +4.4. Ablation Study +In this section, we perform ablation studies to ver- +ify the effectiveness of different modules and design +choices of our method. First, we conduct experiments +on different hyperparameter α and β to see how the +sample-level and task-level calibration can affect the fi- +nal results. Then, we perform the study on the effec- +tiveness of using the query-guided attentive prototypes +in the NN classification step. +Effect on different hyperparameter α, β. Differ- +ent α and β values correspond to different degrees of +sample-level and task-level calibration applied to the in- +put data. Geometrically, α, β and 1 − α − β can also be +understood as the coordinates of the calibration result +w.r.t to the triangle formed by the three points ¯xi, si, ti. +To quantitatively reveal how these two hyperparameters +can affect the results, we enumerate different α and β +values on both the validation and test sets of different +datasets. From the results in Figure 4, it can be found +the accuracy near the origin of the figures are smaller, +which means performing calibration can improve upon +using the original features for classification, i.e., α and +β is zero. Also, different datasets prefer different α and +β combinations for achieving higher performance. For +example, miniImageNet shows better results when α+β +is around 0.9 and CUB prefers a relatively smaller cal- +ibration, i.e., α + β is around 0.6. For tieredImageNet, +better results are obtained around the topper left of the +figure, showing the task-level calibration is more help- +ful than the sample-level. Overall, the trend on the test +set is consistent with the validation set. From above ex- +periments, it shows the sample-level and task-level cali- +bration are consistently effective, while how to selecting +the good α and β values are dataset dependent. There- +fore, for our best results, we use the α and β selected +based on the validation set and report their performance +on the test set. +Effect on using attentive prototypes in NN classifi- +cation. To improve the conventional prototype based +NN classificaiton, we propose to compute the query- +guided attentive prototypes to represent the support +class. To verify the effectiveness of this scheme, we per- +form ablation study for 5-way 5-shot tasks on different +tasks using different prototype computation schemes. +Specifically, we take the calibrated support features +and compute the prototypes for the support classes by +performing the conventional average operation or our +query-guided attentive averaging (Eq. 13). The results +10 + +Figure 4: The effect of different α and β on the validation (top) and test (bottom) set of different datasets. Accuracy (%) for 5-way 1-shot task on +miniImageNet, tieredImageNet and CUB are measured. The warmer color corresponds to higher accuracy. +Table 4: Ablation study on using the query-guided attentive proto- +types in NN classification. Accuray (%) on the test set of miniIma- +geNet, tieredImageNet and CUB are measured. +Model +miniImageNet tieredImageNet +CUB +5-way 5-shot +5-way 5-shot +5-way 5-shot +Average +84.11 +88.54 +91.27 +Attentive +84.37 +88.67 +91.36 +in Table 4 show that the attentive prototypes can lead to +better performance. Hence, we adopt the attentive pro- +totypes in our NN-based classification. +5. Conclusion +In this paper, we propose a simple yet effective frame- +work, named P3DC-Shot, for few-shot classification. +Without any retraining and expensive computation, our +prior-driven discrete data calibration method can effi- +ciently calibrate the support samples based on prior- +information from the base classes to obtain the less- +biased support data for NN-based classification. 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Hinton, Visualizing data using t-sne, +Journal of Machine Learning Research 9 (2008) 2579–2605. +13 + diff --git a/29AyT4oBgHgl3EQf1vl1/content/tmp_files/load_file.txt b/29AyT4oBgHgl3EQf1vl1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4a99b80e2898ce18e517294a6194edf7c491cc8 --- /dev/null +++ b/29AyT4oBgHgl3EQf1vl1/content/tmp_files/load_file.txt @@ -0,0 +1,1249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf,len=1248 +page_content='P3DC-Shot: Prior-Driven Discrete Data Calibration for Nearest-Neighbor Few-Shot Classification Shuangmei Wanga,∗, Rui Maa,b,∗, Tieru Wua,b,∗∗, Yang Caoa,∗∗ aJilin University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2699 Qianjin Street, Changchun, 130012, China bEngineering Research Center of Knowledge-Driven Human-Machine Intelligence, MOE, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2699 Qianjin Street, Changchun, 130012, China Abstract Nearest-Neighbor (NN) classification has been proven as a simple and effective approach for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The query data can be classified efficiently by finding the nearest support class based on features extracted by pretrained deep models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' However, NN-based methods are sensitive to the data distribution and may produce false prediction if the samples in the support set happen to lie around the distribution boundary of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To solve this issue, we present P3DC-Shot, an improved nearest-neighbor based few-shot classification method empowered by prior- driven data calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Inspired by the distribution calibration technique which utilizes the distribution or statistics of the base classes to calibrate the data for few-shot tasks, we propose a novel discrete data calibration operation which is more suitable for NN-based few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, we treat the prototypes representing each base class as priors and calibrate each support data based on its similarity to different base prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we perform NN classification using these discretely calibrated support data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Results from extensive experiments on various datasets show our efficient non-learning based method can outperform or at least comparable to SOTA methods which need additional learning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Keywords: Few-Shot Learning, Image Classification, Prototype, Calibration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Introduction Deep learning has triggered significant breakthroughs in many computer vision tasks, such as image classifi- cation [1, 2, 3], object detection [4, 5, 6], and seman- tic segmentation [7, 8, 9] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One key factor for the success of deep learning is the emergence of large-scale datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', ImageNet [2], MSCOCO [10], Cityscapes [11], just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' However, it is difficult and expensive to collect and annotate sufficient data sam- ples to train a deep model with numerous weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The data limitation has become a main bottleneck for more broader application of deep leaning, especially for the tasks involving rarely seen samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' On the other hand, human can learn to recognize novel visual concepts from only a few samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' There is still a notable gap This work is supported in part by the National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2020YFA0714103) and the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 61872162 and 62202199).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ∗Co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ∗∗Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' between human intelligence and the deep learning based artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Few-shot learning (FSL) aims to learn neural models for novel classes with only a few samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Due to its ability for generalization, FSL has attracted extensive interests in recent years [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Few-shot classification is the most widely studied FSL task which attempts to recognize new classes or classify data in an unseen query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Usually, few-shot classification is formulated in a meta-learning frame- work [15, 16, 17, 18, 19, 20, 21, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In the meta-training stage, the N-way K-shot episodic training paradigm is often employed to learn generalizable clas- sifiers or feature extractors for data of the base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, in the meta-testing stage, the meta-learned clas- sifiers can quickly adapt to a few annotated but unseen data in a support set and attain the ability to classify the novel query data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Although meta-learning has shown the effectiveness for few-shot classification, it is unclear how to set the optimal class number (N) and per-class sample number (K) when learning the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Also, the learned classifier may not perform well when the sample number K used in meta-testing does not match Preprint submitted to Elsevier January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='00740v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='CV] 2 Jan 2023 the one used in the meta-training [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' On the other hand, nearest-neighbor (NN) based clas- sification has been proven as a simple and effective ap- proach for FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Based on features obtained from the meta-learned feature extractor [15, 16] or the pretrained deep image models [25], the query data can be effi- ciently classified by finding the nearest support class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, the prediction is determined by measuring the similarity or distance between the query feature and the prototypes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', average or centroid) of the support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' From the geometric view, NN-based classi- fication can be solved using a Voronoi Diagram (VD) which is a partition of the space formed by the support features [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Given a query feature, its class can be predicted by computing the closest Voronoi cell that corresponds to a certain support class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' With proper VD construction and feature distance metrics, the state-of- the-art performance can be achieved for few-shot clas- sification [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' However, due to the limited number of support samples, NN-based few-shot classification is sensitive to the distribution of the sampled data and may produce false prediction if the samples in the support set happen to lie around the distribution boundary of differ- ent classes (see Figure 1 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To solve above issues, various efforts have been paid to more effectively utilize the knowledge or priors from the base classes for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One natural way is to learn pretrained classifiers or image encoders with the abundant labeled samples of base classes and then adapt them the novel classes via transfer learning [29, 30, 31, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Meanwhile, it has been shown that variations in selecting the base classes can lead to dif- ferent performance on the novel classes [32, 33, 34] and how to select the base classes for better feature repre- sentation learning still needs more investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' On the other hand, a series of works [35, 36, 37, 38] per- form data calibration to the novel classes so that the re- sults are less affected by the limited number of support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One representative is Distribution Calibration (DC) [38] which assumes the features of the data fol- low the Gaussian distribution and transfers the statis- tics from the similar base classes to the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, DC trains a simple logistic regression classifier to classify the query features using features sampled from the calibrated distributions of the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Although DC has achieved superior performance than previous meta-learning [19, 21, 22] or transfer-learning [29, 30, 31, 23] based methods, it relies on the strong as- sumption for Gaussian-like data distribution and it can- not be directly used for NN-based few-shot classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In this paper, we propose P3DC-Shot, an improved Support sample Query sample Calibrated support sample Figure 1: When samples in the support set lie around the distribution boundary of different classes, the NN classifier may produce false pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' By performing discrete calibration for each support sample using priors from the base classes, the calibrated support data is trans- formed closer to the actual class centroid and can lead to less-biased NN classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The colored regions represent the underlying data distribution of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The gray lines are the predicted deci- sion boundaries by the NN classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' NN-based few-shot classification method that employs prior information from base classes to discretely cali- brate or adjust the support samples so that the calibrated data is more representative for the underlying data dis- tribution (Figure 1 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Our main insight is even the novel classes have not been seen before, they still share similar features to some base classes, and the prior in- formation from the base classes can serve as the context data for the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' When only a few support samples are available for the novel classes, performing prior-driven calibration can alleviate the possible bias introduced by the few-shot support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' With the calibrated support samples, the query data can be more accurately classified by a NN-based classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, for the prior information, we compute the prototype, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', the average of features, for each base class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we propose three different schemes for se- lecting the similar prototypes to calibrate the support data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Firstly, we propose the sample-level calibration which selects the top M most similar base prototypes for each support sample and then apply weighted averaging between each support sample and selected prototypes to obtain the calibrated support sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Secondly, to uti- lize more context from the base classes, we propose the task-level calibration which combines the most similar base prototypes for each support sample into a union and performs the calibration for the support samples us- ing each prototype in the union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In addition, we pro- pose a unified calibration scheme that combines the two above schemes so that the calibration can exploit dif- ferent levels of prior information from the base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To utilize the calibrated support samples for the NN- based classification, we further obtain the prototypes of 2 the support class using an attention-weighted averaging, while the attention weights are computed between the query sample and each calibrated support sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Fi- nally, the classification of a query sample is simply de- termined by finding its nearest support prototype mea- sured by the cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparing to DC, our P3DC-Shot adopts the simi- lar idea of transferring the information or statistics from the base classes to the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The key differ- ence is our data calibration is performed on each indi- vidual support sample rather than the distribution pa- rameters and we employ the NN-based classification in- stead of the learned classifier as in DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparing to other NN-based few-shot classification methods such as SimpleShot [25], since our support data is calibrated, the NN classification is less affected by the sampling bias for the support data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='g, the calibrated data is more likely to be close to the center of the corresponding novel class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We conduct extensive comparisons with re- cent state-of-the-art few-shot classificaiton methods on miniImageNet [2], tiredImageNet [39] and CUB [40] and the results demonstrate the superiority and general- izability of our P3DC-Shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Ablation studies on differ- ent calibration schemes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', different weights between the sample-level and task-level calibration also show the necessity of combining two schemes for better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In summary, our contributions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We propose P3DC-Shot, a prior-driven dis- crete data calibration strategy for nearest-neighbor based few-shot classification to enhance the model’s robustness to the distribution of the sup- port samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Without additional training and expensive compu- tation, the proposed method can efficiently cali- brate each support sample using information from the prototypes of the similar base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We conduct extensive evaluations on three discrete calibration schemes on various datasets and the re- sults show our efficient non-learning based method can outperform or at least comparable to SOTA few-shot classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Related Work In this section, we first review the representative meta-learning and transfer learning based few-shot clas- sification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we summarize the nearest- neighbor and data calibration based approaches which are most relevant to our P3DC-Shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Meta-learning based few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Meta- learning [41] has been widely adopted for few-shot clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The core idea is to leverage the episodic training paradigm to learn generalizable classifiers or feature extractors using the data from the base classes in an optimization-based framework [18, 19, 20, 21, 22], as well as learn a distance function to measure the similarity between the support and query samples through metric-learning [42, 15, 17, 43, 44, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For example, MAML [19] is one of the most representa- tive optimization-based meta-learning method for few- shot classification and its goal is to learn good net- work initialization parameters so that the model can quickly adapt to new tasks with only a small amount of new training data from the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For metric- learning based methods such as the Matching Networks [15], Prototypical Networks [16] and Relation Net- works [17], the network is trained to either learn an embedding function with a given distance function or learn both the embedding and the distance function in a meta-learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Unlike the optimization and metric-learning based methods which require so- phisticated meta-learning steps, our method can directly utilize the features extracted by the pretrained models and perform the prior-driven calibration to obtain less- biased support features for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Transfer learning based few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Transfer learning [45, 46, 47] is a classic machine learn- ing or deep learning technique that aims to improve the the learning of a new task through the transfer of knowledge from one or more related tasks that have al- ready been learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Pretraining a deep network on the base dataset and transferring knowledge to the novel classes via fine-tuning [31, 48, 30] has been shown as the strong baseline for the few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To learn better feature representations which can lead to improved few-shot fine-tuning performance, Mangla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' [29] propose S2M2, the Self-Supervised Manifold Mixup, to apply regularization over the feature mani- fold enriched via the self-supervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In addition to training new linear classifiers based on the pretrained weights learned from the base classes, Meta-Baseline [23] performs meta-learning to further optimize the pre- trained weights for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' On the other hand, it has been shown the results of the transfer learn- ing based methods depend on different selections of the base classes for pretraining [32, 33], while how to se- lect the base classes to achieve better performance is still challenging [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In comparison, our P3DC-shot does not need the additional cost for feature represen- tation learning and can more effectively utilize the base classes in a NN-based classification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Nearest neighbor based few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' NN-based classification has also been investigated for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The main idea is to compute the 3 prototypes of the support samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', the mean or cen- troid of the support features, and classify the query sam- ple using metrics such as L2 distance, cosine similarity or a learned distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In SimpleShot [25], it shows nearest neighbor classification with features sim- ply normalized by L2 norm and measured by Euclidean distance can achieve competitive few-shot classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Instead of performing nearest neighbor classifi- cation on the image-level features, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' [49] intro- duces a Deep Nearest Neighbor Neural Network which performs nearest neighbor search over the deep local descriptors and defines an image-to-class measure for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' From a geometric view, Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' [50] utilize the Cluster-induced Voronoi Diagram (CIVD) to incorporate cluster-to-point and cluster-to- cluster relationships to the nearest neighbor based clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Similar to above methods, our method is based on the nearest prototype classification, while we perform the prior-driven data calibration to obtain less-biased support data for the prototype computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Meanwhile, computing the attentive or reweighted pro- totypes [51, 52, 53] that are guided by the base classes or query samples has also been investigated recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We follow the similar idea and compute the attention- weighted prototypes for NN-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Data calibration for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Due to the limited number of samples, the prototypes or cen- troids computed from the few-shot support data may be biased and cannot represent the underlying data distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Simply performing NN-based classification on these biased prototypes will lead to inaccurate classi- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Several methods have been proposed to cali- brate or rectify the data to obtain better samples or pro- totypes of the support class [35, 36, 37, 54, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Using the images in the base classes, RestoreNet [35] learns a class agnostic transformation on the feature of each image to move it closer to the class center in the fea- ture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To reduce the bias caused by the scarcity of the support data, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', [36] employ the pseudo- labeling to add unlabelled samples with high prediction confidence into the support set for prototype rectifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In [37], Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' propose a Pair-wise Similar- ity Module to generate calibrated class centers that are adapted to the query sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Instead of calibrating in- dividual support samples, Distribution Calibration (DC) [38] aims to calibrate the underlying distribution of the support classes by transferring the Gaussian statistics from the base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' With sufficient new support data sampled from the calibrated distribution, an additional classifier is trained in [38] to classify the query sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In contrast to these methods, we do not require additional training or assumption of the underlying dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Instead, we directly use the prototypes of the base classes to calibrate each support sample individ- ually and we adopt the NN-based classification which makes the whole pipeline discrete and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One recent work that is similar to ours is Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' [54] which proposes the Task Centroid Projection Removing (TCPR) module and transforms all support and query features in a given task to alleviate the sample selection bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparing to [54], we only calibrate the support samples using the priors from the base classes and keep the query samples unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Method To effectively utilize the prior knowledge from the base classes, we first propose two independent calibra- tion strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', sample-level calibration and task- level calibration, which exploit different levels of infor- mation from the base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we combine the sample-level and task-level calibration together to ob- tain the final calibrated support samples which will be used for the nearest neighbor classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Figure 2 shows an illustration of the P3DC-Shot pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Given a pretrained feature extractor F and a set of prototypes of base classes, we perform the prior- driven discrete calibration to the normalized features of the support data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Initially, the query sample in green is closer to the support sample in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' After the proposed calibration using the related base class proto- types, the query sample becomes closer to the calibrated support sample in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In the following, we provide technical details of the P3DC-Shot for few-shot classi- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Problem Statement In this paper, we focus on the few-shot image clas- sification which aims to classify the new image sam- ples from the novel classes with just a few labeled im- age samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Normally, the new data sample is called a query sample and the labelled samples are called sup- port samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' With the aid of a set of base classes rep- resented by their prototypes Pb = {pb i }nb i=1, our goal is to calibrate the support samples from novel-class so that they can be better matched with the query samples by a nearest neighbor classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Here, all data samples are represented by the features computed from a pretrained feature extractor F(·) : X → Rd, while X is the domain of the image space and d is the dimension of the feature space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' pb i is the prototype of a base class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' which is com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='puted as the average feature of the samples within the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='L2 norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Calibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Support data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Query data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Final calibrated support features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Endpoint of sample-level calibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Endpoint of task-level calibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='All base class prototypes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Selected prototypes for a sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Selected prototypes for a task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='̅������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='Figure 2: An illustration of the P3DC-Shot pipeline for the 2-way 1-shot scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that the direct interpolation of the three triangle vertices return a feature on the triangle plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' After normalization, the final calibrated features ¯xu 1 and ¯xu 2 are on the hypersphere of the normalized space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' nb is the number of all base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For simplic- ity, we directly use xi to represent the feature F(xi) of an image xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We follow the conventional few-shot learning setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', build a series of N-way K-shot tasks where N is the number of novel classes and K is the number of sup- port samples in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Formally, each task con- sists of a support set S = {(xi, yi)}N×K i=1 and a query set Q = {qi}N×K+N×Q i=N×K+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Here, yi is the label of the corre- sponding sample, which is known for the support set and unknown for the query set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Q is the number of query sample for each novel class in the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Given a support feature xi, we perform our prior-driven calibra- tion to obtain the calibrated support feature xc i = C(xi), where C(·) : Rd → Rd conducts feature transformation based on the information from the base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we predict the label of a query feature by performing nearest neighbor classification w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t the novel class pro- totypes computed from the calibrated support feature(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Prior-Driven Discrete Data Calibration Before we perform calibration to the support data, we first apply L2 normalization to the support and query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' It is shown in SimpleShot [25] that using L2-normalized feature with a NN-based classifier can lead to competitive results for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Hence, we obtain ¯xi for a support feature xi by: ¯xi = normalize(xi) = xi ∥xi∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (1) Similarly, the normalization of the query features are also computed: ¯qi = normalize(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' By working with the normalized features, we can obviate the absolute scales of the features and focus on the similarities and differences on their directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that, the normal- ized features are used in the feature combination step (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 7, 10 and 11) for obtaining the interpolation be- tween the normalized features and in the NN-based clas- sification step (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 12) for performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Next, we propose the sample-level and task-level cal- ibration, and their combination to utilize the priors from the base classes for obtaining the less-biased support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Sample-Level Calibration According to previous works [55, 38] which also use the information from base classes for classifying the new classes, the base classes with higher similarities to the query classes are more important than other base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Hence, we first propose to perform calibration based on the top similar base classes for each support sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Moreover, following DC [38], we apply the Tukeys’s Ladder of Powers transformation [56] to the features of the support samples before the calibration: ˜xi = � xλ i if λ � 0 log(xi) if λ = 0 (2) Here, λ is a hyperparameter which controls the distri- bution of the transformed feature, with a smaller λ can lead to a less skewed feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='5 and obtain the transformed support feature ˜xi from the original feature xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we select the top M base classes with higher similarities to a transformed support feature ˜xi: ΛM i = {pb j| j ∈ topM(Si)}, (3) where Si = {< ˜xi, pb j > | j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' nb}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (4) Here, ΛM i stores the M nearest base prototypes with re- spect to a transformed support feature vector ˜xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' topM(·) is an operator that returns the index of top M elements from Si, the similarity set of ˜xi, while the similarity be- tween ˜xi and a base prototype pb j is computed by the 5 inner product < ·, · >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In DC [38], the distributions of the base and novel classes are assumed as Gaussian distribution and the statistics (mean and co-variance) of the base classes are used to calibrate the distribution of the novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In contrast, we directly use the sim- ilar base prototypes to calibrate each support feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, the calibration for ˜xi driven by base proto- types pb j ∈ ΛM i is computed as: si = ˜xi + � j∈ΛM i wijpb j, (5) where the weights of the M nearest base classes proto- types in ΛM i are obtained by applying Softmax to the similarities between ˜xi and these prototypes: wij = e<˜xi,pb j> � k∈ΛM i e<˜xi,pb k> , j ∈ ΛM i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (6) It should be noted that, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 5, the support feature ˜xi is a transformed feature, while the base prototypes are in the original feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' This setting is the same as DC does for calibrating the distribution of the novel classes and it can be understood as follows: 1) the trans- formation can initially reduce the skewness of the few- shot-sampled support features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2) the term wijpb j can be regarded as the projection of ˜xi w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t prototype pb j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3) ˜xi is calibrated based on its projects to all of its similar base prototypes in ΛM i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Finally, the sample-level calibration for a normalized support sample ¯xi is defined as: ¯xs i = normalize((1 − α)¯xi + α¯si), (7) where α ∈ [0, 1] is a parameter to linearly combine the normalized support feature ¯xi and normalized base- prototypes-driven calibration ¯si = norm(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' As shown in Figure 2, ¯xi and ¯si form a line in the normalized fea- ture space and ¯xs i is the normalization of a in-between point on this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In general, the sample-level calibra- tion can rectify each support sample based on its own top M most similar base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Task-Level Calibration By performing the sample-level calibration, the bias induced by the few-shot support samples can be reduced to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' However, when the sampling bias is too large, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', the support sample is lying near the boundary of a class, the set of similar base classes ΛM i obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3 may also be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To alleviate such bias, we propose the task-level calibration which utilizes the base prototypes related to all support samples when calibrating each individual support feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Concretely, for a support set S = {(xi, yi)}N×K i=1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t a task T , we col- lect the top M similar base prototypes for each support sample and form a union of related base prototypes for T : ΛT = N×K � i=1 ΛM i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (8) Then, for a transformed support sample ˜xi obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2, the calibration using all of the task-related base prototypes is computed by: ti = ˜xi + � j∈ΛT wi jpb j, (9) where wi j is calculated in the similar way as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 6, but the similarities are computed using the prototypes from ΛT instead of ΛM i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' By involving more prototypes to cal- ibrate the support samples, the bias caused by only using nearby prototypes for a near-boundary support sample can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we define the task-level calibration for a nor- malized support sample ¯xi as: ¯xt i = normalize((1 − β)¯xi + β¯ti), (10) where ¯ti is the normalization of ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Similar to the sample- level calibration, ¯xi and ¯ti also form a line in the normal- ized feature space, while the calibration for each support sample is based on the union of all related base proto- types ΛT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Unified Model The sample-level and task-level calibration utilize different levels of information from the base classes to rectify the support samples in a discrete manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To fur- ther attain the merits of both calibration schemes, we propose a unified model which linearly combines the sample-level and task-level calibration: xc i = ¯xu i = normalize((1 − α − β)¯xi + α¯si + β¯ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (11) Here, ¯xu i which is also denoted as xc i , is the final calibra- tion for a normalized support sample ¯xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Geometrically, xc i can be understood as the normalization of an interpo- lated feature point xu i locating in the triangle formulated by the three vertices ¯xi, ¯si and ¯ti, while 1 − α − β, α and β are the barycentric coordinates of xu i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Different α and β values can lead to different calibration effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' When β = 0, the unified model degenerates to the sample- level calibration, while when α = 0, the model becomes to the task-level calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We quantitatively evaluate the effects of different α and β values in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Nearest Prototype Classifier With the calibrated support set Sc = {(xc i , yi)}N×K i=1 , we compute the prototypes {pn}N n=1 for the novel classes and perform cosine similarity based nearest classification for a query feature q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To simplify the notation, we fur- ther represent Sc = {Sc n}N n=1, while Sc n = {(xc k, yk = n)}K k=1 is the support set for a novel class CLS n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For the 1-shot case, each calibrated support sample becomes one prototype and the class of the query fea- ture is predicted by the nearest prototype classifier: y∗ = max pn cos(¯q, pn), (12) where pn = xc n is the calibrated prototype for novel class CLS n and ¯q is the normalization of query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For the multi-shot case, one way to obtain the pro- totype for a novel class is simply to compute the av- erage of all support features for the given class as in Prototypical Networks [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' However, merely using the unweighted average of the support features as prototype does not consider the importance of the support samples w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Therefore, we adopt the idea of attentive prototype which is proposed in recent works [51, 53] for query-guided prototype computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In our implemen- tation, we define the attention-weighted prototype as: pq n = � xc k∈Scn akxc k, (13) where ak = e � xcm∈Scn e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' (14) Here, xc k and xc m are the calibrated support samples be- longing to the CLS n’s support set Sc n and ak is the atten- tion weight computed by applying Softmax to the sim- ilarities between query q and these calibrated support samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' pq n is the CLS n’s prototype guided by query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 12, the prediction for a query q is obtained by finding the novel class with the nearest pro- totype pq n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Experiments In this section, we perform quantitative compar- isons between our P3DC-Shot and state-of-the-art few-shot classification methods on three represen- tative datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We also conduct ablation studies on evaluating different hyperparameters and design choices for our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Our code is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='com/breakaway7/P3DC-Shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Datasets We evaluate our prior-driven data calibration strate- gies on three popular datasets for benchmarking few shot classificaiton: miniImageNet [2], tieredImageNet [39] and CUB [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' miniImageNet and tieredImageNet contain a broad range of classes including various an- imals and objects, while CUB is a more fine-grained dataset that focuses on various species of birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, the miniImageNet [2] is derived from the ILSVRC-2012 [58] and it contains a subset of 100 classes, each of which consisting of 600 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We follow the split used in [18] and obtain 64 base, 16 val- idation and 20 novel classes for miniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comar- ing to miniImageNet, the tieredImageNet [39] is a larger subset of [58] which contains 608 classes and therefore more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We follow [39] and split the tiered- ImageNet into 351, 97, and 160 classes for base, vali- dation, and novel classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For CUB [40], it is the short name for Caltech-UCSD Birds 200 dataset, which contains a total of 11,788 images covering 200 categories of different bird species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We split the CUB dataset into 100 base, 50 validation and 50 novel classes following [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that the set formed by the base classes can also be regarded as the train set and the novel classes correspond to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Implementation Details For each image in the dataset, we represent it as a 640-dimensional feature vector which is extracted us- ing the WideResNet [59] pretrained by the S2M2 [29] work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Our calibration pipeline can efficiently proceed in four steps: 1) find the M = 5 nearby base prototypes for each support sample xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 2) compute the endpoint of the sample-level calibration for xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 3) col- lect all nearby base prototypes for all support samples in the task and compute the endpoint of the task-level calibration for xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4) combine the sample-level and task-level calibration and obtain the final calibrated support sample xc i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The parameter α and β for weighting the sample-level and task-level calibration are selected based on the best results obtained on the validation set for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' All experiments are conducted on a PC with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='70GHz CPU and 16G memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' No GPU is needed during the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' On average, for a 5- way 5-shot task, it takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='027 seconds to calibrate the support samples and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='002 seconds for performing the nearest prototype classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparison and Evaluation To evaluate the performance of our P3DC-Shot, we first conduct quantitative comparisons with some rep- resentative and state-of-the-art few-short classification 7 Table 1: Quantitative comparison on the test set of miniImageNet, tieredImageNet and CUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The 5-way 1-shot and 5-way 5-shot classification accuracy (%) with 95% confidence intervals are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Best results are highlighted in bold and second best are in italic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The last line shows the α and β selected based on the valiation set for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' * 8 and 20 are the number of ensembles in DeepVoro and DeepVoro++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' † The results of [54] on tieredImageNet are obtained using its released code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Methods miniImageNet tieredImageNet CUB 5-way 1-shot 5-way 5-shot 5-way 1-shot 5-way 5-shot 5-way 1-shot 5-way 5-shot Meta-learning (metric-learning) MatchingNet [15] (2016) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='92 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='71 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='89 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='58 ProtoNet [16] (2017) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='82 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='92 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='88 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='51 RelationNet [17] (2018) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='83 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='66 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='78 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='91 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='63 Meta-learning (optimization) MAML [19] (2017) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='70 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='84 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='10 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='84 LEO [21] (2019) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='08 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='12 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='15 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='09 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='22 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='16 DCO [22] (2019) 62.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='53 Transfer learning Baseline++ [31] (2019) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='10 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='43 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='17 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='81 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='78 Negative-Cosine [57] (2020) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} 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+page_content='43 S2M2R [29] (2020) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='30 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='52 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='34 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 Nearest neighbor SimpleShot [25] (2019) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='14 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='22 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='15 DeepVoro(8)∗ [50] (2022) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='29 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='29 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='22 DeepVoro++(20)∗ [50] (2022) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='46 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 Data calibration RestoreNet [35] (2020) 59.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='31 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='46 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='24 MCL-Katz+PSM [37] (2022) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='03 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='03 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='90 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='08 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='89 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='08 S2M2+TCPR† [54] (2022) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='41 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='27 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='31 P3DC-Shot (α = 0, β = 0) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='30 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='43 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='22 P3DC-Shot (α = 1, β = 0) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='33 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='24 P3DC-Shot (α = 0, β = 1) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='31 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='33 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 P3DC-Shot (α = 1 3, β = 1 3) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='30 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='43 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 P3DC-Shot (selected α, β) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='44 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='30 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='32 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='43 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='9) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='3) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we compare with different data trans- formation or calibration schemes and provide qualita- tive visualization for showing the difference of our cali- bration results w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In addition, we eval- uate the generalizability of our method by performing classification tasks with different difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Quantitative comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' As there are numerous efforts have been paid to the few-shot classification, we mainly compare our P3DC-Shot with representative and SOTA works which cover different types of few- shot learning schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The compared methods include the metric-learning based meta-learning [15, 16, 17], optimization-based meta-learning [19, 21, 22], transfer learning [31, 57, 29], nearest neighbor [25, 50] and cal- ibration [35, 38, 37, 54] based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For certain methods such as [29, 28], we only compare with their basic versions and do not consider their model trained with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that as not every method has conducted experiments on all three datasets, we mainly compare with their reported results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One excep- tion is for [54], we compare with its results generated using its released code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For our method, we report the results of our model with different hyperparameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In particular, we consider the case when α and β are both zero, which makes our method a simple NN-based method with no data calibration and only shows the effect for using the query-guided prototype computation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' We also compare with the results of α or β is 1, or both of them are equal to 1 3, which correspond to the cases that the endpoint of the sample-level or task-level calibration or the barycenter of the calibration triangle (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In the end, we provide our best results with the α or β se- lected based on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For each dataset, we evaluate on the 5-way 1-shot and 5-way 5-shot classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For each set- ting, 2,000 testing tasks, each of which contains 5 × K (K = 1 or 5) samples for the support set and 5 × 15 8 •• ． 口� 本 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='. -\xad ． ． .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ，，， ＂ •• ·护h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='． 心 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=".了 '." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' -� ～ ．沁：．．“心.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=". °'“)．一." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='.,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ·- --■一一- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 一护 ． ＿ ． I I ． .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='. ． ． ． .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' lJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ． ． ． ． .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='. ． ．． ． ． ． ． ． ． •• 气 ． 它v 二, □ 女 ．． ． ． 炉 女 ． ． 女 0 v ． `i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='．． ． ． ． ． ． ． 觅 ． ． ． 炉 •• ．． ＇， ． ． ． ． ． ． .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' ． •• (a) (b) (c) Figure 3: T-SNE visualization of the calibration on example support samples from the test set of miniImageNet (a), tieredImageNet (b), and CUB (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The colored dots are data from the same underlying classes as the selected sample and the star is the center of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Given a support sample (represented in square), the upside down triangle is our calibration result and the lozenge is the calibration result of DC [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' samples for the query set, are randomly generated from the test split of the corresponding dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Table 1 shows the quantitative comparison results on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' It can be seen that our best results outperform most meth- ods in the 5-way 1-shot setting and are comparable to the SOTA methods [28, 38] for the 5-way 5-shot set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that although [37] achieves best results on the CUB dataset, it is inferior on miniImageNet and tiered- ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Moreover, since [37] follows a metric-based few-shot learning pipeline, it still requires to train the feature extractor and the metric module for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For [28], it performs generally well on all three datasets, but as an ensemble-based method, its computation time is much longer than our method, especially when the ensemble number is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In contrast, our method does not require any training and only needs to perform an efficient calibration step for each testing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Also, from results of our method with different α and β values in Table 1, it can be found when α and β is zero, the query-guided prototype computation can lead to better performance than the simple NN-based Sim- pleShot [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' When either the sample-level or task-level calibration is applied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', α or β is not zero, the results are better than the non-calibrated version, showing the calibration can indeed reduce the bias for the support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Meanwhile, which calibration type is more suitable is depending on the underlying data distribu- tion of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' By selecting the α and β based on the validation set of each dataset, the results are further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In the ablation study, we perform more ex- periments and analysis of different α and β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparison with different data transformation or calibration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To further verify the effectiveness Table 2: Comparison with different data transformation or calibration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Accuracy (%) for 5-way 1-shot task on the test set of mini- ImageNet are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Model miniImageNet CUB 5-way 1-shot 5-way 1-shot NN 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='50 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='40 L2N+NN 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='61 CL2N+NN 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='96 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='54 DC+L2N+NN 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='49 P3DC-Shot 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='68 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='86 (selected α, β) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='9) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4) of our prior-driven data calibration, we compare with several NN-based baseline methods which perform dif- ferent data transformation or calibration schemes and the results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In this experiment, all methods are based on the pretrained WideResNet fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Also, only the 5-way 1-shot classification ac- curacy is measured so that the comparison is focused on feature transformation instead of the prototype com- putation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The first baseline is NN, which is a naive inner product based nearest neighbor classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, L2N and CL2N represent L2 normalization and centered L2 normalization which have been shown as effective in SimpleShot [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In addition, another base- line that follows the data calibration scheme in DC [38] is compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Comparing to the original DC, this base- line directly takes the calibrated and then normalized features and employs NN for classification instead of training new classifiers using the sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' From Table 2, it can be observed the data normalization or cal- ibration can significantly improve the NN-based classi- 9 ★Table 3: Generalizability test on different N in N-way 1-shot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Accuracy (%) on the test set of miniImageNet are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For our P3DC-Shot, the same α = 0 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='9 selected based on the validation set for the 5-way 1-shot case are used for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Models 5-way 7-way 9-way 11-way 13-way 15-way 20-way RestroreNet [35] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='56 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='55 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='54 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='98 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='34 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='52 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='48 L2N+NN 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='86 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='45 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='25 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='80 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='12 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='06 CL2N+NN 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='96 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='69 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='23 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='93 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='36 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='85 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='65 P3DC-Shot 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='68 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='58 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='03 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='75 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='21 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='43 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='33 fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In addition, our data calibration achieves the best results comparing to other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The main rea- son is the L2N and CL2N only perform transformation rather than calibration using the base priors, while the modified DC does not consider the attentive similarity between the support samples and the base classes when performing the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Visualization of the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To qualitatively verify the effectiveness of our calibration, we show the T-SNE [60] visualization of the calibration results for some example support samples in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The results of calibrating the same sample using DC [38] are also compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' It can be seen from Figure 3 that our calibra- tion can more effectively transform the support samples closer to the center of the underlying classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For DC, the calibration may be minor or even be far away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The reason is still due to it treats the nearby base classes with the same weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' In contrast, our cal- ibration pays more attention to the similar base classes when determining the weights for combining the base prototypes (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 5 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Generalizability test on different N in N-way clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Following [35], we conduct a series of N- way 1-shot experiments on miniImageNet to test the generalizability of the proposed calibration for differ- ent classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Table 3 shows the results of the baseline methods [35], L2N and CL2N and ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Note that with the N increases, there are more data samples in a test task and the classification becomes more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' It can be observed that our P3DC-Shot achieves con- sistent best results comparing to the baseline methods, verifying our method is generalizable to classification tasks with different difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Ablation Study In this section, we perform ablation studies to ver- ify the effectiveness of different modules and design choices of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' First, we conduct experiments on different hyperparameter α and β to see how the sample-level and task-level calibration can affect the fi- nal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Then, we perform the study on the effec- tiveness of using the query-guided attentive prototypes in the NN classification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Effect on different hyperparameter α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Differ- ent α and β values correspond to different degrees of sample-level and task-level calibration applied to the in- put data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Geometrically, α, β and 1 − α − β can also be understood as the coordinates of the calibration result w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='t to the triangle formed by the three points ¯xi, si, ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To quantitatively reveal how these two hyperparameters can affect the results, we enumerate different α and β values on both the validation and test sets of different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' From the results in Figure 4, it can be found the accuracy near the origin of the figures are smaller, which means performing calibration can improve upon using the original features for classification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', α and β is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Also, different datasets prefer different α and β combinations for achieving higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For example, miniImageNet shows better results when α+β is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='9 and CUB prefers a relatively smaller cal- ibration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', α + β is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' For tieredImageNet, better results are obtained around the topper left of the figure, showing the task-level calibration is more help- ful than the sample-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Overall, the trend on the test set is consistent with the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' From above ex- periments, it shows the sample-level and task-level cali- bration are consistently effective, while how to selecting the good α and β values are dataset dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' There- fore, for our best results, we use the α and β selected based on the validation set and report their performance on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Effect on using attentive prototypes in NN classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To improve the conventional prototype based NN classificaiton, we propose to compute the query- guided attentive prototypes to represent the support class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' To verify the effectiveness of this scheme, we per- form ablation study for 5-way 5-shot tasks on different tasks using different prototype computation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Specifically, we take the calibrated support features and compute the prototypes for the support classes by performing the conventional average operation or our query-guided attentive averaging (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The results 10 Figure 4: The effect of different α and β on the validation (top) and test (bottom) set of different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Accuracy (%) for 5-way 1-shot task on miniImageNet, tieredImageNet and CUB are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' The warmer color corresponds to higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Table 4: Ablation study on using the query-guided attentive proto- types in NN classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Accuray (%) on the test set of miniIma- geNet, tieredImageNet and CUB are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Model miniImageNet tieredImageNet CUB 5-way 5-shot 5-way 5-shot 5-way 5-shot Average 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='11 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='54 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='27 Attentive 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='37 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='67 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='36 in Table 4 show that the attentive prototypes can lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Hence, we adopt the attentive pro- totypes in our NN-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Conclusion In this paper, we propose a simple yet effective frame- work, named P3DC-Shot, for few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Without any retraining and expensive computation, our prior-driven discrete data calibration method can effi- ciently calibrate the support samples based on prior- information from the base classes to obtain the less- biased support data for NN-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Exten- sive experiments show that our method can outperform or at least comparable to SOTA methods which need ad- ditional learning steps or more computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' One lim- itation of our method is we rely on the whole valida- tion set to select the good hyperparameters α and β to determine which degree of the sample-level and task- level calibration is more suitable for the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Investigating a more general scheme to combine the sample-level and task-level calibration is an interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Moreover, when exploring the combina- tion schemes, we only focus on exploring the inner area of the calibration triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' It is worthy to extend the parameter search to a larger area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=', by extrapolation of the calibration triangle, to find whether better results can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Simonyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AyT4oBgHgl3EQf1vl1/content/2301.00740v1.pdf'} +page_content=' Zisserman, Very deep 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CONFERENCE, APR 6–10, 2022, LAS VEGAS, US +1 +A Data-Driven Gaussian Process Filter for +Electrocardiogram Denoising +Mircea Dumitru, Qiao Li, Erick Andres Perez Alday, Ali Bahrami Rad, Gari D. Clifford, Reza Sameni* +Abstract—Objective: Gaussian Processes (GP)-based filters, +which have been effectively used for various applications includ- +ing electrocardiogram (ECG) filtering can be computationally +demanding and the choice of their hyperparameters is typically +ad hoc. Methods: We develop a data-driven GP filter to address +both issues, using the notion of the ECG phase domain — a time- +warped representation of the ECG beats onto a fixed number +of samples and aligned R-peaks, which is assumed to follow a +Gaussian distribution. Under this assumption, the computation of +the sample mean and covariance matrix is simplified, enabling an +efficient implementation of the GP filter in a data-driven manner, +with no ad hoc hyperparameters. The proposed filter is evaluated +and compared with a state-of-the-art wavelet-based filter, on +the PhysioNet QT Database. The performance is evaluated by +measuring the signal-to-noise ratio (SNR) improvement of the +filter at SNR levels ranging from –5 to 30 dB, in 5 dB steps, +using additive noise. For a clinical evaluation, the error between +the estimated QT-intervals of the original and filtered signals is +measured and compared with the benchmark filter. Results: It is +shown that the proposed GP filter outperforms the benchmark +filter for all the tested noise levels. It also outperforms the state- +of-the-art filter in terms of QT-interval estimation error bias +and variance. Conclusion: The proposed GP filter is a versatile +technique for preprocessing the ECG in clinical and research +applications, is applicable to ECG of arbitrary lengths and +sampling frequencies, and provides confidence intervals for its +performance. +Index Terms—ECG Bayesian filter, Gaussian processes, ECG +denoising, ECG wavelet denoising, QT-interval estimation +I. INTRODUCTION +Electrocardiogram (ECG) denoising is a recurrent problem +in traditional and wearable cardiac monitors. The problem +has been addressed by various approaches, including model- +based and non-model-based filters. A powerful non-parametric +framework for ECG filtering is via Gaussian process (GP) +models [1], [2], which considers the ECG beats as GPs with +common parameters. The choice of the beats GP hyperparam- +eters, namely the mean and kernel functions is non-evident and +ad hoc. For GP models with no beat assumptions, beside the +ambiguity in parameter selection, the GP filter implementation +involves the inversion of large covariance matrices, which +precludes the use of this framework for long ECG records. +In this paper, ECG filtering is addressed via a data-driven +non-parametric GP model. The novelty of the proposed filter +is that it requires no ad hoc GP model hyperparameters and it +is computationally efficient, making it suitable for any length +ECG records; it is based on the assumption that each phase +The authors are with the Department of Biomedical Informatics, School +of Medicine, Emory University. G. D. Clifford is also with the Biomedical +Engineering Department, Georgia Institute of Technology. Corresponding +author: R. Sameni (email: rsameni@dbmi.emory.edu). +domain beat — a time-warped (stretched or squeezed) repre- +sentation of the ECG beats onto a fixed number of samples +and aligned R-peaks — is an ensemble of an underlying +GP. The mean and the kernel function are set via the phase +domain sample mean and covariance matrix, computed via +the available ensembles, which are transformed back to the +time-domain and used to derive the posterior mean using the +Bayesian formalism. +This proposed filter is data-driven, does not presume any +parametric model for the underlying GP, and is computation- +ally efficient. +The filter is evaluated in terms of signal-to-noise ratio (SNR) +improvement, using as benchmark a wavelet-based ECG de- +noiser that was demonstrated in [3] to outperform adaptive +filters [4], Tikhonov regularization and Extended Kalman +filters [5], in terms of SNR improvement. The proposed filter’s +clinical performance is evaluated by measuring the QT-interval +error between the clean ECG and its corresponding filtered +version. +II. GAUSSIAN PROCESS-BASED ECG FILTERING +A. The mathematical model +The ECG measurement x(t) is assumed to be an additive +mixture of a clean ECG s(t), assumed to be a GP contami- +nated by additive white noise: +x(t) = s(t) + n(t), +t ∈ {t1 . . . tN} +∆= TN, +(1) +where n(t) ∼ N(0, vn), vn denotes the noise variance and +N denotes the number of measurements. The signal x(t) is +assumed to be baseline-wander (BW) and powerline noise +removed, which are relatively straightforward, with classical +filtering pipelines (cf. Section III-A). Therefore, the filter +design objective is focused on in-band ECG noise removal. +For the beat i, Ti = +� +ti1 . . . tiRi . . . tiNi +� +denotes the set of +time samples, ti1 representing the first sample, tiRi the sample +corresponding to the R-peak and tiNi the last sample. We +further define xi = [x(t)]i∈Ti, si = [s(t)]i∈Ti, ni = [n(t)]i∈Ti +as vectorial representations of the measurement, clean ECG +and noise, respectively. Therefore, xi = si + ni. +Next, we define matrices Θi ∈ RT ×Ni to map the time +domain beats xi, si and ni to the phase domain beats +ξi = Θixi, ςi = Θisi, ηi = Θini, +(2) +with aligned R-peaks and the same number of samples T +(Fig. 1). The Θi matrices are defined by considering T knots + +2 +EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US +0 +25 +50 +75 +100 +125 +150 +175 +200 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +amplitude [mv] +x[1] +x[2] +x[3] +x[4] +x[5] +x[6] +0 +50 +100 +150 +200 +time [samples] +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +amplitude [mV] +ξ[1] +ξ[2] +ξ[3] +ξ[4] +ξ[5] +ξ[6] +Fig. 1. Time-domain measurements beats (top) and the corresponding phase +domain ECG beats (bottom), with the same number T of samples for the +first 6 beats of sel100 record from QTDB [6] with 0 dB Gaussian additive +noise. Transformation matrices Θi are defined via (3). +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Fig. 2. +Corner detail example of transformation matrix Θi (left), ΘT +i +(middle) and the corresponding (diagonal) Gramian Gi = ΘT +i Θi (right). +equidistantly distributed in the interval [1, Ni] and assigning +Θi(j, k) = +� +1, if j − 1 ≤ (k − 1) Ni−1 +T −1 < j +0, otherwise +, +(3) +with j = 1, . . . , Ni − 1, k = 1, . . . , T and T ≥ maxi {Ni}. +With this choice, the corresponding Gramian matrices Gi, are +diagonal matrices (Fig. 2), +Gi = ΘT +i Θi = diag [gi] and diag +� +ΘiΘT +i +� += 1T , +(4) +with gi ∈ RNi and 1T ∈ RT . Therefore, Gi is invertible and +the back transformation from the phase to the time domain is +given by Ψi = G−1 +i ΘT +i . +From (1) and (2), the ECG beats satisfy ξi = ςi + ηi. +As shown in Fig. 1, in the phase domain the beats have +been normalized in lengths and the R-peaks are aligned. +Therefore, the phase-domain sample variations are only due to +the stochastic inter-beat variations of the ECG beats and noise. +As our working model, we assume that the phase domain beats +ξi to be ensembles of an underlying GP +ξi ∼ N (µξ, Kξ) . +(5) +Moreover, from the time domain noise assumption and (2), +the phase domain noise beats also have a zero-mean normal +distribution ηi ∼ N +� +0, vnΘiΘT +i +� +. Therefore, the phase do- +main ECG beats follow ςi ∼ N +� +µξ, Kξ − vnΘiΘT +i +� +, where +the model parameters µξ and Kξ can be estimated by the +sample mean ¯µξ := B−1 �B +i=1 ξi and the sample covariance +¯ +Kξ := B−1 �B +i=1(ξi− ¯µξ)(ξi− ¯µξ)T , where B is the number +of beats. Therefore, the time domain (clean) ECG beats follow +a Normal distribution si ∼ N (µsi, Ksi) with parameters +µsi = Ψi ¯µξ, Ksi = Ψi +� ¯ +Kξ − ˆvnΘiΘT +i +� +ΨT +i , +(6) +where ˆvn represents the noise variance estimate and the +covariance matrix corresponding to time domains beats xi is +given by +Kxi = Ψi ¯ +KξΨi. +(7) +Finally, the filtered beats are defined as the time domain +posterior mean, using (6) and (7): +ˆsi = µsi + KsiK−1 +xi (xi − µsi) , +(8) +In the sequel, we refer to µsi and ˆsi as prior-based and +posterior-based GP filter results. +B. The GP filter with diagonal covariance matrix +The direct implementation of the filter in (8) requires the +inversion of covariance matrices that typically have huge +condition numbers. The matrix inversion can be avoided if +we consider the diagonal case of ¯ +Kξ: +¯kξ = diag +� ¯ +Kξ +� +, kηi +(4)= ˆvn1T +(9) +In this case, the corresponding time domain matrices are also +diagonal and can be computed via +kxi = +� +ΘT +i ¯kξ +� +⊘ g2 +i , ksi = +� +ΘT +i +�¯kξ − kηi +�� +⊘ g2 +i +(10) +with ◦ and ⊘ denoting the Hadamard product and division, +respectively (element-wise product and division), g2 +i := gi◦gi, +the time domain (prior) mean computed via +µsi = +� +ΘT +i ¯µξ +� +⊘ gi, +(11) +and the corresponding filter given by +ˆsi = µsi + ksi ⊘ kxi ◦ (xi − µsi) . +(12) +The overall algorithm for GP ECG filtering is summarized in +Algorithm 1 and is available online in our Git repository [7]. +Algorithm 1 GP ECG filtering +1: {tiRi}i = RPeakDetector(x) +[Section III-B] +2: ˆvn = NoiseVartianceEstimator(x) [Section II-D] +Input: x, {tiRi}i +Output: {�si}i +3: function GPDIAG(x, {tiRi}i, ˆvn) +⊲ GP diagonal filter +4: +for all beats do +⊲ phase domain computations +5: +compute transformation matrices Θi via (3) +6: +compute the vectors gi = diag +� +ΘT +i Θi +� +7: +compute kηi via (9) +8: +compute the phase beats ξi via (2) +9: +end for +10: +compute phase domain sample mean ¯ +µξ +11: +compute phase domain sample variance vector ¯kξ +12: +for all beats do +⊲ time domain computations +13: +compute ECG prior mean µsi via (11) +14: +compute ECG variance ksi via (10) +15: +compute measurements variance kxi via (10) +16: +compute the filtered ECG ˆsi via (12) +17: +end for +18: end function + +A DATA-DRIVEN ECG GAUSSIAN PROCESS FILTER, M. DUMITRU ET AL. +3 +C. Computational cost and model selection +The direct implementation of a GP filter (without the hereby +proposed phase-domain model) would be as follows [1], [2]: +�s = µs + KsK−1 +x (x − µs) , +(13) +with the computational complexity O(N 3), dominated by the +inversion of the measurement covariance matrix Kx. In this +approach the model’s hyperparameters are the mean µs, the +covariance matrix Ks) and the noise variance vn (or more +generally the noise covariance matrix) and optimizing them +via classical methods (e.g. maximum evidence, leave-one- +out cross validation, [8, Ch. 5]) adds to the computational +complexity. For long ECGs, the application of this model +is not possible. Previous research considered the GP beat- +wise formulation and adopted a model-based approach to +confine the structure of the covariance matrices [1], [2], but the +choice of the particular model-based mean and kernel function +families remains ad-hoc and difficult to justify. +The proposed model infers the GP mean and covariance +matrix in a data-driven way, based on the sample mean and +covariance matrix from the phase domain (6) and (7), and +in the diagonal case, Algorithm 1, does not require any +inversion. The fundamental assumption allowing the data- +driven computation is the assumption that the phase domain +beats ξi are ensembles from the same underlying GP, (5). +D. Hyperparameter selection +The number of phase domain beat samples T is chosen +greater than the longest beat in the time domain; this allows the +choice of the transformation and back transformation matrices +such that the time-phase-time transition can be done without +(transformation) errors. The noise variance ˆvn can be com- +puted via maximum evidence or practically from the baseline +segment of the ECG beats, where the heart is electrically silent +and only the noise is exhibited in the ECG. +III. RESULTS +A. Baseline wander removal +The BW is removed via two successively zero-phase first +order forward-backward lowpass filters (filtfilt in MAT- +LAB/Python SciPy) with cut-off frequencies set at fc = +5.0 Hz and fc = 80.0 Hz, respectively. While the resulting +passband frequency range is rather narrow and eliminates some +ECG-related components, it enables us to assess the filtering +performance for the dominant ECG frequency band. +B. R-peak detection and heartbeat segmentation +The proposed filter requires the ECG R-peaks. The beats +are defined relative to the R-peaks, segmenting the mea- +surements at the midpoints between successive R-peaks. The +R-peak estimation is done using a modified version of the +Pan–Tompkins algorithm [9]. Specifically, the version used in +this paper estimates the R-peaks by successively applying a +band pass filter, an outlier saturation filter via the hyperbolic +tangent function, a square root moving average filter and a +thresholding. +C. Evaluation +The PhysioNet QT Database (QTDB) [6] is used to evaluate +the developed filter. QTDB consists of 15 minutes 2-lead +ECGs sampled at fs = 250 Hz. The baseline wander was +removed as detailed in Section III-B. The required software +for preprocessing and R-peak detection were adopted from the +Open-Source Electrophysiological Toolbox (OSET) [10]. +The benchmark filter is a wavelet denoiser with a Symlet–5 +mother wavelet, soft thresholding, Stein’s unbiased risk es- +timate (SURE) shrinkage rule, rescaling using a single-level +noise level estimation and four levels of decomposition. In a +previous study, this combination was proved to outperform +other ECG filtering schemes [3]. The filter evaluation is +measured in terms of SNR improvement and QT-interval +estimation error. +D. SNR improvement performance +The ECG records were contaminated by additive white +Gaussian noise at SNR levels ranging from –5 to 30 dB, in +5 dB steps. An example of the noisy and filtered ECG are +shown in Fig. 3. The average and standard deviation of the +SNR improvement is reported for each noise level, for the +proposed and benchmark methods in Fig. 4. Accordingly, the +proposed posterior-based filter improves the SNR for every +level of noise tested and outperforms the prior-based and the +benchmark filter for all tested levels of noise. +E. Clinical parameters preservation +The accuracy of QT-interval estimation is considered to test +the quality of the proposed methods for clinical ECG parame- +ters. For this, the QT-interval estimation error (∆QT) between +the QT-interval estimated from the filtered ECG and the QT- +interval estimated from the noiseless ECG is measured and +compared between the benchmark and the proposed method at +variable input noise levels. The QT-interval estimation method +used is adopted from [11]. Fig. 5 shows the median and the +interquartile range (IQR) of ∆QT for the benchmark wavelet +and the proposed filter, measured over QTDB. Accordingly, +compared with the benchmark method, the GP posterior filter +is reducing the median error for all levels of input noise. +IV. DISCUSSION AND CONCLUSION +In this work we addressed the problem of ECG denoising +via a data-driven based GP model, with beat-wise computa- +tions. Compared with the existing non-parametric ECG filters, +the proposed filter makes no ad hoc assumptions about the GP +model and can be used for ECG records of arbitrary length, +since the computational cost has been significantly reduced +as compared with conventional GP filters. The proposed filter +is efficient in terms of SNR improvement, outperforming the +benchmark performances for all tested noise levels (Fig. 4) and +also clinically, with an improved QT-interval estimation error +compared with the benchmark wavelet denoiser, for all tested +levels of noise (Fig. 5). Another advantage of the proposed +filter is its Bayesian formulation, which allows us to quantify +the filter’s uncertainty (via the estimated variances). It also + +4 +EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US +−0.6 +−0.3 +0.0 +0.3 +0.6 +0.9 +amplitude [mV] +x +GP Prior [ΔSNR=12.0] +−0.6 +−0.3 +0.0 +0.3 +0.6 +0.9 +amplitude [mV] +GP Po terior [ΔSNR=13.3] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +time [ ] +−0.6 +−0.3 +0.0 +0.3 +0.6 +0.9 +amplitude [mV] +Wavelet [ΔSNR=7.1] +(a) input SNR = 0 dB +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +x +GP P io [ΔSNR=7.3] +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +GP Poste io [ΔSNR=10.6] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +time [s] +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +Wavelet [ΔSNR=6.1] +(b) input SNR = 5 dB +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +x +GP Prior [ΔSNR=2.4] +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +GP Po terior [ΔSNR=8.1] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +time [ ] +−0.40 +−0.14 +0.12 +0.38 +0.64 +0.90 +amplitude [mV] +Wavelet [ΔSNR=5.1] +(c) input SNR = 10 dB +Fig. 3. The sel100 recording from the PhysioNet QTDB [6]. From top to bottom the measurements x vs. the prior estimate (11), the posterior estimate +(12), and the wavelet denoiser (Section III-C), at different input SNR levels. The post-filtering SNR improvement is noted in each case. +−5 +0 +5 +10 +15 +20 +25 +30 +Input SNR [dB] +−25 +−20 +−15 +−10 +−5 +0 +5 +10 +15 +20 +SNR improvemen [dB] +Wavele [benchmark] +GP Prior +GP Pos erior +Fig. 4. Mean and standard deviation SNR improvement using the proposed +GP filter and the benchmark wavelet denoiser [3] across all samples of the +PhysioNet QTDB [6], in leads I and II, with 5 repetitions using different noise +instances per record. +-5 +0 +5 +10 +15 +20 +25 +30 +Input SNR [dB] +−100 +−50 +0 +50 +100 +150 +200 +QT es ima ion error [ms] +Wavele +GP Fil er +Fig. 5. +The median and the interquartile range for ∆QT estimations +corresponding to the proposed and benchmark filters across all samples of +the PhysioNet QTDB [6]. +provides a framework that allows for synthetic ECG generation +via data-driven learned parameters, which can be used in +generative models for producing synthetic ECG records for +data greedy machine learning and deep learning applications. +In future studies, the fundamental assumption of the model, +namely the same underlying Gaussian distribution for all the +beats in the phase domain can be relaxed, by clustering the +beats and assuming different underlying distributions for the +beats in each cluster. Also, comparison with expert annotated +QT-interval (and other clinical parameters) is required and +statistical hypothesis testing should be performed to investigate +if the differences are statistically insignificant. The proposed +filter requires the R-peaks for aligning the ECG beats in the +phase-domain, which requires investigating to what extend the +filtering performance is susceptible to mis-detection of the +R-peaks and morphological variations due to ectopic beats. +The Python codes corresponding to the Algorithm 1 and the +reported results are available in [7]. +V. ACKNOWLEDGEMENTS +The authors acknowledge support from the National Insti- +tute of Biomedical Imaging and Bioengineering under the NIH +grant R01EB030362, and the National Center for Advancing +Translational Sciences under the NIH Award UL1TR002378. +REFERENCES +[1] B. Rivet, M. Niknazar, and C. Jutten, “Non parametric modelling of +ECG: Applications to denoising and single sensor fetal ECG extraction,” +in LVA/ICA 2012 - 10th International Conference on Latent Variable +Analysis and Signal Separation, vol. LNCS 7191. +Tel-Aviv, Israel: +Springer, Mar 2012, pp. 470–477. +[2] M. Niknazar, B. Rivet, and C. Jutten, “Fetal ECG extraction from a +single sensor by a non-parametric modeling,” in 2012 Proceedings of +the 20th European Signal Processing Conference, 2012, pp. 949–953. +[3] R. Sameni, “Online filtering using piecewise +smoothness priors: +Application to normal and abnormal electrocardiogram denoising,” +Signal Processing, vol. 133, pp. 52–63, Apr. 2017. [Online]. Available: +https://doi.org/10.1016/j.sigpro.2016.10.019 +[4] P. Laguna, R. Jane, O. Meste, P. Poon, P. Caminal, H. Rix, and +N. Thakor, “Adaptive filter for event-related bioelectric signals using an +impulse correlated reference input: comparison with signal averaging +techniques,” IEEE Transactions on Biomedical Engineering, vol. 39, +no. 10, pp. 1032–1044, 1992. +[5] R. Sameni, M. B. Shamsollahi, C. Jutten, and G. D. Clifford, “A +nonlinear bayesian filtering framework for ECG denoising,” Biomedical +Engineering, IEEE Transactions on, vol. 54, no. 12, pp. 2172–2185, +December 2007. [Online]. Available: https://doi.org/10.1109/TBME. +2007.897817 +[6] P. Laguna, R. Mark, A. Goldberg, and G. Moody, “A database for +evaluation of algorithms for measurement of QT and other waveform +intervals in the ECG,” in Computers in Cardiology 1997. +IEEE, 1997. +[Online]. Available: https://doi.org/10.1109/cic.1997.648140 +[7] M. +Dumitru, +Data +Driven +Gaussian +Process +filter +for +ECG, +2022. [Online]. Available: https://github.com/alphanumericslab/OSET/ +tree/master/UnderDevelopment/DataDrivenGPFilter +[8] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine +Learning, ser. Adaptive computation and machine learning. +The MIT +Press, 2005. +[9] J. Pan and W. J. Tompkins, “A Real-Time QRS Detection Algorithm,” +Biomedical Engineering, IEEE Transactions on, vol. BME-32, no. 3, pp. +230–236, 1985. +[10] R. Sameni, The Open-Source Electrophysiological Toolbox (OSET), +version +3.14, +2018. +[Online]. +Available: +https://github.com/ +alphanumericslab/OSET +[11] Q. Li, M. Dumitru, and E.A. Perez Alday, et al., “QT-Interval Estimation +Improved with Fusion of Multiple Automated Algorithms,” in Interna- +tional Society for Computerized Electrocardiology (ISCE), April 6-10, +2022, Las Vegas, NV, 2022. + diff --git a/2dE0T4oBgHgl3EQfugHS/content/tmp_files/load_file.txt b/2dE0T4oBgHgl3EQfugHS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f97f76580bba48dc754ccd4ee9edb53617b8509c --- /dev/null +++ b/2dE0T4oBgHgl3EQfugHS/content/tmp_files/load_file.txt @@ -0,0 +1,424 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf,len=423 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='02607v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='SP] 6 Jan 2023 EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US 1 A Data-Driven Gaussian Process Filter for Electrocardiogram Denoising Mircea Dumitru, Qiao Li, Erick Andres Perez Alday, Ali Bahrami Rad, Gari D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Clifford, Reza Sameni* Abstract—Objective: Gaussian Processes (GP)-based filters, which have been effectively used for various applications includ- ing electrocardiogram (ECG) filtering can be computationally demanding and the choice of their hyperparameters is typically ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Methods: We develop a data-driven GP filter to address both issues, using the notion of the ECG phase domain — a time- warped representation of the ECG beats onto a fixed number of samples and aligned R-peaks, which is assumed to follow a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Under this assumption, the computation of the sample mean and covariance matrix is simplified, enabling an efficient implementation of the GP filter in a data-driven manner, with no ad hoc hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The proposed filter is evaluated and compared with a state-of-the-art wavelet-based filter, on the PhysioNet QT Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The performance is evaluated by measuring the signal-to-noise ratio (SNR) improvement of the filter at SNR levels ranging from –5 to 30 dB, in 5 dB steps, using additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' For a clinical evaluation, the error between the estimated QT-intervals of the original and filtered signals is measured and compared with the benchmark filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Results: It is shown that the proposed GP filter outperforms the benchmark filter for all the tested noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' It also outperforms the state- of-the-art filter in terms of QT-interval estimation error bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Conclusion: The proposed GP filter is a versatile technique for preprocessing the ECG in clinical and research applications, is applicable to ECG of arbitrary lengths and sampling frequencies, and provides confidence intervals for its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Index Terms—ECG Bayesian filter, Gaussian processes, ECG denoising, ECG wavelet denoising, QT-interval estimation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' INTRODUCTION Electrocardiogram (ECG) denoising is a recurrent problem in traditional and wearable cardiac monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The problem has been addressed by various approaches, including model- based and non-model-based filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' A powerful non-parametric framework for ECG filtering is via Gaussian process (GP) models [1], [2], which considers the ECG beats as GPs with common parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The choice of the beats GP hyperparam- eters, namely the mean and kernel functions is non-evident and ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' For GP models with no beat assumptions, beside the ambiguity in parameter selection, the GP filter implementation involves the inversion of large covariance matrices, which precludes the use of this framework for long ECG records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' In this paper, ECG filtering is addressed via a data-driven non-parametric GP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The novelty of the proposed filter is that it requires no ad hoc GP model hyperparameters and it is computationally efficient, making it suitable for any length ECG records;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' it is based on the assumption that each phase The authors are with the Department of Biomedical Informatics, School of Medicine, Emory University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Clifford is also with the Biomedical Engineering Department, Georgia Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Corresponding author: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Sameni (email: rsameni@dbmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' domain beat — a time-warped (stretched or squeezed) repre- sentation of the ECG beats onto a fixed number of samples and aligned R-peaks — is an ensemble of an underlying GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The mean and the kernel function are set via the phase domain sample mean and covariance matrix, computed via the available ensembles, which are transformed back to the time-domain and used to derive the posterior mean using the Bayesian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' This proposed filter is data-driven, does not presume any parametric model for the underlying GP, and is computation- ally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The filter is evaluated in terms of signal-to-noise ratio (SNR) improvement, using as benchmark a wavelet-based ECG de- noiser that was demonstrated in [3] to outperform adaptive filters [4], Tikhonov regularization and Extended Kalman filters [5], in terms of SNR improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The proposed filter’s clinical performance is evaluated by measuring the QT-interval error between the clean ECG and its corresponding filtered version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' GAUSSIAN PROCESS-BASED ECG FILTERING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The mathematical model The ECG measurement x(t) is assumed to be an additive mixture of a clean ECG s(t), assumed to be a GP contami- nated by additive white noise: x(t) = s(t) + n(t), t ∈ {t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' tN} ∆= TN, (1) where n(t) ∼ N(0, vn), vn denotes the noise variance and N denotes the number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The signal x(t) is assumed to be baseline-wander (BW) and powerline noise removed, which are relatively straightforward, with classical filtering pipelines (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, the filter design objective is focused on in-band ECG noise removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' For the beat i, Ti = � ti1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' tiRi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' tiNi � denotes the set of time samples, ti1 representing the first sample, tiRi the sample corresponding to the R-peak and tiNi the last sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' We further define xi = [x(t)]i∈Ti, si = [s(t)]i∈Ti, ni = [n(t)]i∈Ti as vectorial representations of the measurement, clean ECG and noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, xi = si + ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Next, we define matrices Θi ∈ RT ×Ni to map the time domain beats xi, si and ni to the phase domain beats ξi = Θixi, ςi = Θisi, ηi = Θini, (2) with aligned R-peaks and the same number of samples T (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The Θi matrices are defined by considering T knots 2 EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US 0 25 50 75 100 125 150 175 200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4 −0.' metadata={'source': 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200 time [samples] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 amplitude [mV] ξ[1] ξ[2] ξ[3] ξ[4] ξ[5] ξ[6] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Time-domain measurements beats (top) and the corresponding phase domain ECG beats (bottom), with the same number T of samples for the first 6 beats of sel100 record from QTDB [6] with 0 dB Gaussian additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Transformation matrices Θi are defined via (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 0 5 10 15 20 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0 5 10 15 20 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0 5 10 15 20 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Corner detail example of transformation matrix Θi (left), ΘT i (middle) and the corresponding (diagonal) Gramian Gi = ΘT i Θi (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' equidistantly distributed in the interval [1, Ni] and assigning Θi(j, k) = � 1, if j − 1 ≤ (k − 1) Ni−1 T −1 < j 0, otherwise , (3) with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' , Ni − 1, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' , T and T ≥ maxi {Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' With this choice, the corresponding Gramian matrices Gi, are diagonal matrices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 2), Gi = ΘT i Θi = diag [gi] and diag � ΘiΘT i � = 1T , (4) with gi ∈ RNi and 1T ∈ RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, Gi is invertible and the back transformation from the phase to the time domain is given by Ψi = G−1 i ΘT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' From (1) and (2), the ECG beats satisfy ξi = ςi + ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 1, in the phase domain the beats have been normalized in lengths and the R-peaks are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, the phase-domain sample variations are only due to the stochastic inter-beat variations of the ECG beats and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' As our working model, we assume that the phase domain beats ξi to be ensembles of an underlying GP ξi ∼ N (µξ, Kξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' (5) Moreover, from the time domain noise assumption and (2), the phase domain noise beats also have a zero-mean normal distribution ηi ∼ N � 0, vnΘiΘT i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, the phase do- main ECG beats follow ςi ∼ N � µξ, Kξ − vnΘiΘT i � , where the model parameters µξ and Kξ can be estimated by the sample mean ¯µξ := B−1 �B i=1 ξi and the sample covariance ¯ Kξ := B−1 �B i=1(ξi− ¯µξ)(ξi− ¯µξ)T , where B is the number of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Therefore, the time domain (clean) ECG beats follow a Normal distribution si ∼ N (µsi, Ksi) with parameters µsi = Ψi ¯µξ, Ksi = Ψi � ¯ Kξ − ˆvnΘiΘT i � ΨT i , (6) where ˆvn represents the noise variance estimate and the covariance matrix corresponding to time domains beats xi is given by Kxi = Ψi ¯ KξΨi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' (7) Finally, the filtered beats are defined as the time domain posterior mean, using (6) and (7): ˆsi = µsi + KsiK−1 xi (xi − µsi) , (8) In the sequel, we refer to µsi and ˆsi as prior-based and posterior-based GP filter results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The GP filter with diagonal covariance matrix The direct implementation of the filter in (8) requires the inversion of covariance matrices that typically have huge condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The matrix inversion can be avoided if we consider the diagonal case of ¯ Kξ: ¯kξ = diag � ¯ Kξ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' kηi (4)= ˆvn1T (9) In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' the corresponding time domain matrices are also diagonal and can be computed via kxi = � ΘT i ¯kξ � ⊘ g2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' ksi = � ΘT i �¯kξ − kηi �� ⊘ g2 i (10) with ◦ and ⊘ denoting the Hadamard product and division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' respectively (element-wise product and division),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' g2 i := gi◦gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' the time domain (prior) mean computed via µsi = � ΘT i ¯µξ � ⊘ gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' (11) and the corresponding filter given by ˆsi = µsi + ksi ⊘ kxi ◦ (xi − µsi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' (12) The overall algorithm for GP ECG filtering is summarized in Algorithm 1 and is available online in our Git repository [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Algorithm 1 GP ECG filtering 1: {tiRi}i = RPeakDetector(x) [Section III-B] 2: ˆvn = NoiseVartianceEstimator(x) [Section II-D] Input: x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' {tiRi}i Output: {�si}i 3: function GPDIAG(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' {tiRi}i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' ˆvn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='⊲ GP diagonal filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='for all beats do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='⊲ phase domain computations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute transformation matrices Θi via (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute the vectors gi = diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='ΘT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='i Θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute kηi via (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute the phase beats ξi via (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute phase domain sample mean ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='µξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute phase domain sample variance vector ¯kξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='for all beats do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='⊲ time domain computations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute ECG prior mean µsi via (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute ECG variance ksi via (10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute measurements variance kxi via (10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='compute the filtered ECG ˆsi via (12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='18: end function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='A DATA-DRIVEN ECG GAUSSIAN PROCESS FILTER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' DUMITRU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Computational cost and model selection The direct implementation of a GP filter (without the hereby proposed phase-domain model) would be as follows [1], [2]: �s = µs + KsK−1 x (x − µs) , (13) with the computational complexity O(N 3), dominated by the inversion of the measurement covariance matrix Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' In this approach the model’s hyperparameters are the mean µs, the covariance matrix Ks) and the noise variance vn (or more generally the noise covariance matrix) and optimizing them via classical methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' maximum evidence, leave-one- out cross validation, [8, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 5]) adds to the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' For long ECGs, the application of this model is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Previous research considered the GP beat- wise formulation and adopted a model-based approach to confine the structure of the covariance matrices [1], [2], but the choice of the particular model-based mean and kernel function families remains ad-hoc and difficult to justify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The proposed model infers the GP mean and covariance matrix in a data-driven way, based on the sample mean and covariance matrix from the phase domain (6) and (7), and in the diagonal case, Algorithm 1, does not require any inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The fundamental assumption allowing the data- driven computation is the assumption that the phase domain beats ξi are ensembles from the same underlying GP, (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Hyperparameter selection The number of phase domain beat samples T is chosen greater than the longest beat in the time domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' this allows the choice of the transformation and back transformation matrices such that the time-phase-time transition can be done without (transformation) errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The noise variance ˆvn can be com- puted via maximum evidence or practically from the baseline segment of the ECG beats, where the heart is electrically silent and only the noise is exhibited in the ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Baseline wander removal The BW is removed via two successively zero-phase first order forward-backward lowpass filters (filtfilt in MAT- LAB/Python SciPy) with cut-off frequencies set at fc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 Hz and fc = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' While the resulting passband frequency range is rather narrow and eliminates some ECG-related components, it enables us to assess the filtering performance for the dominant ECG frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' R-peak detection and heartbeat segmentation The proposed filter requires the ECG R-peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The beats are defined relative to the R-peaks, segmenting the mea- surements at the midpoints between successive R-peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The R-peak estimation is done using a modified version of the Pan–Tompkins algorithm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Specifically, the version used in this paper estimates the R-peaks by successively applying a band pass filter, an outlier saturation filter via the hyperbolic tangent function, a square root moving average filter and a thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Evaluation The PhysioNet QT Database (QTDB) [6] is used to evaluate the developed filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' QTDB consists of 15 minutes 2-lead ECGs sampled at fs = 250 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The baseline wander was removed as detailed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The required software for preprocessing and R-peak detection were adopted from the Open-Source Electrophysiological Toolbox (OSET) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The benchmark filter is a wavelet denoiser with a Symlet–5 mother wavelet, soft thresholding, Stein’s unbiased risk es- timate (SURE) shrinkage rule, rescaling using a single-level noise level estimation and four levels of decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' In a previous study, this combination was proved to outperform other ECG filtering schemes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The filter evaluation is measured in terms of SNR improvement and QT-interval estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' SNR improvement performance The ECG records were contaminated by additive white Gaussian noise at SNR levels ranging from –5 to 30 dB, in 5 dB steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' An example of the noisy and filtered ECG are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The average and standard deviation of the SNR improvement is reported for each noise level, for the proposed and benchmark methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Accordingly, the proposed posterior-based filter improves the SNR for every level of noise tested and outperforms the prior-based and the benchmark filter for all tested levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Clinical parameters preservation The accuracy of QT-interval estimation is considered to test the quality of the proposed methods for clinical ECG parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' For this, the QT-interval estimation error (∆QT) between the QT-interval estimated from the filtered ECG and the QT- interval estimated from the noiseless ECG is measured and compared between the benchmark and the proposed method at variable input noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The QT-interval estimation method used is adopted from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 5 shows the median and the interquartile range (IQR) of ∆QT for the benchmark wavelet and the proposed filter, measured over QTDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Accordingly, compared with the benchmark method, the GP posterior filter is reducing the median error for all levels of input noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION In this work we addressed the problem of ECG denoising via a data-driven based GP model, with beat-wise computa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Compared with the existing non-parametric ECG filters, the proposed filter makes no ad hoc assumptions about the GP model and can be used for ECG records of arbitrary length, since the computational cost has been significantly reduced as compared with conventional GP filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The proposed filter is efficient in terms of SNR improvement, outperforming the benchmark performances for all tested noise levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 4) and also clinically, with an improved QT-interval estimation error compared with the benchmark wavelet denoiser, for all tested levels of noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Another advantage of the proposed filter is its Bayesian formulation, which allows us to quantify the filter’s uncertainty (via the estimated variances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' It also 4 EXTENDED VERSION OF A POSTER PRESENTED AT THE 46TH ISCE CONFERENCE, APR 6–10, 2022, LAS VEGAS, US −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' From top to bottom the measurements x vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' the prior estimate (11), the posterior estimate (12), and the wavelet denoiser (Section III-C), at different input SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The post-filtering SNR improvement is noted in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' −5 0 5 10 15 20 25 30 Input SNR [dB] −25 −20 −15 −10 −5 0 5 10 15 20 SNR improvemen [dB] Wavele [benchmark] GP Prior GP Pos erior Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Mean and standard deviation SNR improvement using the proposed GP filter and the benchmark wavelet denoiser [3] across all samples of the PhysioNet QTDB [6], in leads I and II, with 5 repetitions using different noise instances per record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 5 0 5 10 15 20 25 30 Input SNR [dB] −100 −50 0 50 100 150 200 QT es ima ion error [ms] Wavele GP Fil er Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The median and the interquartile range for ∆QT estimations corresponding to the proposed and benchmark filters across all samples of the PhysioNet QTDB [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' provides a framework that allows for synthetic ECG generation via data-driven learned parameters, which can be used in generative models for producing synthetic ECG records for data greedy machine learning and deep learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' In future studies, the fundamental assumption of the model, namely the same underlying Gaussian distribution for all the beats in the phase domain can be relaxed, by clustering the beats and assuming different underlying distributions for the beats in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Also, comparison with expert annotated QT-interval (and other clinical parameters) is required and statistical hypothesis testing should be performed to investigate if the differences are statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The proposed filter requires the R-peaks for aligning the ECG beats in the phase-domain, which requires investigating to what extend the filtering performance is susceptible to mis-detection of the R-peaks and morphological variations due to ectopic beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' The Python codes corresponding to the Algorithm 1 and the reported results are available in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge support from the National Insti- tute of Biomedical Imaging and Bioengineering under the NIH grant R01EB030362, and the National Center for Advancing Translational Sciences under the NIH Award UL1TR002378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Rivet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Niknazar, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE0T4oBgHgl3EQfugHS/content/2301.02607v1.pdf'} +page_content=' Jutten, “Non parametric modelling of ECG: Applications to 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Lloyd1, and Naratip Santitissadeekorn1 +1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK +January 16, 2023 +Abstract +Data assimilation techniques, such as ensemble Kalman filtering, have been shown to be a +highly effective and efficient way to combine noisy data with a mathematical model to track +and forecast dynamical systems. However, when dealing with high-dimensional data, in many +situations one does not have a model, so data assimilation techniques cannot be applied. In +this paper, we use dynamic mode decomposition to generate a low-dimensional, linear model +of a dynamical system directly from high-dimensional data, which is defined by temporal and +spatial modes, that we can then use with data assimilation techniques such as the ensemble +Kalman filter. +We show how the dynamic mode decomposition can be combined with the +ensemble Kalman filter (which we call the DMDEnKF) to iteratively update the current state +and temporal modes as new data becomes available. We demonstrate that this approach is able +to track time varying dynamical systems in synthetic examples, and experiment with the use +of time-delay embeddings. We then apply the DMDEnKF to real world seasonal influenza-like +illness data from the USA Centers for Disease Control and Prevention, and find that for short +term forecasting, the DMDEnKF is comparable to the best mechanistic models in the ILINet +competition. +Keywords +Dynamic mode decomposition; Ensemble Kalman filter; Data-driven modelling; Data assimilation; +Dynamical systems +1 +Introduction +Data assimilation refers to the collection of methods that integrate vast data sets with sophisticated +mathematical models, to track and forecast systems that may evolve or change [38]. The majority +of its applications lie in the earth sciences [51], however due to the generality of its techniques they +have also been successfully applied in a wide range of areas from medicine [18] to ecology [40]. +The Kalman filter [36] is one such data assimilation technique widely used throughout industry [3] +that optimally combines predictions from a linear model with Gaussian data. Whilst traditionally +applied to a model’s state, the parameters of the model can simultaneously be filtered, leading to +1 +arXiv:2301.05504v1 [math.DS] 13 Jan 2023 + +what is known as the joint state-parameter estimation problem [33]. If the system being filtered is +nonlinear, alternative versions of the Kalman filter can be utilized such as the extended Kalman filter +[53], unscented Kalman filter [59] or ensemble Kalman filter (EnKF) [23]. The EnKF represents +the distribution of a system’s state with an ensemble of random samples, that can then be used to +estimate useful statistics like the state’s covariance via the sample covariance or a point estimate +of the state via the sample mean [23] and is well-suited for high-dimensional problems. All of these +methods require a model of the system, however if no model exists then one must be generated and +the most generalizable way to do this is via data-driven modelling. +Dynamic mode decomposition (DMD) is a data-driven modelling technique for identifying low di- +mensional, spatial and temporal patterns within a dynamical system directly from high-dimensional +data [54]. It does this by postulating the state vector is evolved via a linear system and looking for +a low-dimensional approximation of the eigenvalues (temporal modes) and corresponding eigenvec- +tors (spatial modes). Spatial modes can be thought of as modes that decompose state variables +into separate components that evolve together linearly in time. The corresponding temporal modes +describe whether a spatial mode is growing, decaying or stationary in time. DMD has been used +to approximate dynamical systems from measurement data in a multitude of fields, ranging from +epidemiology [49], finance [43] and neuroscience [12]. Due to its popularity, it has been extended +to systems that are nonlinear in their recorded measurement functions via Extended/Kernel DMD +[61]/[44], with one such extension Hankel-DMD [2] employing time-delay embeddings of the original +observables. In the presence of measurement noise, the standard DMD has been shown to induce a +systematic bias by asymmetrically attributing all noise to the model’s target output measurements +and none to its inputs during training [15]. This systematic bias, prompted the creation of noise +handling variants of DMD that directly account for the noise term [15], the Forward Backward +DMD [15] that performs DMD forwards and backward in time and combines the results, and To- +tal DMD (TDMD) [31] that minimizes the total least squares error as opposed to minimizing the +ordinary least squares error. +The aim of this paper is to develop an algorithm that iteratively improves the temporal modes +(eigenvalues) and state estimates produced by DMD with the EnKF as new data becomes available. +This would be highly useful for dynamical systems that make a change from growing or decaying +behaviour over time. While estimating just the state of the system using the DMD modes can be +done using a standard Kalman filter, without also filtering the model’s temporal mode, estimates +are likely to suffer if the system changes over time. Methods already exist that combine DMD with +the Kalman filter [45] or extended Kalman filter [46], which apply filtering to estimate the entire +system dynamics matrix. The filtering in our work is instead focused on efficiently tracking the +system’s temporal modes, and forecasting the system’s future states. DMD produces a linear model +which makes it a natural fit for the Kalman filter, however when a system’s state and temporal +modes are estimated simultaneously the filtering process becomes nonlinear. Hence, we need to use +a filter designed for a nonlinear model, and we chose the EnKF due to its versatility, scalability to +large dynamical systems, and ease of implementation [52]. While any DMD variant that produces +temporal modes would be compatible with the DMDEnKF framework, we use TDMD to remain +consistent with the EnKF’s assumption that noise is present in the data. In tandem, we apply +the DMDEnKF using a total least squares version of Hankel-DMD, henceforth referred to as the +Hankel-DMDEnKF, to investigate the effect time-delay embeddings have on our framework. +2 + +To demonstrate the DMDEnKF method, we first test it on synthetically generated datasets. Ini- +tially, on a simple noisy oscillating system with a decreasing period of oscillation, we use the DM- +DEnKF to track the system’s temporal modes and compare results with the Hankel-DMDEnKF, +other iterative DMD variants, and “gold standard” filtering methods. Next, we simulate a pan- +demic and evaluate the DMDEnKF’s ability to track the system’s temporal modes and generate +multistep ahead forecasts. +Figure 1: ILI consultations as a percentage of total weekly GP consultations in the US from 2003 to end of +2018. The data shows the annual peaks in ILI consultations that vary in size, timing and shape, which +would make them difficult to model with a simple SIR-type model. +Finally, we apply the DMDEnKF and Hankel-DMDEnKF to real seasonal influenza-like illness +(ILI) data in the United States from the Centers for Disease Control and Prevention (CDC) ILINet +[25] shown in Figure 1, with the aim of investigating their forecasting skills for ILI consultation +rates. ILI is defined as a fever with a cough or sore throat that has no known cause other than +influenza [25] and infects between 9 and 35 million people in the US each year [24]. Due to its +prevalence, a multitude of methods have already been developed to model the spread of ILI [14, 47] +and the approaches these models take can broadly be classified as either mechanistic or statistical +[37]. Mechanistic methods [5, 48] make explicit hypotheses about what is driving the spread of an +infectious disease, before then fitting parameters in the proposed models to the data. They have +the advantage of being highly interpretable making them useful when trying to understand how +one can control the spread of a disease [39], however can make assumptions that are oversimplified +[4]. For example, a simple SIR-type model would struggle to describe specific behaviours like the +drop in ILI consultations around Christmastime seen in Figure 1. +Statistical methods [11, 60] +are generally more versatile as they require fewer domain-specific assumptions, but both methods +achieve a similar predictive skill in real time on the ILINet dataset [50]. The DMDEnKF attempts +to find a middle ground between the two methods, remaining versatile by virtue of being purely +data-driven but also providing some level of interpretability via the associated DMD modes. +The remainder of this paper will be structured as follows. First, a brief summary of DMD, Hankel- +3 + +8 +7 +6 +4 +三 +%3 +2 +1 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +DateDMD and EnKF algorithms for completeness. +After which, the DMDEnKF algorithm will be +described in full. We will then apply the DMDEnKF and Hankel-DMDEnKF to synthetic data +and compare their performance against other pre-existing, iterative DMD variants. Finally, we will +use the DMDEnKF and Hankel-DMDEnKF on ILINet data to forecast the rate of ILI consultations +up to 4 weeks into the future and examine their performance. +2 +DMDEnKF +2.1 +Dynamic Mode Decomposition (DMD) +Consider an n dimensional state xk ∈ Rn measured at regular time intervals k = 1, ..., m. Assuming +this time-series data was generated by a linear dynamical system, the consecutive states xk and +xk+1 are connected via +xk+1 = Axk +(2.1) +for some unknown matrix A ∈ Rn×n. By denoting +X = +� +�� +| +| +| +x1 +x2 +... +xm−1 +| +| +| +� +�� , +X′ = +� +�� +| +| +| +x2 +x3 +... +xm +| +| +| +� +�� , +(2.2) +equation (2.1) can be written succinctly over all consecutive data pairs as +X′ = AX. +(2.3) +To minimize the mean squared error term �m−1 +k=1 ∥xk+1 − Axk∥2 +2, the standard DMD defines +A = X′X+, +(2.4) +where X+ is the Moore-Penrose pseudoinverse [6] of X. Efficiently solving for the eigendecom- +position of A is the primary purpose of DMD, as these eigenvalues/eigenvectors correspond to +spatio-temporal patterns in the data. +The DMD method starts by applying the Singular Value Decomposition (SVD) to the data matrix +X, representing it as the matrix multiplication of 2 real-valued, orthonormal matrices (complex and +unitary if X ∈ Cn×m) U ∈ Rn×n, V ∈ Rm×m and a rectangular diagonal matrix with decreasing +non-negative real values (Σ ∈ Rn×m) in the form +X = UΣV∗. +(2.5) +The best rank r approximation of a matrix according to the Eckart-Young Theorem [22] is obtained +by truncating its SVD, hence by truncating equation (2.5) to a suitable rank r [26] we can compress +the data matrix with minimal loss of information, which we write as +X ≈ UrΣrV∗ +r. +(2.6) +By performing this compression, we are implicitly assuming that there exists a low dimensional +(≤ r), linear structure within the high-dimensional data. +4 + +The Moore-Penrose pseudoinverse can be found directly from the SVD computed in equation (2.5) +as VΣ−1U∗. We use the rank r truncated matrices from equation (2.6) for reasons of efficiency, +setting +A = X′X+, +≈ X′VrΣ−1 +r U∗ +r. +(2.7) +This approximation of A now acts only on an r dimensional subspace defined by Col(Ur). Hence, +we can restrict A onto this r dimensional subspace (representing the largest r POD modes of X) +and denote the restricted A ∈ Rr×r as +˜A = U∗ +rAUr, +≈ U∗ +rX′VrΣ−1 +r . +(2.8) +We calculate the eigenvalues (λi), and corresponding eigenvectors (vi) of ˜A, and define +Λ = +� +��� +λ1 +0 +0 +0 +... +0 +0 +0 +λr +� +��� , +W = +� +�� +| +| +| +v1 +v2 +... +vr +| +| +| +� +�� . +(2.9) +Reconstructing the eigenvalues/eigenvectors of the original operator A will provide insights into +the structure of the system [1] and allow us to propagate it forward in time. The eigenvalues of ˜A +(Λ) can be shown to be equal to the eigenvalues of the original operator A [34], however recovering +the original eigenvectors is more involved and can be done using either projected or exact DMD. +We use the exact DMD method introduced by Tu et al. [34] as it finds the exact DMD modes (Φ) +for all eigenvectors with non-zero λi, where Φ is defined as +Φ = X′VrΣ−1 +r W. +(2.10) +DMD modes with zero eigenvalues have no effect on the system’s dynamics, so this restriction of +exact DMD is of little consequence. This method finds A such that AX = X′ exactly provided +r ≥ rank(X) and X and X′ are linearly consistent [34]. +With Λ and Φ in hand, we can construct a r dimensional approximation of A, however still need +to find the initial phase and amplitude of each mode. The standard method [54] for computing this +vector (b) is to rewrite the initial state x1 in a basis of the DMD modes via +b = Φ+x1. +(2.11) +It is worth noting that there exist alternative methods for example [13, 35] that focus on optimizing +b over all data points with additional conditions. +To summarise, the final solution to the discrete system can be written as +xk = ΦΛkb. +(2.12) +In the remainder of the paper, we call Λ the temporal modes and Φ the spatial modes. +5 + +2.1.1 +Hankel-DMD +Hankel-DMD first augments the original, measured state xk ∈ Rn, by appending to it measurements +of the state at the previous d − 1 time steps +h(xk) = +� +xkT +xk−1T +. . . +xk−(d−1)T �T +, +(2.13) +to form a new state h(xk) ∈ Rdn. This is known as a time-delay embedding, and we refer to d +as the delay-embedding dimension. Taking time-delay embeddings, h(xk), to be our new states, +matrices X and X′ from equation (2.2) now become +X = +� +��� +xd +. . . +xm−1 +... +... +... +x1 +. . . +xm−d +� +��� , +X′ = +� +��� +xd+1 +. . . +xm +... +... +... +x2 +. . . +xm−(d−1) +� +��� . +(2.14) +With X and X′ defined above, Hankel-DMD proceeds exactly as the standard DMD algorithm, +generating eigenvalues Λ, DMD modes Φ and their initial states b as described above. The original +system can be reconstructed/forecast for all time steps from d onwards, by applying equation (2.12) +and restricting the result to the first n rows. +2.1.2 +Iterative DMD Variants +There exists other variants of DMD that are designed to be applied iteratively, and in this paper we +will compare these with the DMDEnKF in their ability to track a system’s eigenvalues and make +future state predictions. Streaming DMD [32] is an adaption of the standard DMD algorithm to +efficiently process new data as it becomes available, and the noise aware variant Streaming TDMD +[30] is the first variant we wish to compare against. The second method we will use for comparison +is Windowed DMD [62], where the standard DMD described above is applied over a sliding window +of the w most recent data snapshots only. The final method we will be comparing against is Online +DMD [62], specifically the variant of this algorithm that places an exponentially decaying weight ρ +on the importance of past measurements. +2.2 +Ensemble Kalman Filter (EnKF) +Consider a discrete-time, nonlinear dynamical system with a stochastic perturbation +xk = F(xk−1) + wk, +wk ∼ N(0, Qk), +(2.15) +where F is a nonlinear function F : Rn → Rn, xk ∈ Rn is the system’s state, wk ∈ Rn is a +stochastic perturbation and N is the normal distribution with mean 0 and covariance matrix Qk. +A measurement equation that relates what we observe to the true state of the system is given by +yk = H(xk) + vk, +vk ∼ N(0, Rk), +(2.16) +where H : Rn → Rl is the system’s observation operator, yk ∈ Rl is an observation of the system, +vk ∈ Rl is the noise in the observation and N is the normal distribution with mean 0 and covariance +matrix Rk. We focus on the instance relevant to our use case where H is linear, so can be represented +by a matrix H ∈ Rl×n. +6 + +In general, filtering methods aim to combine information from the state-transition model (2.15) and +observation model (2.16) to compute the conditional density p(xk|Yk), where Yk = (y1, ..., yk). +The Kalman filter is the optimal filter if F and H are both linear and the stochastic perturbations +are normal [36]. The EnKF was developed to deal with the filtering problem where either the linear +or normal assumption (or both) is violated [23]. It exploits the Kalman formulation to propagate +an ensemble of the state into a region of high probability in such a way that the ensemble spread +would be consistent with the linear and normal model. +To begin the EnKF algorithm, an initial ensemble of N state estimates ˆx(1) +0 ,..., ˆx(N) +0 +is required. If +an ensemble is not available, one can be generated from initial state estimates ˆx0 and covariance +matrix P0 by taking N independent draws from N(ˆx0, P0). +Algorithm +The EnKF then acts as follows [23]: +Step 1: Propagate forward in time each ensemble member using equation (2.15) for i = 1, ..., N +via +ˆx(i) +k|k−1 = F(ˆx(i) +k−1|k−1) + w(i) +k . +(2.17) +The notation ˆx(i) +k|k−1 denotes the state estimate at time k of the ith ensemble member ˆx(i) +k +using +only information up to time k − 1, and ˆx(i) +k−1|k−1 represents the same ensemble member at time +k − 1 using information up to time k − 1. Each w(i) +k +is independently drawn from N(0, Qk). The +current covariance matrix can also now be estimated via the sample covariance of the ensemble, +which we denote as ˆPk|k−1. This can then be used to estimate the Kalman Gain matrix ˆKk as +ˆKk = ˆPk|k−1HT (HˆPk|k−1HT + Rk)−1. +(2.18) +Step 2: Calculate the measurement innovation utilizing equation (2.16). +From measurement yk, we again use i = 1, ..., N and generate simulated measurements +y(i) +k += yk + v(i) +k +(2.19) +where each v(i) +k +is an independent draw from N(0, Rk). These simulated measurements y(i) +k +are +combined with the ensemble members ˆx(i) +k|k−1 from equation (2.17) to define N measurement inno- +vations +e(i) +k = y(i) +k − Hˆx(i) +k|k−1. +(2.20) +The e(i) +k +represent samples from the distribution of the distance of the model’s prediction from the +measured value. +Step 3: Combine the model estimates in equation (2.17) and measurement innovation of equation +(2.20) via the estimated Kalman gain from (2.18) to update each ensemble member’s state estimate +ˆx(i) +k|k = ˆx(i) +k|k−1 + ˆKke(i) +k . +(2.21) +We can generate a point estimate for the state ˆxk using the mean of the N updated ensemble +members. This process then repeats every time a new state measurement becomes available, with +the updated ensemble from the previous data point becoming the initial ensemble for the new one. +We combine these 2 previously described techniques to form the DMDEnKF. This new, hybrid +method uses DMD to generate a low dimensional model of a dynamical system that is then itera- +tively improved by the EnKF as new data emerges. +7 + +2.3 +DMDEnKF +We now describe how we carry out filtering of the temporal modes and state of the system, while +keeping the spatial modes found by one’s chosen version of DMD on the “spin-up” fixed. We note +that once we allow the temporal modes to vary with the spatial modes being fixed, these are no +longer eigenvalues/eigenvectors, and we then call them temporal modes. Consider an n dimensional +state xk ∈ Rn measured at regular time intervals k = 1, ..., m and then measured iteratively at times +k = m + 1, .... +Algorithm +Step 1: Perform the chosen version of DMD on the dataset x1, ..., xm, defining X, X′ as before in +equation (2.2) to obtain the expression +xk = ΦΛkb, +(2.22) +where +Λ = +� +��� +λ1 +0 +0 +0 +... +0 +0 +0 +λr +� +��� , +Φ = +� +�� +| +| +| +d1 +d2 +... +dr +| +| +| +� +�� , +b = +� +��� +b1 +... +br +� +��� , +(2.23) +and defining λi, di, bi as the ith temporal mode, DMD mode, initial condition triplet of the r +retained modes. This acts as a spin-up process to generate a model we can then filter using the +EnKF. +Step 2: Define the matrices required for the EnKF’s ensemble initialisation, propagation via +equation (2.15), and measurement using equation (2.16). +First, rewrite each of the r temporal modes in polar coordinates as +λi = τieθii, +(2.24) +where τi ≥ 0, 0 ≤ θi < 2π and i2 = −1. As xk ∈ Rn, the temporal modes in the DMD model’s +spectrum will either be real or in a complex conjugate pair. When filtering, we view the temporal +modes as a time varying parameter. +However, we must enforce that the real temporal modes +remain real and complex conjugate pairs remain intact, as this ensures the state output by the +model will still be real. We do this by defining the filterable model parameters µi as new variables +for i = 1, ..., r +µi = +� +τi, +if θi = 0, or ∄ j for j < i such that λ∗ +j = λi, +θi, +otherwise. +(2.25) +Written in this way, these µi’s represent all the possible degrees of freedom in the model’s temporal +modes under the additional constraint of producing a real state estimate. By maintaining a note +of the positional indexes of each complex conjugate pair produced in the initial DMD, it is possible +to recover the λi representation from the µi’s. While this transformation technically requires the +full list of µi’s, we informally write Λ(µi) = λi to symbolize the reversion from µi’s back to λi’s. +Ensemble initialisation: We can now define an augmented joint parameter state z0 ∈ Rn+r to +be used as the initial state for the EnKF +z0 = +� +− +xm +− +µ1 +. . . +µr +�T +. +(2.26) +8 + +We denote the joint parameter state at time m + k as zk ∈ Rn+r. To generate an initial ensemble +from this state, we first define sample covariance C = (1/m)(X′ − ΦΛΦ+X)(X′ − ΦΛΦ+X)T , +which represents the current state uncertainty based on prediction errors in the spin-up DMD. We +then form the initial covariance matrix +P0 = +� +C +0 +0 +α2Ir +� +, +(2.27) +where α2 > 0, Ir is the r-dimensional identity matrix, and the α2Ir term determines the initial +uncertainty in the spin-up DMD’s temporal modes. Take independent draws from N(z0, P0) until +the ensemble is sufficiently large. The optimal ensemble size will vary from problem to problem, +adding ensemble members will increase accuracy but at the cost of computational efficiency. +Propagation: Using the notation zi +k to signify the ith element of zk, we define the matrix Λzk ∈ +Rr×r for state zk as +Λzk = +� +��� +Λ(zn+1 +k +) +0 +0 +0 +... +0 +0 +0 +Λ(zn+r +k +) +� +��� . +(2.28) +The EnKF’s propagation equation can be written as +zk+1 = +� +ΦΛzkΦ+ +0 +0 +Ir +� +zk + wk. +(2.29) +For convenience, we introduce notation zi:j +k ∈ Rj−i+1 to denote the ith through to the jth element +of zk where i ≤ j +zi:j +k = +� +zi +k +. . . +zj +k +�T +. +(2.30) +Equation (2.29) propagates z1:n +k +representing the state in the DMD framework xm+k forward in +time using the standard DMD equation with the updated temporal modes from Λzk. The vector +zn+1:n+r +k +represents the current estimate of the temporal modes in their µi representation and is +unchanged other than the addition of noise by the propagation equation, for although we assume +the temporal modes vary in time no direction of drift in the parameters is explicitly foreknown. +The vector wk ∈ Rn+r is a normally distributed variable wk ∼ N(0, Qk), and this represents the +uncertainty within the model of the system. We construct Qk as follows, +Qk = +� +α1In +0 +0 +α2Ir +� +, +(2.31) +where α1 and α2 are constants determined by the user such that α2 ≪ α1. This construction +with Qk a diagonal matrix assumes model errors for each element of zk are uncorrelated with +one another. +The condition α2 ≪ α1 ensures that the state of the DMD system z1:n +k +changes +significantly faster than its temporal modes zn+1:n+r +k +, as parameters by definition should vary +slowly in time. Furthermore, for the temporal mode’s moduli being filtered, it prevents the strictly +positive modulus dropping below 0. +Measurement: We write the EnKF’s measurement equation as +yk = +� +In +0 +0 +0 +� +zk + vk, +(2.32) +9 + +where yk ∈ Rn are observations of the DMD state xm+k, and vk ∈ Rn is a normally distributed +variable ∼ N(0, Rk) representing the noise in the measurements. We assume new measurements +yk to be available for the full DMD state z1:n +k +but not its temporal modes zn+1:n+r +k +, as this is +consistent with the format of the data used to generate the spin-up DMD model. We also assume +uncorrelated measurement noise on each dimension of the state, so choose a diagonal matrix Rk. +Step 3 State measurements xm+k at times k = 1, ... are being iteratively generated. By setting +yk = xm+k as each new measurement arrives, we can iteratively apply the EnKF to produce a +hybrid estimate for zk that combines model predictions from zk−1 and noisy measurement yk. +A brief summary of how the EnKF does this is provided in Section 2.2, and a more expansive +description can be found at [42]. +Step 4: The state of the original system xm+k can be reconstructed from zk by simply taking it’s +first n elements z1:n +k . Predictions p steps ahead at time m + k can also be forecast from zk via +xm+k+p = ΦΛp +zkΦ+z1:n +k . +(2.33) +The Hankel-DMDEnKF is defined algorithmically in exactly the same way, with the only difference +being that Hankel-DMD is applied over the “spin-up” period as opposed to standard DMD. +3 +Synthetic Applications +3.1 +Comparison against other iterative DMD variants +To test the DMDEnKF, we first apply it to data generated from a synthetic system with time +varying eigenvalues, which we aim to track. The dynamics of this system are governed by the 2 +dimensional rotation matrix, where the angle of rotation θk increases linearly from π/64 to π/8 +over the course of 500 time steps. The evolution of the state xk of the system can hence be written +as +xk+1 = +� +cos(θk) +− sin(θk) +sin(θk) +cos(θk) +� +xk, +x1 = +� +1 +0 +� +, +(3.1) +where θk = π/64 + (k−1)(7π/64) +499 +and k = (1, ..., 500). +We assume noisy measurement values yk to be available for the state at each time step, such that +yk = xk + vk, +vk ∼ N(0, σ2I2), +(3.2) +where each experiment σ = 0.05 or 0.5 to simulate a low or high level of measurement noise +respectively. +The 500 values of yk (shown in Figure 2) are used to train the DMDEnKF and +Hankel-DMDEnKF, with the first 100 time steps being used for the spin-up process described in +Step 1 of the DMDEnKF algorithm to produce the output described in equation (2.23). +We will also train the iterative variants of DMD described at the end of Section 2.1 (Streaming +TDMD1, Windowed DMD and Online DMD) on this dataset to compare their ability to track the +1As the synthetic dataset is small, it is computationally tractable to apply batch methods to the data. Hence, +instead of applying the true Streaming TDMD algorithm, we use batch TDMD over all data up to the current +time step as a proxy for Streaming TDMD utilizing code from the PyDMD library [19]. +As Streaming TDMD +approximates the results of TDMD with the only differences occurring due to additional data compression steps in +Streaming TDMD’s algorithm, we believe this to be an acceptable substitution. +10 + +Figure 2: Time series for a synthetic system with a linearly increasing eigenvalue argument, showing the +state’s first dimension with no, low (σ = 0.05) and high (σ = 0.5) measurement noise. +system’s time varying eigenvalues against that of the DMDEnKF. Within the Windowed DMD +algorithm, we replace DMD with TDMD to allow for this method to effectively handle the noise +in the data, henceforth referring to this amalgamation of the two methods as Windowed TDMD. +To implement Online DMD, we use code made available by its creators here [29]. Computational +parameters were set as follows; window size w = 10 for Windowed TDMD, exponential decay +rate ρ = 0.9 for Online DMD, delay-embedding dimension d = 50 for the Hankel-DMDEnKF and +spin-up time steps m = 100 for the DMDEnKF as previously stated. +At each time step k, the system’s true eigenvalues can be written in modulus-argument form as +λk = 1e±θki, +(3.3) +and for each time step where the models are defined their estimates of the system’s eigenvalues can +also be written as +ˆλk = ˆτke±ˆθki. +(3.4) +We start by comparing the errors in each method’s estimate of the constant eigenvalue modulus +(ˆτk−1). A thousand runs of the synthetic data were generated for each value of σ, and the difference +of each method’s eigenvalue modulus and argument from their true values at every time step after +the spin-up period were collected. When any of the methods failed to identify the eigenvalues as a +complex conjugate pair at a given time step in a run, the dominant eigenvalue’s modulus was used +for modulus error calculations. The average errors in the eigenvalue modulus estimates are shown +in Table 1. +For all levels of measurement noise, Streaming TDMD estimated the eigenvalue modulus the most +accurately. This is due to the method’s assumption of a stationary system, hence assigning an +equal weight to the importance of each data point, which works well in the case of estimating a +constant parameter. At low levels of measurement noise as seen in the first column of Table 1, +Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly +11 + +2 +dimension +0 +1 +XkTrue State +yk for = 0.05 +-2 +yk for o = 0.5 +0 +100 +200 +300 +400 +500 +TimestepsIterative DMD +Mean Eigenvalue Modulus Error +Variant +σ = 0.05 +σ = 0.5 +Windowed TDMD +9.82 × 10−3 +1.39 +Online DMD +6.04 × 10−3 +3.06 × 10−1 +Streaming TDMD +2.31 × 10−4 +2.50 × 10−3 +DMDEnKF +8.07 × 10−3 +1.89 × 10−2 +Hankel-DMDEnKF +9.49 × 10−3 +1.38 × 10−2 +Table 1: Mean absolute errors in the synthetic system’s eigenvalue modulus estimates produced by each +iterative DMD variant over all time steps over the course of all 1000 runs. Measurement noise is set to +either low levels with σ = 0.05 (left) or high levels with σ = 0.5 (right). Streaming TDMD scored +significantly lower errors than all other methods, and as noise levels increased, errors in Windowed TDMD +and Online DMD grew significantly larger than those produced by the DMDEnKF and Hankel-DMDEnKF. +well with mean eigenvalue modulus errors below 0.01. As errors in the eigenvalue modulus grow +exponentially when forecasting future states, these 4 methods could produce acceptable short term +forecasts but would quickly diverge from the true state as the forecast horizon was extended. At +high levels of noise shown in the second column of Table 1, Windowed TDMD and Online DMD’s +eigenvalue modulus estimates degrade significantly, making them unsuitable for forecasting in this +scenario. The errors in the DMDEnKF and Hankel-DMDEnKF remain fairly small, however are +still an order of magnitude greater than those produced by Streaming TDMD. +A typical trajectory of the eigenvalue argument estimates (ˆθk) for each method over the course +of one run from the end of the spin-up period onwards can be seen in Figures 3a and 3c. The +error distributions for each method’s eigenvalue argument estimates (ˆθk − θk) over all 1000 runs +are plotted in Figures 3b and 3d. +At low levels of noise as seen in Figures 3a and 3b, all 5 methods on average underestimated the +eigenvalue argument of the system. This is to be expected as the eigenvalue argument is increasing +with time, meaning that all but the last data pair available to each method would have been +generated using an argument smaller than its current value. Streaming TDMD exhibited the worst +performance, again due to its equal weighting of every data point, however in this instance being a +negative quality as it hampers the model’s ability to adapt to fresh data that reflects the changing +parameter. Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed +similarly. Online DMD produced a tighter error distribution, but with a slightly larger bias than +Windowed TDMD. This suggests that Online DMD’s soft thresholding reduces the model volatility +caused by measurement noise compared to the hard cut-off employed by Windowed TDMD. For +this same reason however, Online DMD is slower to adapt to new measurements than Windowed +TDMD, leading to a larger bias below the system’s true eigenvalue argument. The DMDEnKF +and Hankel-DMDEnKF performed very similar to Windowed TDMD at this noise level, however +tweaks to the magnitude of the DMDEnKF’s system uncertainty matrix can be made to balance +the speed of model innovation with its volatility and produce distributions closer to that of Online +DMD if required. +At higher noise levels shown in Figures 3c and 3d, the performance of Windowed TDMD and Online +DMD significantly degrades. Placing a larger weight on more recent samples allowed these methods +12 + +(a) Eigenvalue argument trajectory for σ = 0.05. +(b) Error distribution for σ = 0.05. +(c) Eigenvalue argument trajectory for σ = 0.5. +(d) Error distribution for σ = 0.5. +Figure 3: +Estimates of the synthetic system’s eigenvalue argument produced by each iterative DMD +variant. Presented are typical trajectories of the eigenvalue argument at each time step over the course of 1 +experiment’s run (left) and error distributions of the difference between the true system’s eigenvalue +argument and the estimated eigenvalue argument over all time steps over the course of all 10 runs (right). +Measurement noise is set to either low levels with σ = 0.05 (top) or high levels with σ = 0.5 (bottom). The +DMDEnKF and Hankel-DMDEnKF experience similar errors to Online DMD and Windowed TDMD at +low measurement noise, but track the eigenvalue argument much more accurately than them for high +measurement noise. +to quickly adapt to changes in the system’s parameters, however as the noise increases this induces +an extreme volatility in their respective models. +The performance of Streaming TDMD is not +largely changed from the low noise case, still lagging behind the true system values but somewhat +insulated from the noise by its symmetric treatment of all data points. Here the benefit of explicit +inclusion of measurement noise in the DMDEnKF framework becomes apparent, as at this noise +level the DMDEnKF and Hankel-DMDEnKF are the only techniques tested capable of producing +an accurate eigenvalue argument estimate. +Furthermore, here we see the first significant difference in the performance of the DMDEnKF and +Hankel-DMDEnKF, as the DMDEnKF’s error distribution has a thin tail extending down to −π/8 +which is not present in the error distribution of the Hankel-DMDEnKF. These additional errors are +caused by the spin-up of the DMD for the DMDEnKF method occasionally failing to identify the +system’s eigenvalues as a complex conjugate pair (empirically, this happens ∼ 3% of the time), due +to the increased noise in the data. When this happens, the DMDEnKF catastrophically fails for +the EnKF is unable to generate complex eigenvalues from real ones regardless of how many future +time steps are filtered due to its formulation in equation (2.25). This failure of the DMDEnKF can +be mitigated in the following way. If the errors produced by the DMDEnKF during the filtering +stage are deemed too large (e.g. exceed a given threshold) for a prolonged period of time, then the +spin-up DMD process can be rerun on an extended dataset consisting of the original spin up data, +13 + +0.5 +True Eigenvalue +Streaming TDMD +igenvalue Argument +0.4 +Windowed TDMD +Online DMD +0.3 +Hankel-DMDEnKF +DMDEnKF +0.2 +0.1 +0.0 +100 +200 +300 +400 +500 +TimestepsStreaming TDMD +70 +WindowedTDMD +60 +Online DMD +Hankel-DMDEnKE +50 +DMDEnKF +Density +40 +30 +20 +10 +%.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +Distance from True Eigenvalues Argument0.5 +TrueEigenvalue +Streaming TDMD +0.4 +Windowed TDMD +OnlineDMD +0.3 +Hankel-DMDEnKF +DMDEnKF +0.2 +0.1 +0.0 +100 +200 +300 +400 +500 +TimestepsStreaming TDMD +70 +Windowed TDMD +60 +Online DMD +Hankel-DMDEnKF +50 +DMDEnKF +Density +40 +30 +20 +10 +-%.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +Distance from True Eigenvalues Argumentplus the newly available data used so far in the filtering step. By including more data in the spin-up +process, the spin-up DMD model is more likely to successfully capture the signal component in the +data as a pose to measurement noise, and hence produce eigenvalues with the same structure as +those of the true system. Time-delay embeddings make the SVD step in the DMD algorithm more +robust to measurement noise [16]. Hence, while the Hankel-DMDEnKF is similarly restricted by +the eigenvalues it can produce at the filtering stage, in all 1000 runs of the synthetic data the spin +up Hankel-DMD was able to identify the system’s eigenvalues to be a complex conjugate pair, so +this was not an issue for the Hankel-DMDEnKF. +3.2 +Comparing against DMD with a particle filter +Having compared the performance of the DMDEnKF against other iterative DMD variants, we +now focus on evaluating the filtering component of the algorithm. Since the linear DMD model +acts nonlinearly in the filter when applied to both the model’s state and eigenvalues, we compare +the EnKF filter with a particle filter. Particle filters [27] have been show to converge to the optimal +filter as the number of particles tends to infinity for general nonlinear models with non-Gaussian +noise [17]. However, particle filters are restricted to low dimensional systems only, as the number of +particles required scales approximately exponentially with the dimension of the state [57]. Hence, +we compare the DMDEnKF and Hankel-DMDEnKF with a DMD plus particle filter which we will +take to be the “gold standard” estimation to assess how well the EnKF does with the nonlinear +filtering problem. +We use the same synthetic system (3.1) with a linearly increasing eigenvalue argument as in the +previous subsection to generate data with high levels of measurement noise (σ = 0.5); a trajectory +of which can be seen in Figure 2. Again, the time-delay embedding dimension d = 50 for the +Hankel-DMDEnKF, and the first 100 time steps are used to train a spin-up DMD model, with the +next 400 used to filter the state and spin-up model’s eigenvalues. +The DMDEnKF’s filter state thus has dimension 4 (2 state dimensions and 2 temporal modes), +while the Hankel-DMDEnKF’s filter state is of dimension 102 (100 state dimensions and 2 temporal +modes). To generate a “gold standard” solution, at the filtering step we use a particle filter with +10,000 particles, applying multinomial importance resampling [20] every time the effective sample +size falls below half the number of particles to avoid sample degeneracy [21]. For the DMDEnKF +and Hankel-DMDEnKF at the filtering step, we run the EnKF with varying numbers of ensemble +members (N), to see if as N increases their estimates mean and covariance will tend to that of +the particle filter ensemble. We generated 1000 runs of the synthetic data to apply the particle +filter/EnKF with each value of N to and collected the errors in the eigenvalue argument estimates +for each method at every time step. +As can be seen in Figure 4a, the DMD particle filter with 10,000 particles produces an extremely +tight error distribution that is slightly biased to produce estimates below that of the true eigen- +value’s argument. This is to be expected, as mentioned in the previous subsection, due to the +system’s eigenvalue argument constantly increasing. There is also a thin tail in the error distribu- +tion that extends down to −π/8. This is again a result of the spin up DMD sometimes failing to +identify a complex conjugate eigenvalue pair, trapping the particle filter in the faulty assumption +that the eigenvalues are real. +14 + +(a) Error distributions. +(b) Mean squared errors. +Figure 4: Error distributions (left) and mean squared errors (right) for estimates of the synthetic system’s +eigenvalue arguments produced by the DMDEnKF and Hankel-DMDEnKF with varying numbers of +ensemble members (N) against those produced by a particle filter with 10,000 particles. Increasing N +quickly leads to error levels in the DMDEnKF and Hankel-DMDEnKF that are similar to those produced by +their respective “gold standards”. +For low numbers of ensemble members (N = 5), the DMDEnKF and Hankel-DMDEnKF are centred +at a similar value to the “gold standard”. However, they produce a far larger spread with long tails +in both directions that imply a lack of robustness with this few ensemble members. With only a +small increase to N = 10, both methods become more stable, as although they still have a larger +variance than the particle filter, the long positive tails from N = 5 have been eliminated. A similar +pattern occurs as we move to N = 20, with more ensemble members resulting in a tighter error +distribution. At this point, the Hankel-DMDEnKF’s distribution can be distinguished from that +of the DMDEnKF and DMD particle filter by its aforementioned lack of a persistent thin negative +tail. By N = 40, the main peaks of the DMDEnKF, Hankel-DMDEnKF and “gold standard” are +almost indistinguishable on the graphical scale, with the DMDEnKF and DMD particle filter both +sharing a thin negative tail. +Figure 4b shows how the mean squared error for the eigenvalue arguments predicted by the DM- +DEnKF and Hankel-DMDEnKF are affected by varying the number of ensemble members. For +the DMDEnKF, errors initially sharply decline as N is increased, however on this small synthetic +system returns diminish quickly after N = 20. By N = 50, we achieve a mean squared error with +the DMDEnKF only ∼ 3% larger than that of the “gold standard”, despite using 200 times fewer +particles. When comparing the Hankel-DMDEnKF to the “gold standard”, the errors in the DMD +particle filter’s eigenvalue estimates are skewed by the runs in which the spin up DMD was unable +to identify a complex conjugate eigenvalue pair, as Hankel-DMD did not encounter this problem +on these synthetic examples. To attempt to fairly compare the filtering methods, we remove all +runs in which the spin up DMD failed in this way, before again calculating the mean squared error +for the DMD particle filter and recording it in Figure 4b. A similar pattern of reducing errors with +diminishing returns can be seen for the Hankel-DMDEnKF as ensemble size is increased, and by +N = 50 its mean squared error is within 5% of the newly calculated DMD particle filter’s score. +Our results show that in this simple, synthetic case at least, the EnKF is an efficient and effective +solution to the nonlinear filtering problem that arise within the DMDEnKF framework. +15 + +No. of Ensemble Members = 5 +No. of Ensemble Members = 10 +DMDParticleFilter +DMDParticle Filter +20 +with10,000particles +with10,000particles +15 +Hankel-DMDEnKF +Hankel-DMDEnKF +DMDEnKF +DMDEnKF +10 +5 +Densit +No. of Ensemble Members = 20 +No. of Ensemble Members = 40 +DMDParticleFilter +DMDParticleFilter +20 +with10,000 particles +with10,000 particles +15 +Hankel-DMDEnKF +Hankel-DMDEnKF +DMDEnKF +DMDEnKF +10 +5 +0 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.20.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance from TrueEigenvalues ArgumentHankel-DMDEnKF +10-1 +DMDEnKF +Mean Squared Error +10-2 +DMD Particle Filter +with 10,000 particles +10-3 +DMD Particle Filter +with 10,000 particles +and failed runs removed +10 +20 +30 +40 +50 +No. of Ensemble Members3.3 +Tracking a synthetically generated pandemic +Lastly, we test the DMDEnKF’s performance on synthetic data designed to simulate a simple +pandemic with a state xk representing the level of infection in 3 different population classes. The +system’s dynamics are governed by a matrix A ∈ R3×3 that we randomly generate with non- +negative elements each being drawn from the Uniform distribution U[0, 1). The (i, j)th element of +A represents how the level of infection in class j at time k will affect the level of infection in class i +at time k + 1. To control whether the synthetic pandemic is spreading or dying off, we then define +a new matrix ˆA = A +λ1 where λ1 is the largest eigenvalue of A, thus ensuring the spectral radius +ρ(ˆA) = 1. By introducing a constant γ, we can replace A with γ ˆA causing the state to grow if +γ > 1 or decay for γ < 1. To simulate a pandemic, we linearly decrease γ from 1.01 to 0.99 over the +course of the experiment’s 1000 time steps. The initial state used is a vector of ones. The system +that generates the synthetic data can be written as +xk+1 = γk ˆAxk, +x1 = +� +1 +1 +1 +�T +, +(3.5) +where the state xk ∈ R3, γk = 1.01 − 0.02(k−1) +999 +and k = (1, ..., 1000). We assume not to have access +to the true state of the system xk but instead noisy measurements yk defined by +yk = xk + vk, +(3.6) +The constant σ that governs the level of measurement noise is set to σ = 0.05 to represent low +noise and σ = 0.5 for high noise as in (3.2). Figure 5 shows the values of the system’s three state +dimensions and the respective available measurements over the course of one run. +Figure 5: +Time series for a synthetic system that simulates a pandemic, showing all 3 dimensions of the +state with no, low (σ = 0.05) and high (σ = 0.5) measurement noise. +All five DMD variants tested had their computational parameters set to the same values as those +used in the synthetic experiments in Section 3.1. The only small difference was that Streaming +TDMD, Windowed TDMD, the DMDEnKF and Hankel-DMDEnKF truncated the data by remov- +ing the smallest singular value to reduce model instability caused by what was often a very low +16 + +16 +Xk True State +14 +ykforg=0.05 +12 +yk for g= 0.5 +Values +10 +8 +State +6 +4 +0 +0 +200 +400 +600 +800 +1000 +Timestepssignal-to-noise ratio in this direction. Online DMD did not apply any truncation to the data as the +method was not designed to do so, however it did not appear to suffer from any stability issues as +a consequence. +The first 100 measurements (y1, ..., y100) were used to initialize the models, and as each new data +point (y100, ..., y1000) was successively fed into the models, they produced 50 step ahead forecasts +(ˆx150, ..., ˆx1050). +We generate 1000 data points, however a standard flu season lasts around 20 +weeks. For this reason, we chose to forecast 50 steps ahead to mimic forecasting 1 week ahead in +a more realistic timescale. The relative prediction errors ˆek = ∥xk−ˆxk∥ +∥xk∥ +could then be calculated +for k = (150, ..., 1000) and the mean of these errors was the main metric we used to evaluate the +forecasting skill of each method over the course of one run. +A thousand runs were performed +for both low and high levels of noise and the empirical cumulative distributions of 50 step ahead +forecast mean run relative errors for low noise (σ = 0.05) can be seen in Figure 6. +Figure 6: +Cumulative error distributions of the mean run relative errors for the 50 step ahead forecasts of +each iterative DMD variant. Mean relative errors were calculated over all time steps for each run of the +experiment, with the results from 1000 runs under low levels of measurement noise (σ = 0.05) displayed. +Forecast errors had a wide range for some methods, due to exponentially compounding errors caused by +forecasting 50 steps ahead. The DMDEnKF, Hankel-DMDEnKF and Online DMD produced errors orders +of magnitude smaller than those of Streaming TDMD and Windowed TDMD. +The first noteworthy feature of the cumulative error distributions is the wide range in some method’s +forecast errors. This is a result of the 50 step ahead forecasts being produced by training each +model to forecast 1 step ahead, then applying the trained model to the data 50 times. As such, +forecast errors compound exponentially and small errors over a 1-step forecast horizon can become +vast after 50 iterations. Inspecting the individual methods, we see Windowed TDMD to be the +worst performing method. This is due to its aforementioned instability under measurement noise +caused by considering only a small subset of the data at a time. This instability could be reduced +by increasing the window size (w) computational parameter, however as w increases the model’s +ability to track a system that changes with time diminishes. Streaming TDMD had the second- +largest errors, caused by the method’s assumption of a stationary system hindering its ability to +17 + +1.0 +Streaming TDMD +Windowed TDMD +Online DMD +DMDEnKF +0.8 +Hankel-DMDEnKF + Proportion of runs +0.6 +0.4 +0.2 +0.0 +1010 +1029 +1048 +1067 +1086 +10105 +10124 +10143 +50 step ahead forecast mean run relative errorcorrectly adapt to the system’s changing eigenvalues as new data became available. In the majority +of cases, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly well. All +three methods exhibited cumulative error distributions tightly concentrated around a low error +value, however in a few runs, the DMDEnKF became unstable and produced large errors. It is +clear even at low levels of noise that the forecasting performance of Online DMD, the DMDEnKF +and Hankel-DMDEnKF are far superior on this type of system to those of Windowed TDMD and +Streaming TDMD. Hence, we now focus exclusively on these top three performing methods to allow +for a thorough comparison of them on an appropriate error scale. +(a) σ = 0.05 +(b) σ = 0.5 +Figure 7: Error distributions of the mean run relative errors for the 50 step ahead forecasts of Online +DMD, the DMDEnKF and Hankel-DMDEnKF, attained over 1000 runs under low σ = 0.05 (left) and high +σ = 0.5 (right) levels of measurement noise. Similar errors were found at both noise levels, with Online +DMD performing better at low measurement noise and the DMDEnKF/Hankel-DMDEnKF performing +better at high measurement noise. +In Figure 7 for Online DMD, the DMDEnKF and Hankel-DMDEnKF, we plot the distributions of +50 step ahead forecast mean run relative errors at both low and high levels of measurement noise. +At low levels of noise, Online DMD’s errors peak at a lower level than those of the DMDEnKF +and Hankel-DMDEnKF, however as noise levels increase we see the peaks switch sides, and the +DMDEnKF/Hankel-DMDEnKF become the better performing methods. At both noise levels, the +peak in the DMDEnKF’s error distribution is centred at the same value as the Hankel-DMDEnKF’s +peak, however it is less dense due to the additional probability mass stored in the long tail of the +DMDEnKF’s error distribution, which is not present in that of the Hankel-DMDEnKF. +These disproportionately large errors in the DMDEnKF distribution’s tail occur when the spin-up +DMD process fails to produce a model similar enough to the system’s true dynamics. As briefly +touched upon in the first synthetic example, if the spin-up DMD model is sufficiently inaccurate +then it can stop the EnKF from effectively assimilating new data, leading to the catastrophic +failure of the DMDEnKF. In this example, as the signal-to-noise ratio in the direction of the +second-largest singular value was often low, an unfortunate random draw of the system dynamics +(A) and measurement noise (vk) in the spin-up period could produce large errors in DMD’s second +eigenvalue. Empirically, using the interquartile range method to detect outlying forecast errors, +this DMDEnKF failure occurred 5.5% of the time for σ = 0.05. The errors would persist in the +filtering step as new data was processed, whereas other methods were able to recover from poor +initial model estimates more effectively. The quality of the model produced by the initial DMD is +dependent on the quality and volume of the spin-up data, hence this problem was exacerbated and +occurred much more regularly at higher noise levels (21.9% of the time for σ = 0.5). It could be +18 + +35 +Online DMD +Hankel-DMDEnKF +30 +DMDEnKF +25 +10 +5 +0 +3 × 10-2 +4 × 10-2 +6 × 10-2 +10-1 +1052 +1056 +5o step ahead forecast mean run relative errorOnline DMD +16 +Hankel-DMDEnKF +14 +DMDEnKF +12 +8 +6 +4 +2 +0 +10-1 +2 × 10-1 +3 × 10-14× 10-1 +6 × 10-1 +1043 +1047 +5o step ahead forecast mean run relative errormitigated somewhat by increasing the number of time steps in the spin-up stage as described at +the end of Section 3.1, however similarly to Windowed TDMD as the system is assumed to be time +varying there likely exists a point of negative returns once the spin-up period becomes too long due +to the stationarity assumption of batch DMD becoming progressively more violated. +Unlike the DMDEnKF, the Hankel-DMDEnKF and Online DMD do not suffer from a long tail in +their error distributions, and perform consistently well over all 1000 runs. At both noise levels, +their error distributions have a similar variance, with the Hankel-DMDEnKF’s errors being slightly +more tightly grouped than those of Online DMD. Hence, the average error is the main factor when +differentiating between the method’s performance in this example, meaning Online DMD is the +preferred method at low noise and the Hankel-DMDEnKF (or DMDEnKF provided the spin-up +DMD does not catastrophically fail) is more accurate at high noise. As both methods posses useful +yet differing attributes, we generate a typical data trajectory (one for which the DMDEnKF does +not fail) for both low and high measurement noise. We then investigate how each model’s 50 step +ahead forecasts and dominant eigenvalue estimates change over the course of each run, as shown +in Figure 8. +(a) 50 step ahead forecasts for σ = 0.05. +(b) Dominant eigenvalue’s modulus for σ = 0.05. +(c) 50 step ahead forecasts for σ = 0.5. +(d) Dominant eigenvalue’s modulus for σ = 0.5. +Figure 8: Typical trajectories of the 50 step ahead forecasts for the value of the state’s first dimension (left) +and estimates of the dominant eigenvalue’s current modulus (right) under low (σ = 0.05) and high +(σ = 0.5) levels of measurement noise for Online DMD, the DMDEnKF and Hankel-DMDEnKF over the +course of 1 run. Online DMD forecasts 50 steps ahead more accurately at low noise, and the +DMDEnKF/Hankel-DMDEnKF more accurately at high noises, however when signal-to-noise ratio is low +(at the start and end of the experiment) Online DMD’s eigenvalue estimates become unstable. +First, observing the low noise forecasts in Figure 8a, it is clear Online DMD produces forecasts +that are more robust and closer to the true state’s value than those of the DMDEnKF and Hankel- +DMDEnKF. This was to be expected, by virtue of Online DMD’s lower average errors in the error +distributions of Figure 7a at this noise level. As noise is increased, the forecasts in Figure 8c show +the DMDEnKF and Hankel-DMDEnKF becoming the more accurate methods, however Online +19 + +14 +Xk True State +Online DMD +12 +Hankel-DMDEnKF +State in dimension +DMDEnKF +10 +8 +6 +4 +2 +200 +400 +600 +800 +1000 +Timesteps1.010 +True Eigenvalue Mod +OnlineDMD +1.005 +Hankel-DMDEnKE +DMDEnKF +1.000 +0.995 +0.990 +0.985 +200 +400 +600 +800 +1000 +Timesteps18 +XkTrueState +16 +OnlineDMD +14 +Hankel-DMDEnKE +dimension +DMDEnKF +12 +10 +8 +in +State i +6 +4 +0 +200 +400 +600 +800 +1000 +TimestepsModulus +.02 +1.01 +1.00 +PT +0.99 +TrueEigenvalueMod +0.98 +OnlineDMD +Hankel-DMDEnKF +0.97 +DMDEnKF +200 +400 +600 +800 +1000 +TimestepsDMD’s forecasts remains fairly stable, and still appear to be a viable forecasting option. +Analysing the eigenvalue estimates in Figures 8b and 8d, we see that over the middle section of data +where k = (250, ..., 750), Online DMD is able to track the dominant eigenvalue effectively. However, +at the beginning and end of the dataset when states and hence the signal component of each new +data point is small relative to the measurement noise, Online DMD’s eigenvalue estimates become +progressively more unstable. In the low noise case this is not a problem, as Online DMD’s estimates +are significantly more accurate than those of the DMDEnKF/Hankel-DMDEnKF, so even in the +poorly performing sections of the data it’s estimates still better/match those of the DMDEnKF. +For higher noise however, Online DMD provides significantly less robust estimates of the dominant +eigenvalue at the start and end of the datasets than those generated by the DMDEnKF and Hankel- +DMDEnKF. In the epidemiological context of an infectious disease outbreak, which this synthetic +example attempts to mimic, scientists will often try to calculate the basic reproduction number +(R0) [58] using noisy data from the small number of initial infections. If R0 > 1 the number of +infections will grow exponentially if left unchecked, and if R0 < 1 the number of infections will +decay naturally to 0. +Within this example, using the DMDEnKF/Hankel-DMDEnKF one can +quickly determine that initially R0 > 1 and take any required action thanks to the stability of it’s +early eigenvalue estimates, whereas it takes significantly longer and a higher level of infection for +Online DMD to consistently determine if R0 is above or below the growth/decay threshold. +4 +Seasonal Influenza-like Illness Application +4.1 +Problem setup +DMD based methods have previously been applied to infectious disease data [49]. In this case, +DMD modes can be viewed as stationary, spatial modes used to create a reduced order model in +which only the amplitudes and frequencies are time varying [9]. Hence, modelling influenza-like +illness (ILI) data is a prime potential application for the DMDEnKF/Hankel-DMDEnKF. +The CDC’s ILINet data [25] we will be using records the number of ILI General Practitioner (GP) +consultations in the US each week, alongside the number of total GP consultations which can be +used to normalize the ILI data. We use a subset of the data from the start of 2003, the first year +when data is available all year round, to the end of 2018 as seen in Figure 1. We then split each +week’s data into demographics, consisting of 4 age groups (0-4, 5-24, 25-24, 65+) and 10 Health +and Human Services (HHS) regions. Each region consists of the following locations: +• Region 1 - Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont. +• Region 2 - New Jersey, New York, Puerto Rico, and the U.S. Virgin Islands. +• Region 3 - Delaware, District of Columbia, Maryland, Pennsylvania, Virginia, and West +Virginia. +• Region 4 - Alabama, Florida, Georgia, Kentucky, Mississippi, North Carolina, South Carolina, +and Tennessee. +• Region 5 - Illinois, Indiana, Michigan, Minnesota, Ohio, and Wisconsin. +20 + +• Region 6 - Arkansas, Louisiana, New Mexico, Oklahoma, and Texas. +• Region 7 - Iowa, Kansas, Missouri, and Nebraska. +• Region 8 - Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming. +• Region 9 - Arizona, California, Hawaii, and Nevada. +• Region 10 - Alaska, Idaho, Oregon, and Washington. +Whilst ILI consultation data is available over all 40 of these strata, total GP consultation data +is only provided by region. To generate an age breakdown for a region’s total consultations we +linearly interpolate using census data to approximate the US population’s age demographics for +a given week. We then allocate the region’s total consultations to each age group based on the +proportion of the total population they represent. This method assumes that all age groups have +a similar likelihood of attending the GP’s, which may be flawed but we believe it to be sufficient +for the purpose of demonstrating the DMDEnKF on real-world data. +4.2 +Building the spin-up DMD model +The format of the ILI data used in the DMDEnKF is thus a 40 dimensional vector for each week, +recording ILI consultations as a percentage of total GP consultations over every demographic. This +data exists in R40 ++ however DMD computes modes in R40. Hence, to ensure the model’s estimates +are non-negative, we first transform the data by adding a small constant (c = 1) then taking +the natural logarithm of each element. For the Hankel-DMDEnKF, this transformed data is then +delay-embedded with the previous 99 time steps (d = 100) to form a state in R4000. We use data +up to the end of 2012 as training data for the spin-up DMD processes of the DMDEnKF/Hankel- +DMDEnKF detailed in Step 1 of the DMDEnKF algorithm, and then filter the remaining years +from 2013-2018. The transformed, centred data with the split where the spin-up process ends and +the filtering begins marked is shown in Figure 9. +We initially choose to truncate to 8 DMD modes for the DMDEnKF to demonstrate the method. +We discuss the effect of changing the truncation on the DMDEnKF method below, but at 8 DMD +modes approximately the amount of additional variance in the data that is retained by keeping more +modes diminishes significantly. This is evidenced by the “elbow” seen in the cumulative variance +plot of Figure 14a, at the point where the graph transitions from rapidly increasing in variance +with r to a more gradual ascent. We also truncate the Hankel-DMDEnKF to 8 DMD modes, to +allow for a more direct comparison between the two variants. +The spectrum and dominant DMD/Hankel-DMD modes associated with each frequency identified +by the spin-up DMD processes can be seen in Figure 10. All eigenvalues shown in Figures 10a +and 10c had a modulus of ∼ 1, meaning in both cases each mode was expected to persist in the +data without growing exponentially. The major difference between the two methods spectra is that +Hankel-DMD identifies the most dominant mode to have a period of one year, whereas DMD does +not detect any modes with this period. Annual peaks in ILI consultations occurring at a relatively +similar time each year indicates that the data contains a strong mode of period one year, and this +is supported by Fourier analysis [10] which also identifies one year as the dominant period. Hence, +DMD is missing the yearly mode present in the data which Hankel-DMD is able to detect, and +21 + +Figure 9: ILI consultations as a percentage of total weekly GP consultations across 4 age brackets and the +10 HHS regions in the US, log transformed and centred. The peaks of varying size, timing and shape in +Figure 1 are visible here as vertical red areas of varying width and intensity that encompass most +demographics. +this is likely due to Hankel-DMD’s aforementioned enhanced robustness to measurement noise. +There are two clear patterns in the structure of the dominant DMD and Hankel-DMD modes seen +in Figures 10b and 10d. Firstly, their strata generally move together. This is shown by the vast +majority of entries for each DMD mode, and entries within the same delay-embedded week (denoted +by a vertical slice through the mode) for Hankel-DMD modes, sharing the same sign. This implies +that the percentage of ILI consultations increases and decreases at a similar time across all ages +and regions. Secondly, the variance is higher for the younger age groups. This is demonstrated +by the absolute value of elements in the top two rows of each region generally being larger than +those in the bottom two. From Figure 10b, this is visible trivially for the DMD modes. In Figure +10d, age groups are arranged in ascending order for each region, so this effect is evidenced in the +Hankel-DMD modes by the presence of more intensely coloured horizontal lines of width 2, followed +by less intensely coloured lines of width 2 repeating over each of the 10 regions. This indicates that +there are sharper peaks and deeper troughs in the percentage of ILI consultations for the young, +while the rates for those 25 and over remain more stable. +4.3 +Applying the filter +The filtering steps of the DMDEnKF/Hankel-DMDEnKF are then applied over the remaining data +using the spatial and temporal modes from the spin-up DMD and spin-up Hankel-DMD respectively. +Producing a 4-week ahead ILI forecast for the ILINet data that consistently outperforms a simple +historical baseline prediction is difficult even for state-of-the-art models [50]. As such, to test the +DMDEnKF/Hankel-DMDEnKF we use a forecast horizon of 4 weeks when making predictions. +In Figure 11, the DMDEnKF and Hankel-DMDEnKF’s forecasting of total ILI consultations as +22 + +0-4 +2 +consultations +2 +3 +65 + +centred % ILI +Demographic +25.64 +0-4 +65+ +Re +25.64 +0 +65 ± +Log transformed, +Region +2 +0-4 +6 +25.64 +-2 +65 + +2003 +2005 +2007 +2009 +2011 +Split +2015 +2017 +2019 +201 +Date(a) DMD Eigenvalue Spectrum. +(b) Dominant DMD Modes. +(c) Hankel-DMD Eigenvalue Spectrum. +(d) Dominant Hankel-DMD Modes. +Figure 10: Eigenvalue Spectrum (left) and DMD modes in descending order of dominance (right) generated +by the DMD (top)/Hankel-DMD (bottom) applied to the data in Figure 9 up to the spin-up date. In both +cases, all eigenvalues lie approximately on the unit circle, and dominant modes feature the same sign for +most demographics with a magnitude that varies with age. The DMD modes are more interpretable, but +Hankel-DMD identifies the mode with period 1 year, which DMD does not. +23 + +1 +0.5 +Imaginary +0 +-0.5 +-1 +-1 +-0.5 +0 +0.5 +1 +RealMode l: +Eigenvalue Period = 1.9 Years +0-4 +5-24 +25-64 +65 + +Mode 2: +EigenvaluePeriod=0.6Years +0-4 +0.3 +5-24 +25-64 +0.2 +65 + +es +Mode 3: +Eigenvalue Period = 4.4 Years +0.1 +0-4 +5-24 +0.0 +25-64 +-0.1 +65 + +Mode 4: +Eigenvalue Period = 22.5'Years +-0.2 +0-4 +5-24 +25-64 +65 + +1 +2 +3 +4 +5 +6 +8 +9 +ion +Region +Region +Region +ion +ion +Region +Region +Region +Regi +Regi +Regi +Regions1 +0.5 +Imaginary +0 +-0.5 +-1 +-1 +-0.5 +0.5 +1 +RealMode l: +Eigenvalue Period = l.0 Years +21 +28 +35 +Mode 2: +Eigenvalue Period = 13.5 Years +07 +0.04 +14 +Age/Regions +21 +28 +0.02 +35 +Mode 3: +Eigenvalue Period = 1.6 Years +07 +0.00 +14 +-0.02 +21 +28 +35 +-0.04 +Mode 4: +Eigenvalue Period = 0.6 Years +07: +14 +21 +28 +35 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Weeks Delay-Embededa percentage of total weekly GP consultations can be seen in full. Up until 2012, we generate +4-week ahead reconstructions using the spin-up DMD models only, estimating each state by taking +the state measurement from 4 weeks prior, then iteratively applying the model to it 4 times. The +4-week ahead DMD reconstruction in Figure 11a captures more fluctuations in the data than that +of Hankel-DMD, however these high frequency fluctuations can also indicate the effect of noise in +the measurements. The Hankel-DMD reconstruction shown in Figure 11b is much less sensitive +to noise, although fails to identify the sharper peaks in the data, which suggest it may be over- +smoothing. From 2012 onwards the filtering begins, and forecasts are generated as described in the +DMDEnKF algorithm using equation (2.33). The DMDEnKF forecasts become significantly more +stable, while the Hankel-DMDEnKF forecasts improve in capturing the true shape of the data, +however both suffered from some degree of lag in their predictions. +During this second section of the data, the models are producing actual forecasts, as the DM- +DEnKFs only have access to data up to 4 weeks prior to the prediction target’s date. Hence, it +is in this section of the data we compare the models’ performance against that of the historical +baseline. The historical baseline prediction was created in a similar manner to that used in [8], +taking ILI consultation rates from the same week of every previous year in the data (excluding the +pandemic year of 2009) and then producing a probability distribution for the current week’s con- +sultations via Gaussian kernel density estimation (KDE) [55]. KDE Bandwidths were determined +using Silverman’s rule of thumb [56], and when point estimates were required they were taken as +the median of the distribution. The results of the comparisons can be seen in Figure 12 and Table +2. Here it is worth noting that although we use data dated 4 weeks prior to the prediction date, +in reality this data is often subject to revisions so the ILINet data as it currently stands would not +necessarily be available in real time [50]. +4.4 +Evaluating the DMDEnKF’s performance +Figure 12 demonstrates graphically the 4-step ahead DMDEnKF, Hankel-DMDEnKF and historical +baseline forecasts. The DMDEnKFs more successfully capture the shape and height of each flu sea- +son’s peak, however tend to predict the peaks late, whilst the historical baseline consistently under- +predicts the peak rates but is fairly accurate on the timings. The Hankel-DMDEnKF’s forecasts are +smoother than those of the DMDEnKF, however do not capture smaller details within the shape of +the peaks. We also plot the 95% confidence intervals for the DMDEnKF and Hankel-DMDEnKF’s +forecasts in Figure 12, generated using the ensemble that is maintained and propagated in the +EnKF framework. At all times, the real data lies within the DMDEnKF’s confidence interval, +which is not true for the Hankel-DMDEnKF. The DMDEnKF’s confidence interval is significantly +wider than that of the Hankel-DMDEnKF, and this is due to Hankel-DMD’s robustness to noise, +meaning that when the ensemble is propagated through the model, a large amount of probability +mass is concentrated in a small area of the state space. This then leads to the Hankel-DMDEnKF +underestimating the uncertainty in the system, and hence some real data values falling outside the +boundaries of it’s 95% confidence interval. +To numerically compare performance, we used metrics designed for the Forecast the Influenza +Season Collaborative Challenge (FluSight), in which multiple teams would submit predictions about +the weekly ILINet consultation rates for the upcoming flu season at a national and HHS regional +level [7], [8]. The FluSight challenge evaluated models abilities to generate 1-4-week ahead forecasts +24 + +(a) DMDEnKF 4-week ahead forecast. +(b) Hankel-DMDEnKF 4-week ahead forecast. +Figure 11: ILI consultations as a percentage of total weekly GP consultations forecast 4 weeks ahead using +the DMDEnKF (top) and Hankel-DMDEnKF (bottom). The DMD reconstruction captures the shape of the +data well but is unstable, whereas the Hankel-DMD reconstruction is less sensitive to noise but +over-smooths. The DMDEnKF and Hankel-DMDEnKF forecasts help reduce these issues present in their +respective reconstructions, but both suffer from some degree of lag in their predictions. +25 + +10 +Spin-up DMD +Real Data +DMDEnKE +8 +Iconsultations +4 +% +2 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +Date10 +Real Data +Spin-up Hankel-DMD +Hankel-DMDEnKF +8 +Iconsultations +三 +% +2 +0 +2004 +2006 +2008 +2010 +2012 +2014 +2016 +2018 +DateFigure 12: ILI consultations as a percentage of total weekly GP consultations, forecast 4 weeks ahead using +the DMDEnKF (top) and Hankel-DMDEnKF (bottom). A 95% confidence interval for each forecast, and +historical baseline predictions are also shown. The Hankel-DMDEnKF forecasts are smoother than those of +the DMDEnKF, but both forecasts contain some lag. The real data always lies within the DMDEnKF’s +confidence interval but not the Hankel-DMDEnKF’s, however this is likely due to the DMDEnKF’s +confidence interval being significantly wider than that of the Hankel-DMDEnKF. +26 + +18 +Real Data +16 +Historical Baseline +DMDEnKF +95% CI +14 +consultations +12 +10 +8 +% +6 +4 +2 +0 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +Date18 +Real Data +Historical Baseline +16 +Hankel-DMDEnKF +95% CI +14 +consultations +12 +10 +8 +三 +% +6 +4 +2 +0 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +DateForecast +Log Score +Mean Squared Error +Historical Baseline +0.28 +1.24 +1-week ahead DMDEnKF +0.49 +0.33 +2-week ahead DMDEnKF +0.38 +0.61 +3-week ahead DMDEnKF +0.32 +0.87 +4-week ahead DMDEnKF +0.27 +1.16 +1-week ahead Hankel-DMDEnKF +0.41 +0.49 +2-week ahead Hankel-DMDEnKF +0.33 +0.70 +3-week ahead Hankel-DMDEnKF +0.29 +0.97 +4-week ahead Hankel-DMDEnKF +0.23 +1.26 +Table 2: The log scores and mean squared errors for the DMDEnKF and Hankel-DMDEnKF with differing +forecast horizons, and the historical baseline prediction. The DMDEnKF achieves a higher log score and +mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where it attains a +similar level of forecast skill. The Hankel-DMDEnKF consistently underperforms against the DMDEnKF +in both metrics over these short forecast horizons. Scores are calculated over the 6 flu seasons from +2012/13 to 2017/18. +known as short-term targets over the course of a flu season, as well as other longer term targets, +known as seasonal targets, before the season had begun. The DMDEnKF is primarily intended to +be a tool for tracking and short-term forecasting, hence we focus on forecasting these short-term +targets only. For this purpose we used two different metrics, the log probability measure (log score) +slightly adjusted from the FluSight challenge as used in [50] and the mean squared error due to its +popular use in regression problems. The log score represents the geometric average probability of +each model’s prediction being accurate, with accuracy deemed as a forecast within +/ − 0.5 of the +true ILI consultation rate. The higher the log score, the better the forecast. Metrics are calculated +from week 40 to week 20 of the following year to prioritize evaluation of forecasts during the flu +season, and we use the 6 full seasons from 2012/13 to 2017/18. +The results for the historical baseline prediction and DMDEnKF/Hankel-DMDEnKF’s forecasts at +a national level can be seen in Table 2. As one would expect, the accuracy of both DMDEnKFs +degrade as they make predictions further into the future. The DMDEnKF achieves a higher log score +and mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where +it attains a similar level of forecast skill. For forecasts of 5 or more weeks ahead, the DMDEnKF is +unable to outperform the historical baseline in either metric. The Hankel-DMDEnKF consistently +underperforms against the DMDEnKF in both metrics over these short forecast horizons. The top +3 statistical models and top 3 mechanistic models in the FluSight challenge achieved log scores of +0.32 and 0.3 respectively for their 4-week ahead forecasts, hence the DMDEnKF has lower (but +comparable) forecasting skill than current state of the art ILI models. As the forecast horizon is +extended up to 12 weeks ahead, the DMDEnKF’s forecast scores continue to decrease monotonically, +whereas the Hankel-DMDEnKF’s log scores for 9-12 weeks ahead are no worse than those for 5-8 +weeks ahead. As such, the DMDEnKF is preferred for short-term forecasting, while the Hankel- +DMDEnKF is considered superior when forecasting over longer timescales. +Figure 13 shows the log scores for the 4-week ahead DMDEnKF forecast, and how these compare to +27 + +(a) DMDEnKF 4-week ahead forecast. +(b) DMDEnKF 4-week ahead forecast - historical +baseline. +Figure 13: Log scores over all ages and regions for the DMDEnKF’s 4-week ahead forecast (left), followed +by those same scores with the log scores of the historical baseline prediction subtracted (right). The +Hankel-DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so we do not include its +breakdown to avoid redundancy. In the top figure, the generally increasing intensity of red as one moves +down the age groups shows the DMDEnKF performing more accurately for older age groups. The bottom +figure’s varying areas of red/blue shows the DMDEnKF/historical baseline vary in superiority of forecasting +skill depending on the age and region being forecast, with the historical baseline scoring more highly for +most regions. +the scores attained by the historical baseline prediction at an age and regional level. The Hankel- +DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so its breakdown is +rather similar to that of the DMDEnKF, and we do not include it to avoid redundancy. In the +DMDEnKF’s log scores, we see a major pattern in the older age groups scoring higher and hence +being better predicted than the younger demographics. This pattern does not persist when the +historical baseline’s scores are removed, indicating it is a more general trait of the data as opposed +to a specific quality of the DMDEnKF’s modelling technique. There is also a significant difference in +the predictability from region to region. For example, region 1 was the most predictable region for +both the DMDEnKF and historical baseline, which is consistent with the findings in [50]. However, +the DMDEnKF improved on the historical baseline’s forecast for only two of the four age groups +in this region. In [50] it was found that the most overall improvement gained by forecasting for a +region using a model as opposed to the historical baseline prediction also occurred in region 1, so +one would expect to see improvements by the DMDEnKF over the historical baseline in all four age +groups. As log score is heavily influenced by the amount of uncertainty in a forecast, it is possible +that the covariance matrices used in the DMDEnKF were too large for this region. Hence, setting +the magnitude of the DMDEnKF’s variance on a region by region basis could lead to better results +and more accurate forecasts. Region 6 was the worst forecast region by the DMDEnKF, and the +historical baseline predictions were slightly more accurate. Again, this is consistent with [50] where +region 6 was the lowest scoring region for the models. In that work however, region 6 experienced +the second most improvement by using a model over the historical baseline prediction. Hence, for +this region a larger variance within the DMDEnKF may have been more appropriate to account +for its extra unpredictability, further supporting the idea of varying the variance by region in the +future. +28 + +0-4 +0.40 +0.35 +5-24 +0.30 +score +Ages +0.25 +s +25-64 +0.20 +Log +0.15 +65 + +0.10 +0.05 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Region +Region +Region +Region +egion +egion +Region +Region +Region +Region +Re +e +R +R +Regions-0.05 +0-4 +0.00 +5-24 +-0.05 +core +Ages +-0.10S +25-64 +Log +-0.15 +65 + +-0.20 +Region 1 +Region 2. +3 +Region 4 +5 +Region 6. +Region 7. +8 +9 +Region 10 +Region 3 +Region +Region 8 +Region 9 +Regions4.5 +Varying the truncation rank +Having analysed the DMDEnKF and Hankel-DMDEnKF’s ILI forecasting with 8 DMD modes, +we now investigate the effect different truncation ranks (r) have on their performance in Figure +14. From Figure 14a, the subjective process of identifying an “elbow” in the data could lead an +observer to determine a suitable rank for truncation as low as 4 or as high as 12. Application of +the algorithm of Gavish and Donoho for identifying the optimal truncation threshold [26] also finds +the truncation rank to be 12, hence we will focus on investigating values of r in the interval from +4 to 12. +(a) % of total variance in the data. +(b) DMDEnKF log score/mean squared errors for +r = 4, ..., 12. +(c) % of total variance in the delay-embedded data. +(d) Hankel-DMDEnKF log score/mean squared errors +for r = 4, ..., 12. +Figure 14: On the left, the % of the total variance in the data (top) and delay-embedded data (bottom), +dependent on the number of singular values that are retained (r). An “elbow” in the data occurs around +r = 8 where we choose to truncate, however determining the exact position of the “elbow” is subjective and +could be considered anywhere from r = 4 to r = 12. On the right, the log score and mean squared errors for +4-step ahead forecasts generated using the DMDEnKF (top) and Hankel-DMDEnKF (bottom) with differing +values of r. In both cases, log score is maximised and mean squared error minimised for r = 8. +Figures 14b and 14d show how the metrics we use to measure the DMDEnKF and Hankel- +DMDEnKF’s forecasting skill vary with r. An ideal forecast will have a high log score reflecting +a relatively tight and accurate probability distribution, with a low mean squared error indicating +a point estimate close to the true percentage of ILI consultations. For both methods, log score +is maximised and mean squared error minimised by r = 8, indicating this is the optimal rank to +truncate at for our models. For r = 4, we have the simplest model tested, hence it has a low degree +of uncertainty resulting in a relatively high log score, however is too simple to properly model the +system so receives a high mean squared error. By increasing r, we allow the DMDEnKFs more +freedom to capture complexity within the system, resulting in a more accurate representation of +the true dynamics and hence a generally lower mean squared error. When the number of eigen- +29 + +lained +100% +95% +ex +90% +iance +85% +vari +80% +75% + total +70% +Rank cut off +of +65% +at r=8. +% +0 +5 +10 +15 +20 +25 +30 +35 +40 +Singular values retained (r)1.6 +.7 +rror +1.5 +.5 +1.4 +Mean Squared +1.3 +.4 +1.2 +.6 +1.1 +.10 +.12 +.11 +1.0 +.9 +.8 +0.9 +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +0.28 +Log Score 8. +The additional freedom afforded to the DMDEnKF by increasing r also means the model contains +more parameters, each of which have an associated degree of uncertainty. This causes the overall +forecast’s probability distribution to become more spread out, and when no longer offset by the +increased model accuracy up to r = 8, reduces the forecasts log score. +5 +Conclusion +To conclude, we have defined two new algorithms, the DMDEnKF and Hankel-DMDEnKF, that +combine dynamic mode decomposition and Hankel dynamic mode decomposition respectively with +ensemble Kalman filtering, to update state and temporal mode estimates of a dynamical system +as new data becomes available. When applied to simple, synthetic systems with a time varying +parameter and low measurement noise, the DMDEnKFs performed similarly to other iterative DMD +variants tested in tracking the system’s time varying parameter and forecasting future states. As +measurement noise was increased, the DMDEnKFs outperformed the other methods tested in both +metrics, and the Hankel-DMDEnKF produced more stable forecasts than those of the DMDEnKF. +Both DMDEnKFs achieved similar performance levels to their equivalent DMD Particle Filters +(an alteration to the DMDEnKF algorithms where the ensemble Kalman filters were switched for +Particle Filters), while requiring significantly fewer ensemble members. When forecasting influenza- +like illness across age groups and HHS regions in the US using data from the CDC, the DMDEnKF +produced more accurate forecasts than a historical baseline prediction up to 3 weeks ahead, and +forecasts approximately as accurate 4 weeks ahead. The Hankel-DMDEnKF produced less accurate +forecasts for these short-term targets than the DMDEnKF, however in general it’s forecasts were +more stable. Also, the Hankel-DMDEnKF was able to identify the presence of a mode with period 1 +year, which is strongly visible in the data, yet not identified by the DMDEnKF. Both DMDEnKFs +exhibited lower forecasting skill than current state of the art influenza-like illness models. +A natural extension of the DMDEnKF would be to apply extended/kernel DMD in the spin-up +DMD phase, allowing the algorithm to be used more effectively on dynamical systems that act +nonlinearly in their measured states. Instead of taking the observed values alone as the system’s +state xk, these variants use for the state a collection of functions on the observables g(xk), which +often increases the state’s dimension n. The EnKF is well suited to this pairing, as it scales more +computationally efficiently in the state dimension than other Kalman filtering methods [23]. The +best choice of the collection of functions g(xk) as an embedding for nonlinear systems so that DMD +may be effectively utilized is an interesting area of future work. Many methods have been developed +that propose ways of generating g(xk), for example using deep learning [41] or reservoir computing +[28], and this remains a promising avenue for future work. +Code availability +Codes used to produce the results in this paper are available at https://github.com/falconical/DMDEnKF. +30 + +Data availability statement +All data used to produce the results in this paper will be made available upon reasonable request. +Acknowledgements +This work was supported by the UKRI, whose Doctoral Training Partnership Studentship helped +fund Stephen Falconers PhD. He would also like to thank for their valuable discussions, Nadia +Smith and Spencer Thomas from the National Physics Laboratory. +References +[1] H. 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Cattafesta, Online dynamic mode +decomposition for time-varying systems, 2017. +35 + diff --git a/59E5T4oBgHgl3EQfPQ7C/content/tmp_files/load_file.txt b/59E5T4oBgHgl3EQfPQ7C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..222bd0696bd5c17cdc971d98c7a8104a46030e78 --- /dev/null +++ b/59E5T4oBgHgl3EQfPQ7C/content/tmp_files/load_file.txt @@ -0,0 +1,1307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf,len=1306 +page_content='Combining Dynamic Mode Decomposition with Ensemble Kalman Filtering for Tracking and Forecasting Stephen A Falconer1, David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Lloyd1, and Naratip Santitissadeekorn1 1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK January 16, 2023 Abstract Data assimilation techniques, such as ensemble Kalman filtering, have been shown to be a highly effective and efficient way to combine noisy data with a mathematical model to track and forecast dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, when dealing with high-dimensional data, in many situations one does not have a model, so data assimilation techniques cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In this paper, we use dynamic mode decomposition to generate a low-dimensional, linear model of a dynamical system directly from high-dimensional data, which is defined by temporal and spatial modes, that we can then use with data assimilation techniques such as the ensemble Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We show how the dynamic mode decomposition can be combined with the ensemble Kalman filter (which we call the DMDEnKF) to iteratively update the current state and temporal modes as new data becomes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We demonstrate that this approach is able to track time varying dynamical systems in synthetic examples, and experiment with the use of time-delay embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We then apply the DMDEnKF to real world seasonal influenza-like illness data from the USA Centers for Disease Control and Prevention, and find that for short term forecasting, the DMDEnKF is comparable to the best mechanistic models in the ILINet competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Keywords Dynamic mode decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Ensemble Kalman filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Data-driven modelling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Data assimilation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Dynamical systems 1 Introduction Data assimilation refers to the collection of methods that integrate vast data sets with sophisticated mathematical models, to track and forecast systems that may evolve or change [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The majority of its applications lie in the earth sciences [51], however due to the generality of its techniques they have also been successfully applied in a wide range of areas from medicine [18] to ecology [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Kalman filter [36] is one such data assimilation technique widely used throughout industry [3] that optimally combines predictions from a linear model with Gaussian data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Whilst traditionally applied to a model’s state, the parameters of the model can simultaneously be filtered, leading to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05504v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='DS] 13 Jan 2023 what is known as the joint state-parameter estimation problem [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' If the system being filtered is nonlinear, alternative versions of the Kalman filter can be utilized such as the extended Kalman filter [53], unscented Kalman filter [59] or ensemble Kalman filter (EnKF) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The EnKF represents the distribution of a system’s state with an ensemble of random samples, that can then be used to estimate useful statistics like the state’s covariance via the sample covariance or a point estimate of the state via the sample mean [23] and is well-suited for high-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' All of these methods require a model of the system, however if no model exists then one must be generated and the most generalizable way to do this is via data-driven modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Dynamic mode decomposition (DMD) is a data-driven modelling technique for identifying low di- mensional, spatial and temporal patterns within a dynamical system directly from high-dimensional data [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' It does this by postulating the state vector is evolved via a linear system and looking for a low-dimensional approximation of the eigenvalues (temporal modes) and corresponding eigenvec- tors (spatial modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Spatial modes can be thought of as modes that decompose state variables into separate components that evolve together linearly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The corresponding temporal modes describe whether a spatial mode is growing, decaying or stationary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' DMD has been used to approximate dynamical systems from measurement data in a multitude of fields, ranging from epidemiology [49], finance [43] and neuroscience [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Due to its popularity, it has been extended to systems that are nonlinear in their recorded measurement functions via Extended/Kernel DMD [61]/[44], with one such extension Hankel-DMD [2] employing time-delay embeddings of the original observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the presence of measurement noise, the standard DMD has been shown to induce a systematic bias by asymmetrically attributing all noise to the model’s target output measurements and none to its inputs during training [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This systematic bias, prompted the creation of noise handling variants of DMD that directly account for the noise term [15], the Forward Backward DMD [15] that performs DMD forwards and backward in time and combines the results, and To- tal DMD (TDMD) [31] that minimizes the total least squares error as opposed to minimizing the ordinary least squares error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The aim of this paper is to develop an algorithm that iteratively improves the temporal modes (eigenvalues) and state estimates produced by DMD with the EnKF as new data becomes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This would be highly useful for dynamical systems that make a change from growing or decaying behaviour over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' While estimating just the state of the system using the DMD modes can be done using a standard Kalman filter, without also filtering the model’s temporal mode, estimates are likely to suffer if the system changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Methods already exist that combine DMD with the Kalman filter [45] or extended Kalman filter [46], which apply filtering to estimate the entire system dynamics matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The filtering in our work is instead focused on efficiently tracking the system’s temporal modes, and forecasting the system’s future states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' DMD produces a linear model which makes it a natural fit for the Kalman filter, however when a system’s state and temporal modes are estimated simultaneously the filtering process becomes nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, we need to use a filter designed for a nonlinear model, and we chose the EnKF due to its versatility, scalability to large dynamical systems, and ease of implementation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' While any DMD variant that produces temporal modes would be compatible with the DMDEnKF framework, we use TDMD to remain consistent with the EnKF’s assumption that noise is present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In tandem, we apply the DMDEnKF using a total least squares version of Hankel-DMD, henceforth referred to as the Hankel-DMDEnKF, to investigate the effect time-delay embeddings have on our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 2 To demonstrate the DMDEnKF method, we first test it on synthetically generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Ini- tially, on a simple noisy oscillating system with a decreasing period of oscillation, we use the DM- DEnKF to track the system’s temporal modes and compare results with the Hankel-DMDEnKF, other iterative DMD variants, and “gold standard” filtering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Next, we simulate a pan- demic and evaluate the DMDEnKF’s ability to track the system’s temporal modes and generate multistep ahead forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 1: ILI consultations as a percentage of total weekly GP consultations in the US from 2003 to end of 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The data shows the annual peaks in ILI consultations that vary in size, timing and shape, which would make them difficult to model with a simple SIR-type model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Finally, we apply the DMDEnKF and Hankel-DMDEnKF to real seasonal influenza-like illness (ILI) data in the United States from the Centers for Disease Control and Prevention (CDC) ILINet [25] shown in Figure 1, with the aim of investigating their forecasting skills for ILI consultation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' ILI is defined as a fever with a cough or sore throat that has no known cause other than influenza [25] and infects between 9 and 35 million people in the US each year [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Due to its prevalence, a multitude of methods have already been developed to model the spread of ILI [14, 47] and the approaches these models take can broadly be classified as either mechanistic or statistical [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Mechanistic methods [5, 48] make explicit hypotheses about what is driving the spread of an infectious disease, before then fitting parameters in the proposed models to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' They have the advantage of being highly interpretable making them useful when trying to understand how one can control the spread of a disease [39], however can make assumptions that are oversimplified [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For example, a simple SIR-type model would struggle to describe specific behaviours like the drop in ILI consultations around Christmastime seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Statistical methods [11, 60] are generally more versatile as they require fewer domain-specific assumptions, but both methods achieve a similar predictive skill in real time on the ILINet dataset [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF attempts to find a middle ground between the two methods, remaining versatile by virtue of being purely data-driven but also providing some level of interpretability via the associated DMD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The remainder of this paper will be structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' First, a brief summary of DMD, Hankel- 3 8 7 6 4 三 %3 2 1 2004 2006 2008 2010 2012 2014 2016 2018 DateDMD and EnKF algorithms for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' After which, the DMDEnKF algorithm will be described in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We will then apply the DMDEnKF and Hankel-DMDEnKF to synthetic data and compare their performance against other pre-existing, iterative DMD variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Finally, we will use the DMDEnKF and Hankel-DMDEnKF on ILINet data to forecast the rate of ILI consultations up to 4 weeks into the future and examine their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 2 DMDEnKF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 Dynamic Mode Decomposition (DMD) Consider an n dimensional state xk ∈ Rn measured at regular time intervals k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Assuming this time-series data was generated by a linear dynamical system, the consecutive states xk and xk+1 are connected via xk+1 = Axk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1) for some unknown matrix A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By denoting X = � �� | | | x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm−1 | | | � �� , X′ = � �� | | | x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm | | | � �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2) equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1) can be written succinctly over all consecutive data pairs as X′ = AX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3) To minimize the mean squared error term �m−1 k=1 ∥xk+1 − Axk∥2 2, the standard DMD defines A = X′X+, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4) where X+ is the Moore-Penrose pseudoinverse [6] of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Efficiently solving for the eigendecom- position of A is the primary purpose of DMD, as these eigenvalues/eigenvectors correspond to spatio-temporal patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMD method starts by applying the Singular Value Decomposition (SVD) to the data matrix X, representing it as the matrix multiplication of 2 real-valued, orthonormal matrices (complex and unitary if X ∈ Cn×m) U ∈ Rn×n, V ∈ Rm×m and a rectangular diagonal matrix with decreasing non-negative real values (Σ ∈ Rn×m) in the form X = UΣV∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) The best rank r approximation of a matrix according to the Eckart-Young Theorem [22] is obtained by truncating its SVD, hence by truncating equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) to a suitable rank r [26] we can compress the data matrix with minimal loss of information, which we write as X ≈ UrΣrV∗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6) By performing this compression, we are implicitly assuming that there exists a low dimensional (≤ r), linear structure within the high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 4 The Moore-Penrose pseudoinverse can be found directly from the SVD computed in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) as VΣ−1U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We use the rank r truncated matrices from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6) for reasons of efficiency, setting A = X′X+, ≈ X′VrΣ−1 r U∗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='7) This approximation of A now acts only on an r dimensional subspace defined by Col(Ur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, we can restrict A onto this r dimensional subspace (representing the largest r POD modes of X) and denote the restricted A ∈ Rr×r as ˜A = U∗ rAUr, ≈ U∗ rX′VrΣ−1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='8) We calculate the eigenvalues (λi), and corresponding eigenvectors (vi) of ˜A, and define Λ = � ��� λ1 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 0 0 0 λr � ��� , W = � �� | | | v1 v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' vr | | | � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9) Reconstructing the eigenvalues/eigenvectors of the original operator A will provide insights into the structure of the system [1] and allow us to propagate it forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The eigenvalues of ˜A (Λ) can be shown to be equal to the eigenvalues of the original operator A [34], however recovering the original eigenvectors is more involved and can be done using either projected or exact DMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We use the exact DMD method introduced by Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' [34] as it finds the exact DMD modes (Φ) for all eigenvectors with non-zero λi, where Φ is defined as Φ = X′VrΣ−1 r W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10) DMD modes with zero eigenvalues have no effect on the system’s dynamics, so this restriction of exact DMD is of little consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This method finds A such that AX = X′ exactly provided r ≥ rank(X) and X and X′ are linearly consistent [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' With Λ and Φ in hand, we can construct a r dimensional approximation of A, however still need to find the initial phase and amplitude of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The standard method [54] for computing this vector (b) is to rewrite the initial state x1 in a basis of the DMD modes via b = Φ+x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='11) It is worth noting that there exist alternative methods for example [13, 35] that focus on optimizing b over all data points with additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To summarise, the final solution to the discrete system can be written as xk = ΦΛkb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='12) In the remainder of the paper, we call Λ the temporal modes and Φ the spatial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 Hankel-DMD Hankel-DMD first augments the original, measured state xk ∈ Rn, by appending to it measurements of the state at the previous d − 1 time steps h(xk) = � xkT xk−1T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xk−(d−1)T �T , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='13) to form a new state h(xk) ∈ Rdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is known as a time-delay embedding, and we refer to d as the delay-embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Taking time-delay embeddings, h(xk), to be our new states, matrices X and X′ from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2) now become X = � ��� xd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm−d � ��� , X′ = � ��� xd+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' xm−(d−1) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='14) With X and X′ defined above, Hankel-DMD proceeds exactly as the standard DMD algorithm, generating eigenvalues Λ, DMD modes Φ and their initial states b as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The original system can be reconstructed/forecast for all time steps from d onwards, by applying equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='12) and restricting the result to the first n rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 Iterative DMD Variants There exists other variants of DMD that are designed to be applied iteratively, and in this paper we will compare these with the DMDEnKF in their ability to track a system’s eigenvalues and make future state predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Streaming DMD [32] is an adaption of the standard DMD algorithm to efficiently process new data as it becomes available, and the noise aware variant Streaming TDMD [30] is the first variant we wish to compare against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The second method we will use for comparison is Windowed DMD [62], where the standard DMD described above is applied over a sliding window of the w most recent data snapshots only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The final method we will be comparing against is Online DMD [62], specifically the variant of this algorithm that places an exponentially decaying weight ρ on the importance of past measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 Ensemble Kalman Filter (EnKF) Consider a discrete-time, nonlinear dynamical system with a stochastic perturbation xk = F(xk−1) + wk, wk ∼ N(0, Qk), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15) where F is a nonlinear function F : Rn → Rn, xk ∈ Rn is the system’s state, wk ∈ Rn is a stochastic perturbation and N is the normal distribution with mean 0 and covariance matrix Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A measurement equation that relates what we observe to the true state of the system is given by yk = H(xk) + vk, vk ∼ N(0, Rk), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16) where H : Rn → Rl is the system’s observation operator, yk ∈ Rl is an observation of the system, vk ∈ Rl is the noise in the observation and N is the normal distribution with mean 0 and covariance matrix Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We focus on the instance relevant to our use case where H is linear, so can be represented by a matrix H ∈ Rl×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 6 In general, filtering methods aim to combine information from the state-transition model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15) and observation model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16) to compute the conditional density p(xk|Yk), where Yk = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Kalman filter is the optimal filter if F and H are both linear and the stochastic perturbations are normal [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The EnKF was developed to deal with the filtering problem where either the linear or normal assumption (or both) is violated [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' It exploits the Kalman formulation to propagate an ensemble of the state into a region of high probability in such a way that the ensemble spread would be consistent with the linear and normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To begin the EnKF algorithm, an initial ensemble of N state estimates ˆx(1) 0 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', ˆx(N) 0 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' If an ensemble is not available, one can be generated from initial state estimates ˆx0 and covariance matrix P0 by taking N independent draws from N(ˆx0, P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Algorithm The EnKF then acts as follows [23]: Step 1: Propagate forward in time each ensemble member using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', N via ˆx(i) k|k−1 = F(ˆx(i) k−1|k−1) + w(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='17) The notation ˆx(i) k|k−1 denotes the state estimate at time k of the ith ensemble member ˆx(i) k using only information up to time k − 1, and ˆx(i) k−1|k−1 represents the same ensemble member at time k − 1 using information up to time k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Each w(i) k is independently drawn from N(0, Qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The current covariance matrix can also now be estimated via the sample covariance of the ensemble, which we denote as ˆPk|k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This can then be used to estimate the Kalman Gain matrix ˆKk as ˆKk = ˆPk|k−1HT (HˆPk|k−1HT + Rk)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='18) Step 2: Calculate the measurement innovation utilizing equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' From measurement yk, we again use i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', N and generate simulated measurements y(i) k = yk + v(i) k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='19) where each v(i) k is an independent draw from N(0, Rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' These simulated measurements y(i) k are combined with the ensemble members ˆx(i) k|k−1 from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='17) to define N measurement inno- vations e(i) k = y(i) k − Hˆx(i) k|k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20) The e(i) k represent samples from the distribution of the distance of the model’s prediction from the measured value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Step 3: Combine the model estimates in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='17) and measurement innovation of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20) via the estimated Kalman gain from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='18) to update each ensemble member’s state estimate ˆx(i) k|k = ˆx(i) k|k−1 + ˆKke(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='21) We can generate a point estimate for the state ˆxk using the mean of the N updated ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This process then repeats every time a new state measurement becomes available, with the updated ensemble from the previous data point becoming the initial ensemble for the new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We combine these 2 previously described techniques to form the DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This new, hybrid method uses DMD to generate a low dimensional model of a dynamical system that is then itera- tively improved by the EnKF as new data emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 DMDEnKF We now describe how we carry out filtering of the temporal modes and state of the system, while keeping the spatial modes found by one’s chosen version of DMD on the “spin-up” fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We note that once we allow the temporal modes to vary with the spatial modes being fixed, these are no longer eigenvalues/eigenvectors, and we then call them temporal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Consider an n dimensional state xk ∈ Rn measured at regular time intervals k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', m and then measured iteratively at times k = m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='. Algorithm Step 1: Perform the chosen version of DMD on the dataset x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', xm, defining X, X′ as before in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2) to obtain the expression xk = ΦΛkb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='22) where Λ = � ��� λ1 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 0 0 0 λr � ��� , Φ = � �� | | | d1 d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' dr | | | � �� , b = � ��� b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' br � ��� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='23) and defining λi, di, bi as the ith temporal mode, DMD mode, initial condition triplet of the r retained modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This acts as a spin-up process to generate a model we can then filter using the EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Step 2: Define the matrices required for the EnKF’s ensemble initialisation, propagation via equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15), and measurement using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' First, rewrite each of the r temporal modes in polar coordinates as λi = τieθii, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='24) where τi ≥ 0, 0 ≤ θi < 2π and i2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As xk ∈ Rn, the temporal modes in the DMD model’s spectrum will either be real or in a complex conjugate pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' When filtering, we view the temporal modes as a time varying parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, we must enforce that the real temporal modes remain real and complex conjugate pairs remain intact, as this ensures the state output by the model will still be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We do this by defining the filterable model parameters µi as new variables for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', r µi = � τi, if θi = 0, or ∄ j for j < i such that λ∗ j = λi, θi, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='25) Written in this way, these µi’s represent all the possible degrees of freedom in the model’s temporal modes under the additional constraint of producing a real state estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By maintaining a note of the positional indexes of each complex conjugate pair produced in the initial DMD, it is possible to recover the λi representation from the µi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' While this transformation technically requires the full list of µi’s, we informally write Λ(µi) = λi to symbolize the reversion from µi’s back to λi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Ensemble initialisation: We can now define an augmented joint parameter state z0 ∈ Rn+r to be used as the initial state for the EnKF z0 = � − xm − µ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' µr �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='26) 8 We denote the joint parameter state at time m + k as zk ∈ Rn+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To generate an initial ensemble from this state, we first define sample covariance C = (1/m)(X′ − ΦΛΦ+X)(X′ − ΦΛΦ+X)T , which represents the current state uncertainty based on prediction errors in the spin-up DMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We then form the initial covariance matrix P0 = � C 0 0 α2Ir � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='27) where α2 > 0, Ir is the r-dimensional identity matrix, and the α2Ir term determines the initial uncertainty in the spin-up DMD’s temporal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Take independent draws from N(z0, P0) until the ensemble is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The optimal ensemble size will vary from problem to problem, adding ensemble members will increase accuracy but at the cost of computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Propagation: Using the notation zi k to signify the ith element of zk, we define the matrix Λzk ∈ Rr×r for state zk as Λzk = � ��� Λ(zn+1 k ) 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 0 0 0 Λ(zn+r k ) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='28) The EnKF’s propagation equation can be written as zk+1 = � ΦΛzkΦ+ 0 0 Ir � zk + wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='29) For convenience, we introduce notation zi:j k ∈ Rj−i+1 to denote the ith through to the jth element of zk where i ≤ j zi:j k = � zi k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' zj k �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='30) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='29) propagates z1:n k representing the state in the DMD framework xm+k forward in time using the standard DMD equation with the updated temporal modes from Λzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The vector zn+1:n+r k represents the current estimate of the temporal modes in their µi representation and is unchanged other than the addition of noise by the propagation equation, for although we assume the temporal modes vary in time no direction of drift in the parameters is explicitly foreknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The vector wk ∈ Rn+r is a normally distributed variable wk ∼ N(0, Qk), and this represents the uncertainty within the model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We construct Qk as follows, Qk = � α1In 0 0 α2Ir � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='31) where α1 and α2 are constants determined by the user such that α2 ≪ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This construction with Qk a diagonal matrix assumes model errors for each element of zk are uncorrelated with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The condition α2 ≪ α1 ensures that the state of the DMD system z1:n k changes significantly faster than its temporal modes zn+1:n+r k , as parameters by definition should vary slowly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Furthermore, for the temporal mode’s moduli being filtered, it prevents the strictly positive modulus dropping below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Measurement: We write the EnKF’s measurement equation as yk = � In 0 0 0 � zk + vk, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='32) 9 where yk ∈ Rn are observations of the DMD state xm+k, and vk ∈ Rn is a normally distributed variable ∼ N(0, Rk) representing the noise in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We assume new measurements yk to be available for the full DMD state z1:n k but not its temporal modes zn+1:n+r k , as this is consistent with the format of the data used to generate the spin-up DMD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We also assume uncorrelated measurement noise on each dimension of the state, so choose a diagonal matrix Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Step 3 State measurements xm+k at times k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' are being iteratively generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By setting yk = xm+k as each new measurement arrives, we can iteratively apply the EnKF to produce a hybrid estimate for zk that combines model predictions from zk−1 and noisy measurement yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A brief summary of how the EnKF does this is provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2, and a more expansive description can be found at [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Step 4: The state of the original system xm+k can be reconstructed from zk by simply taking it’s first n elements z1:n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Predictions p steps ahead at time m + k can also be forecast from zk via xm+k+p = ΦΛp zkΦ+z1:n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='33) The Hankel-DMDEnKF is defined algorithmically in exactly the same way, with the only difference being that Hankel-DMD is applied over the “spin-up” period as opposed to standard DMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 3 Synthetic Applications 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 Comparison against other iterative DMD variants To test the DMDEnKF, we first apply it to data generated from a synthetic system with time varying eigenvalues, which we aim to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The dynamics of this system are governed by the 2 dimensional rotation matrix, where the angle of rotation θk increases linearly from π/64 to π/8 over the course of 500 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The evolution of the state xk of the system can hence be written as xk+1 = � cos(θk) − sin(θk) sin(θk) cos(θk) � xk, x1 = � 1 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1) where θk = π/64 + (k−1)(7π/64) 499 and k = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We assume noisy measurement values yk to be available for the state at each time step, such that yk = xk + vk, vk ∼ N(0, σ2I2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2) where each experiment σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 to simulate a low or high level of measurement noise respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The 500 values of yk (shown in Figure 2) are used to train the DMDEnKF and Hankel-DMDEnKF, with the first 100 time steps being used for the spin-up process described in Step 1 of the DMDEnKF algorithm to produce the output described in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We will also train the iterative variants of DMD described at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 (Streaming TDMD1, Windowed DMD and Online DMD) on this dataset to compare their ability to track the 1As the synthetic dataset is small, it is computationally tractable to apply batch methods to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, instead of applying the true Streaming TDMD algorithm, we use batch TDMD over all data up to the current time step as a proxy for Streaming TDMD utilizing code from the PyDMD library [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As Streaming TDMD approximates the results of TDMD with the only differences occurring due to additional data compression steps in Streaming TDMD’s algorithm, we believe this to be an acceptable substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 10 Figure 2: Time series for a synthetic system with a linearly increasing eigenvalue argument, showing the state’s first dimension with no, low (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05) and high (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' system’s time varying eigenvalues against that of the DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Within the Windowed DMD algorithm, we replace DMD with TDMD to allow for this method to effectively handle the noise in the data, henceforth referring to this amalgamation of the two methods as Windowed TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To implement Online DMD, we use code made available by its creators here [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Computational parameters were set as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' window size w = 10 for Windowed TDMD, exponential decay rate ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9 for Online DMD, delay-embedding dimension d = 50 for the Hankel-DMDEnKF and spin-up time steps m = 100 for the DMDEnKF as previously stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At each time step k, the system’s true eigenvalues can be written in modulus-argument form as λk = 1e±θki, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3) and for each time step where the models are defined their estimates of the system’s eigenvalues can also be written as ˆλk = ˆτke±ˆθki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4) We start by comparing the errors in each method’s estimate of the constant eigenvalue modulus (ˆτk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A thousand runs of the synthetic data were generated for each value of σ, and the difference of each method’s eigenvalue modulus and argument from their true values at every time step after the spin-up period were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' When any of the methods failed to identify the eigenvalues as a complex conjugate pair at a given time step in a run, the dominant eigenvalue’s modulus was used for modulus error calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The average errors in the eigenvalue modulus estimates are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For all levels of measurement noise, Streaming TDMD estimated the eigenvalue modulus the most accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is due to the method’s assumption of a stationary system, hence assigning an equal weight to the importance of each data point, which works well in the case of estimating a constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At low levels of measurement noise as seen in the first column of Table 1, Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly 11 2 dimension 0 1 XkTrue State yk for = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 2 yk for o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 0 100 200 300 400 500 TimestepsIterative DMD Mean Eigenvalue Modulus Error Variant σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Windowed TDMD 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='82 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='39 Online DMD 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='04 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='06 × 10−1 Streaming TDMD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='31 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='50 × 10−3 DMDEnKF 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='07 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='89 × 10−2 Hankel-DMDEnKF 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='49 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='38 × 10−2 Table 1: Mean absolute errors in the synthetic system’s eigenvalue modulus estimates produced by each iterative DMD variant over all time steps over the course of all 1000 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Measurement noise is set to either low levels with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 (left) or high levels with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Streaming TDMD scored significantly lower errors than all other methods, and as noise levels increased, errors in Windowed TDMD and Online DMD grew significantly larger than those produced by the DMDEnKF and Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' well with mean eigenvalue modulus errors below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As errors in the eigenvalue modulus grow exponentially when forecasting future states, these 4 methods could produce acceptable short term forecasts but would quickly diverge from the true state as the forecast horizon was extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At high levels of noise shown in the second column of Table 1, Windowed TDMD and Online DMD’s eigenvalue modulus estimates degrade significantly, making them unsuitable for forecasting in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The errors in the DMDEnKF and Hankel-DMDEnKF remain fairly small, however are still an order of magnitude greater than those produced by Streaming TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A typical trajectory of the eigenvalue argument estimates (ˆθk) for each method over the course of one run from the end of the spin-up period onwards can be seen in Figures 3a and 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The error distributions for each method’s eigenvalue argument estimates (ˆθk − θk) over all 1000 runs are plotted in Figures 3b and 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At low levels of noise as seen in Figures 3a and 3b, all 5 methods on average underestimated the eigenvalue argument of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is to be expected as the eigenvalue argument is increasing with time, meaning that all but the last data pair available to each method would have been generated using an argument smaller than its current value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Streaming TDMD exhibited the worst performance, again due to its equal weighting of every data point, however in this instance being a negative quality as it hampers the model’s ability to adapt to fresh data that reflects the changing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Windowed TDMD, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Online DMD produced a tighter error distribution, but with a slightly larger bias than Windowed TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This suggests that Online DMD’s soft thresholding reduces the model volatility caused by measurement noise compared to the hard cut-off employed by Windowed TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For this same reason however, Online DMD is slower to adapt to new measurements than Windowed TDMD, leading to a larger bias below the system’s true eigenvalue argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF and Hankel-DMDEnKF performed very similar to Windowed TDMD at this noise level, however tweaks to the magnitude of the DMDEnKF’s system uncertainty matrix can be made to balance the speed of model innovation with its volatility and produce distributions closer to that of Online DMD if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At higher noise levels shown in Figures 3c and 3d, the performance of Windowed TDMD and Online DMD significantly degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Placing a larger weight on more recent samples allowed these methods 12 (a) Eigenvalue argument trajectory for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) Error distribution for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (c) Eigenvalue argument trajectory for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (d) Error distribution for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 3: Estimates of the synthetic system’s eigenvalue argument produced by each iterative DMD variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Presented are typical trajectories of the eigenvalue argument at each time step over the course of 1 experiment’s run (left) and error distributions of the difference between the true system’s eigenvalue argument and the estimated eigenvalue argument over all time steps over the course of all 10 runs (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Measurement noise is set to either low levels with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 (top) or high levels with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF and Hankel-DMDEnKF experience similar errors to Online DMD and Windowed TDMD at low measurement noise, but track the eigenvalue argument much more accurately than them for high measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' to quickly adapt to changes in the system’s parameters, however as the noise increases this induces an extreme volatility in their respective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The performance of Streaming TDMD is not largely changed from the low noise case, still lagging behind the true system values but somewhat insulated from the noise by its symmetric treatment of all data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Here the benefit of explicit inclusion of measurement noise in the DMDEnKF framework becomes apparent, as at this noise level the DMDEnKF and Hankel-DMDEnKF are the only techniques tested capable of producing an accurate eigenvalue argument estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Furthermore, here we see the first significant difference in the performance of the DMDEnKF and Hankel-DMDEnKF, as the DMDEnKF’s error distribution has a thin tail extending down to −π/8 which is not present in the error distribution of the Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' These additional errors are caused by the spin-up of the DMD for the DMDEnKF method occasionally failing to identify the system’s eigenvalues as a complex conjugate pair (empirically, this happens ∼ 3% of the time), due to the increased noise in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' When this happens, the DMDEnKF catastrophically fails for the EnKF is unable to generate complex eigenvalues from real ones regardless of how many future time steps are filtered due to its formulation in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This failure of the DMDEnKF can be mitigated in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' If the errors produced by the DMDEnKF during the filtering stage are deemed too large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' exceed a given threshold) for a prolonged period of time, then the spin-up DMD process can be rerun on an extended dataset consisting of the original spin up data, 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 True Eigenvalue Streaming TDMD igenvalue Argument 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 Windowed TDMD Online DMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 Hankel-DMDEnKF DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 100 200 300 400 500 TimestepsStreaming TDMD 70 WindowedTDMD 60 Online DMD Hankel-DMDEnKE 50 DMDEnKF Density 40 30 20 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 Distance from True Eigenvalues Argument0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 TrueEigenvalue Streaming TDMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 Windowed TDMD OnlineDMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 Hankel-DMDEnKF DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 100 200 300 400 500 TimestepsStreaming TDMD 70 Windowed TDMD 60 Online DMD Hankel-DMDEnKF 50 DMDEnKF Density 40 30 20 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 Distance from True Eigenvalues Argumentplus the newly available data used so far in the filtering step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By including more data in the spin-up process, the spin-up DMD model is more likely to successfully capture the signal component in the data as a pose to measurement noise, and hence produce eigenvalues with the same structure as those of the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Time-delay embeddings make the SVD step in the DMD algorithm more robust to measurement noise [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, while the Hankel-DMDEnKF is similarly restricted by the eigenvalues it can produce at the filtering stage, in all 1000 runs of the synthetic data the spin up Hankel-DMD was able to identify the system’s eigenvalues to be a complex conjugate pair, so this was not an issue for the Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 Comparing against DMD with a particle filter Having compared the performance of the DMDEnKF against other iterative DMD variants, we now focus on evaluating the filtering component of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Since the linear DMD model acts nonlinearly in the filter when applied to both the model’s state and eigenvalues, we compare the EnKF filter with a particle filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Particle filters [27] have been show to converge to the optimal filter as the number of particles tends to infinity for general nonlinear models with non-Gaussian noise [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, particle filters are restricted to low dimensional systems only, as the number of particles required scales approximately exponentially with the dimension of the state [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, we compare the DMDEnKF and Hankel-DMDEnKF with a DMD plus particle filter which we will take to be the “gold standard” estimation to assess how well the EnKF does with the nonlinear filtering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We use the same synthetic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1) with a linearly increasing eigenvalue argument as in the previous subsection to generate data with high levels of measurement noise (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' a trajectory of which can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Again, the time-delay embedding dimension d = 50 for the Hankel-DMDEnKF, and the first 100 time steps are used to train a spin-up DMD model, with the next 400 used to filter the state and spin-up model’s eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF’s filter state thus has dimension 4 (2 state dimensions and 2 temporal modes), while the Hankel-DMDEnKF’s filter state is of dimension 102 (100 state dimensions and 2 temporal modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To generate a “gold standard” solution, at the filtering step we use a particle filter with 10,000 particles, applying multinomial importance resampling [20] every time the effective sample size falls below half the number of particles to avoid sample degeneracy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For the DMDEnKF and Hankel-DMDEnKF at the filtering step, we run the EnKF with varying numbers of ensemble members (N), to see if as N increases their estimates mean and covariance will tend to that of the particle filter ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We generated 1000 runs of the synthetic data to apply the particle filter/EnKF with each value of N to and collected the errors in the eigenvalue argument estimates for each method at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As can be seen in Figure 4a, the DMD particle filter with 10,000 particles produces an extremely tight error distribution that is slightly biased to produce estimates below that of the true eigen- value’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is to be expected, as mentioned in the previous subsection, due to the system’s eigenvalue argument constantly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' There is also a thin tail in the error distribu- tion that extends down to −π/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is again a result of the spin up DMD sometimes failing to identify a complex conjugate eigenvalue pair, trapping the particle filter in the faulty assumption that the eigenvalues are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 14 (a) Error distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) Mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 4: Error distributions (left) and mean squared errors (right) for estimates of the synthetic system’s eigenvalue arguments produced by the DMDEnKF and Hankel-DMDEnKF with varying numbers of ensemble members (N) against those produced by a particle filter with 10,000 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Increasing N quickly leads to error levels in the DMDEnKF and Hankel-DMDEnKF that are similar to those produced by their respective “gold standards”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For low numbers of ensemble members (N = 5), the DMDEnKF and Hankel-DMDEnKF are centred at a similar value to the “gold standard”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, they produce a far larger spread with long tails in both directions that imply a lack of robustness with this few ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' With only a small increase to N = 10, both methods become more stable, as although they still have a larger variance than the particle filter, the long positive tails from N = 5 have been eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A similar pattern occurs as we move to N = 20, with more ensemble members resulting in a tighter error distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At this point, the Hankel-DMDEnKF’s distribution can be distinguished from that of the DMDEnKF and DMD particle filter by its aforementioned lack of a persistent thin negative tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By N = 40, the main peaks of the DMDEnKF, Hankel-DMDEnKF and “gold standard” are almost indistinguishable on the graphical scale, with the DMDEnKF and DMD particle filter both sharing a thin negative tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 4b shows how the mean squared error for the eigenvalue arguments predicted by the DM- DEnKF and Hankel-DMDEnKF are affected by varying the number of ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For the DMDEnKF, errors initially sharply decline as N is increased, however on this small synthetic system returns diminish quickly after N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By N = 50, we achieve a mean squared error with the DMDEnKF only ∼ 3% larger than that of the “gold standard”, despite using 200 times fewer particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' When comparing the Hankel-DMDEnKF to the “gold standard”, the errors in the DMD particle filter’s eigenvalue estimates are skewed by the runs in which the spin up DMD was unable to identify a complex conjugate eigenvalue pair, as Hankel-DMD did not encounter this problem on these synthetic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To attempt to fairly compare the filtering methods, we remove all runs in which the spin up DMD failed in this way, before again calculating the mean squared error for the DMD particle filter and recording it in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A similar pattern of reducing errors with diminishing returns can be seen for the Hankel-DMDEnKF as ensemble size is increased, and by N = 50 its mean squared error is within 5% of the newly calculated DMD particle filter’s score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Our results show that in this simple, synthetic case at least, the EnKF is an efficient and effective solution to the nonlinear filtering problem that arise within the DMDEnKF framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 15 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' of Ensemble Members = 5 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' of Ensemble Members = 10 DMDParticleFilter DMDParticle Filter 20 with10,000particles with10,000particles 15 Hankel-DMDEnKF Hankel-DMDEnKF DMDEnKF DMDEnKF 10 5 Densit No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' of Ensemble Members = 20 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' of Ensemble Members = 40 DMDParticleFilter DMDParticleFilter 20 with10,000 particles with10,000 particles 15 Hankel-DMDEnKF Hankel-DMDEnKF DMDEnKF DMDEnKF 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 Distance from TrueEigenvalues ArgumentHankel-DMDEnKF 10-1 DMDEnKF Mean Squared Error 10-2 DMD Particle Filter with 10,000 particles 10-3 DMD Particle Filter with 10,000 particles and failed runs removed 10 20 30 40 50 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' of Ensemble Members3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 Tracking a synthetically generated pandemic Lastly, we test the DMDEnKF’s performance on synthetic data designed to simulate a simple pandemic with a state xk representing the level of infection in 3 different population classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The system’s dynamics are governed by a matrix A ∈ R3×3 that we randomly generate with non- negative elements each being drawn from the Uniform distribution U[0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The (i, j)th element of A represents how the level of infection in class j at time k will affect the level of infection in class i at time k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To control whether the synthetic pandemic is spreading or dying off, we then define a new matrix ˆA = A λ1 where λ1 is the largest eigenvalue of A, thus ensuring the spectral radius ρ(ˆA) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By introducing a constant γ, we can replace A with γ ˆA causing the state to grow if γ > 1 or decay for γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To simulate a pandemic, we linearly decrease γ from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='99 over the course of the experiment’s 1000 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The initial state used is a vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The system that generates the synthetic data can be written as xk+1 = γk ˆAxk, x1 = � 1 1 1 �T , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) where the state xk ∈ R3, γk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='02(k−1) 999 and k = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We assume not to have access to the true state of the system xk but instead noisy measurements yk defined by yk = xk + vk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6) The constant σ that governs the level of measurement noise is set to σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 to represent low noise and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 for high noise as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 5 shows the values of the system’s three state dimensions and the respective available measurements over the course of one run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 5: Time series for a synthetic system that simulates a pandemic, showing all 3 dimensions of the state with no, low (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05) and high (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' All five DMD variants tested had their computational parameters set to the same values as those used in the synthetic experiments in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The only small difference was that Streaming TDMD, Windowed TDMD, the DMDEnKF and Hankel-DMDEnKF truncated the data by remov- ing the smallest singular value to reduce model instability caused by what was often a very low 16 16 Xk True State 14 ykforg=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 12 yk for g= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Values 10 8 State 6 4 0 0 200 400 600 800 1000 Timestepssignal-to-noise ratio in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Online DMD did not apply any truncation to the data as the method was not designed to do so, however it did not appear to suffer from any stability issues as a consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The first 100 measurements (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', y100) were used to initialize the models, and as each new data point (y100, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', y1000) was successively fed into the models, they produced 50 step ahead forecasts (ˆx150, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', ˆx1050).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We generate 1000 data points, however a standard flu season lasts around 20 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For this reason, we chose to forecast 50 steps ahead to mimic forecasting 1 week ahead in a more realistic timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The relative prediction errors ˆek = ∥xk−ˆxk∥ ∥xk∥ could then be calculated for k = (150, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 1000) and the mean of these errors was the main metric we used to evaluate the forecasting skill of each method over the course of one run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A thousand runs were performed for both low and high levels of noise and the empirical cumulative distributions of 50 step ahead forecast mean run relative errors for low noise (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05) can be seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 6: Cumulative error distributions of the mean run relative errors for the 50 step ahead forecasts of each iterative DMD variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Mean relative errors were calculated over all time steps for each run of the experiment, with the results from 1000 runs under low levels of measurement noise (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05) displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Forecast errors had a wide range for some methods, due to exponentially compounding errors caused by forecasting 50 steps ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF, Hankel-DMDEnKF and Online DMD produced errors orders of magnitude smaller than those of Streaming TDMD and Windowed TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The first noteworthy feature of the cumulative error distributions is the wide range in some method’s forecast errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is a result of the 50 step ahead forecasts being produced by training each model to forecast 1 step ahead, then applying the trained model to the data 50 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As such, forecast errors compound exponentially and small errors over a 1-step forecast horizon can become vast after 50 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Inspecting the individual methods, we see Windowed TDMD to be the worst performing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is due to its aforementioned instability under measurement noise caused by considering only a small subset of the data at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This instability could be reduced by increasing the window size (w) computational parameter, however as w increases the model’s ability to track a system that changes with time diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Streaming TDMD had the second- largest errors, caused by the method’s assumption of a stationary system hindering its ability to 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 Streaming TDMD Windowed TDMD Online DMD DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='8 Hankel-DMDEnKF Proportion of runs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 1010 1029 1048 1067 1086 10105 10124 10143 50 step ahead forecast mean run relative errorcorrectly adapt to the system’s changing eigenvalues as new data became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the majority of cases, Online DMD, the DMDEnKF and Hankel-DMDEnKF all performed similarly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' All three methods exhibited cumulative error distributions tightly concentrated around a low error value, however in a few runs, the DMDEnKF became unstable and produced large errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' It is clear even at low levels of noise that the forecasting performance of Online DMD, the DMDEnKF and Hankel-DMDEnKF are far superior on this type of system to those of Windowed TDMD and Streaming TDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, we now focus exclusively on these top three performing methods to allow for a thorough comparison of them on an appropriate error scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (a) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 (b) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Figure 7: Error distributions of the mean run relative errors for the 50 step ahead forecasts of Online DMD, the DMDEnKF and Hankel-DMDEnKF, attained over 1000 runs under low σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 (left) and high σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 (right) levels of measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Similar errors were found at both noise levels, with Online DMD performing better at low measurement noise and the DMDEnKF/Hankel-DMDEnKF performing better at high measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In Figure 7 for Online DMD, the DMDEnKF and Hankel-DMDEnKF, we plot the distributions of 50 step ahead forecast mean run relative errors at both low and high levels of measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At low levels of noise, Online DMD’s errors peak at a lower level than those of the DMDEnKF and Hankel-DMDEnKF, however as noise levels increase we see the peaks switch sides, and the DMDEnKF/Hankel-DMDEnKF become the better performing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At both noise levels, the peak in the DMDEnKF’s error distribution is centred at the same value as the Hankel-DMDEnKF’s peak, however it is less dense due to the additional probability mass stored in the long tail of the DMDEnKF’s error distribution, which is not present in that of the Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' These disproportionately large errors in the DMDEnKF distribution’s tail occur when the spin-up DMD process fails to produce a model similar enough to the system’s true dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As briefly touched upon in the first synthetic example, if the spin-up DMD model is sufficiently inaccurate then it can stop the EnKF from effectively assimilating new data, leading to the catastrophic failure of the DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In this example, as the signal-to-noise ratio in the direction of the second-largest singular value was often low, an unfortunate random draw of the system dynamics (A) and measurement noise (vk) in the spin-up period could produce large errors in DMD’s second eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Empirically, using the interquartile range method to detect outlying forecast errors, this DMDEnKF failure occurred 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5% of the time for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The errors would persist in the filtering step as new data was processed, whereas other methods were able to recover from poor initial model estimates more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The quality of the model produced by the initial DMD is dependent on the quality and volume of the spin-up data, hence this problem was exacerbated and occurred much more regularly at higher noise levels (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9% of the time for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' It could be 18 35 Online DMD Hankel-DMDEnKF 30 DMDEnKF 25 10 5 0 3 × 10-2 4 × 10-2 6 × 10-2 10-1 1052 1056 5o step ahead forecast mean run relative errorOnline DMD 16 Hankel-DMDEnKF 14 DMDEnKF 12 8 6 4 2 0 10-1 2 × 10-1 3 × 10-14× 10-1 6 × 10-1 1043 1047 5o step ahead forecast mean run relative errormitigated somewhat by increasing the number of time steps in the spin-up stage as described at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1, however similarly to Windowed TDMD as the system is assumed to be time varying there likely exists a point of negative returns once the spin-up period becomes too long due to the stationarity assumption of batch DMD becoming progressively more violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Unlike the DMDEnKF, the Hankel-DMDEnKF and Online DMD do not suffer from a long tail in their error distributions, and perform consistently well over all 1000 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At both noise levels, their error distributions have a similar variance, with the Hankel-DMDEnKF’s errors being slightly more tightly grouped than those of Online DMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, the average error is the main factor when differentiating between the method’s performance in this example, meaning Online DMD is the preferred method at low noise and the Hankel-DMDEnKF (or DMDEnKF provided the spin-up DMD does not catastrophically fail) is more accurate at high noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As both methods posses useful yet differing attributes, we generate a typical data trajectory (one for which the DMDEnKF does not fail) for both low and high measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We then investigate how each model’s 50 step ahead forecasts and dominant eigenvalue estimates change over the course of each run, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (a) 50 step ahead forecasts for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) Dominant eigenvalue’s modulus for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (c) 50 step ahead forecasts for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (d) Dominant eigenvalue’s modulus for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 8: Typical trajectories of the 50 step ahead forecasts for the value of the state’s first dimension (left) and estimates of the dominant eigenvalue’s current modulus (right) under low (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05) and high (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5) levels of measurement noise for Online DMD, the DMDEnKF and Hankel-DMDEnKF over the course of 1 run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Online DMD forecasts 50 steps ahead more accurately at low noise, and the DMDEnKF/Hankel-DMDEnKF more accurately at high noises, however when signal-to-noise ratio is low (at the start and end of the experiment) Online DMD’s eigenvalue estimates become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' First, observing the low noise forecasts in Figure 8a, it is clear Online DMD produces forecasts that are more robust and closer to the true state’s value than those of the DMDEnKF and Hankel- DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This was to be expected, by virtue of Online DMD’s lower average errors in the error distributions of Figure 7a at this noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As noise is increased, the forecasts in Figure 8c show the DMDEnKF and Hankel-DMDEnKF becoming the more accurate methods, however Online 19 14 Xk True State Online DMD 12 Hankel-DMDEnKF State in dimension DMDEnKF 10 8 6 4 2 200 400 600 800 1000 Timesteps1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='010 True Eigenvalue Mod OnlineDMD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='005 Hankel-DMDEnKE DMDEnKF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='985 200 400 600 800 1000 Timesteps18 XkTrueState 16 OnlineDMD 14 Hankel-DMDEnKE dimension DMDEnKF 12 10 8 in State i 6 4 0 200 400 600 800 1000 TimestepsModulus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='00 PT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='99 TrueEigenvalueMod 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='98 OnlineDMD Hankel-DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='97 DMDEnKF 200 400 600 800 1000 TimestepsDMD’s forecasts remains fairly stable, and still appear to be a viable forecasting option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Analysing the eigenvalue estimates in Figures 8b and 8d, we see that over the middle section of data where k = (250, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 750), Online DMD is able to track the dominant eigenvalue effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, at the beginning and end of the dataset when states and hence the signal component of each new data point is small relative to the measurement noise, Online DMD’s eigenvalue estimates become progressively more unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the low noise case this is not a problem, as Online DMD’s estimates are significantly more accurate than those of the DMDEnKF/Hankel-DMDEnKF, so even in the poorly performing sections of the data it’s estimates still better/match those of the DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For higher noise however, Online DMD provides significantly less robust estimates of the dominant eigenvalue at the start and end of the datasets than those generated by the DMDEnKF and Hankel- DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the epidemiological context of an infectious disease outbreak, which this synthetic example attempts to mimic, scientists will often try to calculate the basic reproduction number (R0) [58] using noisy data from the small number of initial infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' If R0 > 1 the number of infections will grow exponentially if left unchecked, and if R0 < 1 the number of infections will decay naturally to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Within this example, using the DMDEnKF/Hankel-DMDEnKF one can quickly determine that initially R0 > 1 and take any required action thanks to the stability of it’s early eigenvalue estimates, whereas it takes significantly longer and a higher level of infection for Online DMD to consistently determine if R0 is above or below the growth/decay threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 4 Seasonal Influenza-like Illness Application 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 Problem setup DMD based methods have previously been applied to infectious disease data [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In this case, DMD modes can be viewed as stationary, spatial modes used to create a reduced order model in which only the amplitudes and frequencies are time varying [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, modelling influenza-like illness (ILI) data is a prime potential application for the DMDEnKF/Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The CDC’s ILINet data [25] we will be using records the number of ILI General Practitioner (GP) consultations in the US each week, alongside the number of total GP consultations which can be used to normalize the ILI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We use a subset of the data from the start of 2003, the first year when data is available all year round, to the end of 2018 as seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We then split each week’s data into demographics, consisting of 4 age groups (0-4, 5-24, 25-24, 65+) and 10 Health and Human Services (HHS) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Each region consists of the following locations: Region 1 - Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 2 - New Jersey, New York, Puerto Rico, and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Virgin Islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 3 - Delaware, District of Columbia, Maryland, Pennsylvania, Virginia, and West Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 4 - Alabama, Florida, Georgia, Kentucky, Mississippi, North Carolina, South Carolina, and Tennessee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 5 - Illinois, Indiana, Michigan, Minnesota, Ohio, and Wisconsin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 20 Region 6 - Arkansas, Louisiana, New Mexico, Oklahoma, and Texas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 7 - Iowa, Kansas, Missouri, and Nebraska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 8 - Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 9 - Arizona, California, Hawaii, and Nevada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 10 - Alaska, Idaho, Oregon, and Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Whilst ILI consultation data is available over all 40 of these strata, total GP consultation data is only provided by region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To generate an age breakdown for a region’s total consultations we linearly interpolate using census data to approximate the US population’s age demographics for a given week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We then allocate the region’s total consultations to each age group based on the proportion of the total population they represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This method assumes that all age groups have a similar likelihood of attending the GP’s, which may be flawed but we believe it to be sufficient for the purpose of demonstrating the DMDEnKF on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 Building the spin-up DMD model The format of the ILI data used in the DMDEnKF is thus a 40 dimensional vector for each week, recording ILI consultations as a percentage of total GP consultations over every demographic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This data exists in R40 + however DMD computes modes in R40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, to ensure the model’s estimates are non-negative, we first transform the data by adding a small constant (c = 1) then taking the natural logarithm of each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For the Hankel-DMDEnKF, this transformed data is then delay-embedded with the previous 99 time steps (d = 100) to form a state in R4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We use data up to the end of 2012 as training data for the spin-up DMD processes of the DMDEnKF/Hankel- DMDEnKF detailed in Step 1 of the DMDEnKF algorithm, and then filter the remaining years from 2013-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The transformed, centred data with the split where the spin-up process ends and the filtering begins marked is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We initially choose to truncate to 8 DMD modes for the DMDEnKF to demonstrate the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We discuss the effect of changing the truncation on the DMDEnKF method below, but at 8 DMD modes approximately the amount of additional variance in the data that is retained by keeping more modes diminishes significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is evidenced by the “elbow” seen in the cumulative variance plot of Figure 14a, at the point where the graph transitions from rapidly increasing in variance with r to a more gradual ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We also truncate the Hankel-DMDEnKF to 8 DMD modes, to allow for a more direct comparison between the two variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The spectrum and dominant DMD/Hankel-DMD modes associated with each frequency identified by the spin-up DMD processes can be seen in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' All eigenvalues shown in Figures 10a and 10c had a modulus of ∼ 1, meaning in both cases each mode was expected to persist in the data without growing exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The major difference between the two methods spectra is that Hankel-DMD identifies the most dominant mode to have a period of one year, whereas DMD does not detect any modes with this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Annual peaks in ILI consultations occurring at a relatively similar time each year indicates that the data contains a strong mode of period one year, and this is supported by Fourier analysis [10] which also identifies one year as the dominant period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, DMD is missing the yearly mode present in the data which Hankel-DMD is able to detect, and 21 Figure 9: ILI consultations as a percentage of total weekly GP consultations across 4 age brackets and the 10 HHS regions in the US, log transformed and centred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The peaks of varying size, timing and shape in Figure 1 are visible here as vertical red areas of varying width and intensity that encompass most demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' this is likely due to Hankel-DMD’s aforementioned enhanced robustness to measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' There are two clear patterns in the structure of the dominant DMD and Hankel-DMD modes seen in Figures 10b and 10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Firstly, their strata generally move together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is shown by the vast majority of entries for each DMD mode, and entries within the same delay-embedded week (denoted by a vertical slice through the mode) for Hankel-DMD modes, sharing the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This implies that the percentage of ILI consultations increases and decreases at a similar time across all ages and regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Secondly, the variance is higher for the younger age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This is demonstrated by the absolute value of elements in the top two rows of each region generally being larger than those in the bottom two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' From Figure 10b, this is visible trivially for the DMD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In Figure 10d, age groups are arranged in ascending order for each region, so this effect is evidenced in the Hankel-DMD modes by the presence of more intensely coloured horizontal lines of width 2, followed by less intensely coloured lines of width 2 repeating over each of the 10 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This indicates that there are sharper peaks and deeper troughs in the percentage of ILI consultations for the young, while the rates for those 25 and over remain more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 Applying the filter The filtering steps of the DMDEnKF/Hankel-DMDEnKF are then applied over the remaining data using the spatial and temporal modes from the spin-up DMD and spin-up Hankel-DMD respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Producing a 4-week ahead ILI forecast for the ILINet data that consistently outperforms a simple historical baseline prediction is difficult even for state-of-the-art models [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As such, to test the DMDEnKF/Hankel-DMDEnKF we use a forecast horizon of 4 weeks when making predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In Figure 11, the DMDEnKF and Hankel-DMDEnKF’s forecasting of total ILI consultations as 22 0-4 2 consultations 2 3 65 + centred % ILI Demographic 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='64 0-4 65+ Re 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='64 0 65 ± Log transformed, Region 2 0-4 6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='64 2 65 + 2003 2005 2007 2009 2011 Split 2015 2017 2019 201 Date(a) DMD Eigenvalue Spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) Dominant DMD Modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (c) Hankel-DMD Eigenvalue Spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (d) Dominant Hankel-DMD Modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 10: Eigenvalue Spectrum (left) and DMD modes in descending order of dominance (right) generated by the DMD (top)/Hankel-DMD (bottom) applied to the data in Figure 9 up to the spin-up date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In both cases, all eigenvalues lie approximately on the unit circle, and dominant modes feature the same sign for most demographics with a magnitude that varies with age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMD modes are more interpretable, but Hankel-DMD identifies the mode with period 1 year, which DMD does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 23 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Imaginary 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 1 RealMode l: Eigenvalue Period = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9 Years 0-4 5-24 25-64 65 + Mode 2: EigenvaluePeriod=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6Years 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 5-24 25-64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 65 + es Mode 3: Eigenvalue Period = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 Years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 0-4 5-24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 25-64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 65 + Mode 4: Eigenvalue Period = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content="5'Years 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 0-4 5-24 25-64 65 + 1 2 3 4 5 6 8 9 ion Region Region Region ion ion Region Region Region Regi Regi Regi Regions1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Imaginary 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 1 RealMode l: Eigenvalue Period = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 Years 21 28 35 Mode 2: Eigenvalue Period = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Years 07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='04 14 Age/Regions 21 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='02 35 Mode 3: Eigenvalue Period = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 Years 07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='00 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='02 21 28 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='04 Mode 4: Eigenvalue Period = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 Years 07: 14 21 28 35 0 10 20 30 40 50 60 70 80 90 Weeks Delay-Embededa percentage of total weekly GP consultations can be seen in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Up until 2012, we generate 4-week ahead reconstructions using the spin-up DMD models only, estimating each state by taking the state measurement from 4 weeks prior, then iteratively applying the model to it 4 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The 4-week ahead DMD reconstruction in Figure 11a captures more fluctuations in the data than that of Hankel-DMD, however these high frequency fluctuations can also indicate the effect of noise in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMD reconstruction shown in Figure 11b is much less sensitive to noise, although fails to identify the sharper peaks in the data, which suggest it may be over- smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' From 2012 onwards the filtering begins, and forecasts are generated as described in the DMDEnKF algorithm using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF forecasts become significantly more stable, while the Hankel-DMDEnKF forecasts improve in capturing the true shape of the data, however both suffered from some degree of lag in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' During this second section of the data, the models are producing actual forecasts, as the DM- DEnKFs only have access to data up to 4 weeks prior to the prediction target’s date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, it is in this section of the data we compare the models’ performance against that of the historical baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The historical baseline prediction was created in a similar manner to that used in [8], taking ILI consultation rates from the same week of every previous year in the data (excluding the pandemic year of 2009) and then producing a probability distribution for the current week’s con- sultations via Gaussian kernel density estimation (KDE) [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' KDE Bandwidths were determined using Silverman’s rule of thumb [56], and when point estimates were required they were taken as the median of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The results of the comparisons can be seen in Figure 12 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Here it is worth noting that although we use data dated 4 weeks prior to the prediction date, in reality this data is often subject to revisions so the ILINet data as it currently stands would not necessarily be available in real time [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 Evaluating the DMDEnKF’s performance Figure 12 demonstrates graphically the 4-step ahead DMDEnKF, Hankel-DMDEnKF and historical baseline forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKFs more successfully capture the shape and height of each flu sea- son’s peak, however tend to predict the peaks late, whilst the historical baseline consistently under- predicts the peak rates but is fairly accurate on the timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMDEnKF’s forecasts are smoother than those of the DMDEnKF, however do not capture smaller details within the shape of the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' We also plot the 95% confidence intervals for the DMDEnKF and Hankel-DMDEnKF’s forecasts in Figure 12, generated using the ensemble that is maintained and propagated in the EnKF framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' At all times, the real data lies within the DMDEnKF’s confidence interval, which is not true for the Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF’s confidence interval is significantly wider than that of the Hankel-DMDEnKF, and this is due to Hankel-DMD’s robustness to noise, meaning that when the ensemble is propagated through the model, a large amount of probability mass is concentrated in a small area of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This then leads to the Hankel-DMDEnKF underestimating the uncertainty in the system, and hence some real data values falling outside the boundaries of it’s 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' To numerically compare performance, we used metrics designed for the Forecast the Influenza Season Collaborative Challenge (FluSight), in which multiple teams would submit predictions about the weekly ILINet consultation rates for the upcoming flu season at a national and HHS regional level [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The FluSight challenge evaluated models abilities to generate 1-4-week ahead forecasts 24 (a) DMDEnKF 4-week ahead forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) Hankel-DMDEnKF 4-week ahead forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 11: ILI consultations as a percentage of total weekly GP consultations forecast 4 weeks ahead using the DMDEnKF (top) and Hankel-DMDEnKF (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMD reconstruction captures the shape of the data well but is unstable, whereas the Hankel-DMD reconstruction is less sensitive to noise but over-smooths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF and Hankel-DMDEnKF forecasts help reduce these issues present in their respective reconstructions, but both suffer from some degree of lag in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 25 10 Spin-up DMD Real Data DMDEnKE 8 Iconsultations 4 % 2 2004 2006 2008 2010 2012 2014 2016 2018 Date10 Real Data Spin-up Hankel-DMD Hankel-DMDEnKF 8 Iconsultations 三 % 2 0 2004 2006 2008 2010 2012 2014 2016 2018 DateFigure 12: ILI consultations as a percentage of total weekly GP consultations, forecast 4 weeks ahead using the DMDEnKF (top) and Hankel-DMDEnKF (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' A 95% confidence interval for each forecast, and historical baseline predictions are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMDEnKF forecasts are smoother than those of the DMDEnKF, but both forecasts contain some lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The real data always lies within the DMDEnKF’s confidence interval but not the Hankel-DMDEnKF’s, however this is likely due to the DMDEnKF’s confidence interval being significantly wider than that of the Hankel-DMDEnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 26 18 Real Data 16 Historical Baseline DMDEnKF 95% CI 14 consultations 12 10 8 % 6 4 2 0 2012 2013 2014 2015 2016 2017 2018 2019 Date18 Real Data Historical Baseline 16 Hankel-DMDEnKF 95% CI 14 consultations 12 10 8 三 % 6 4 2 0 2012 2013 2014 2015 2016 2017 2018 2019 DateForecast Log Score Mean Squared Error Historical Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='24 1-week ahead DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='33 2-week ahead DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='61 3-week ahead DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='87 4-week ahead DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16 1-week ahead Hankel-DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='49 2-week ahead Hankel-DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='70 3-week ahead Hankel-DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='97 4-week ahead Hankel-DMDEnKF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='26 Table 2: The log scores and mean squared errors for the DMDEnKF and Hankel-DMDEnKF with differing forecast horizons, and the historical baseline prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF achieves a higher log score and mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where it attains a similar level of forecast skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMDEnKF consistently underperforms against the DMDEnKF in both metrics over these short forecast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Scores are calculated over the 6 flu seasons from 2012/13 to 2017/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' known as short-term targets over the course of a flu season, as well as other longer term targets, known as seasonal targets, before the season had begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF is primarily intended to be a tool for tracking and short-term forecasting, hence we focus on forecasting these short-term targets only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For this purpose we used two different metrics, the log probability measure (log score) slightly adjusted from the FluSight challenge as used in [50] and the mean squared error due to its popular use in regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The log score represents the geometric average probability of each model’s prediction being accurate, with accuracy deemed as a forecast within +/ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 of the true ILI consultation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The higher the log score, the better the forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Metrics are calculated from week 40 to week 20 of the following year to prioritize evaluation of forecasts during the flu season, and we use the 6 full seasons from 2012/13 to 2017/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The results for the historical baseline prediction and DMDEnKF/Hankel-DMDEnKF’s forecasts at a national level can be seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As one would expect, the accuracy of both DMDEnKFs degrade as they make predictions further into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The DMDEnKF achieves a higher log score and mean squared error than the historical baseline for forecast horizons up to 4 weeks ahead, where it attains a similar level of forecast skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For forecasts of 5 or more weeks ahead, the DMDEnKF is unable to outperform the historical baseline in either metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMDEnKF consistently underperforms against the DMDEnKF in both metrics over these short forecast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The top 3 statistical models and top 3 mechanistic models in the FluSight challenge achieved log scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='32 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 respectively for their 4-week ahead forecasts, hence the DMDEnKF has lower (but comparable) forecasting skill than current state of the art ILI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As the forecast horizon is extended up to 12 weeks ahead, the DMDEnKF’s forecast scores continue to decrease monotonically, whereas the Hankel-DMDEnKF’s log scores for 9-12 weeks ahead are no worse than those for 5-8 weeks ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As such, the DMDEnKF is preferred for short-term forecasting, while the Hankel- DMDEnKF is considered superior when forecasting over longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 13 shows the log scores for the 4-week ahead DMDEnKF forecast, and how these compare to 27 (a) DMDEnKF 4-week ahead forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) DMDEnKF 4-week ahead forecast - historical baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 13: Log scores over all ages and regions for the DMDEnKF’s 4-week ahead forecast (left), followed by those same scores with the log scores of the historical baseline prediction subtracted (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel-DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so we do not include its breakdown to avoid redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the top figure, the generally increasing intensity of red as one moves down the age groups shows the DMDEnKF performing more accurately for older age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The bottom figure’s varying areas of red/blue shows the DMDEnKF/historical baseline vary in superiority of forecasting skill depending on the age and region being forecast, with the historical baseline scoring more highly for most regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' the scores attained by the historical baseline prediction at an age and regional level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' The Hankel- DMDEnKF scored similarly to the DMDEnKF across all ages and regions, so its breakdown is rather similar to that of the DMDEnKF, and we do not include it to avoid redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In the DMDEnKF’s log scores, we see a major pattern in the older age groups scoring higher and hence being better predicted than the younger demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' This pattern does not persist when the historical baseline’s scores are removed, indicating it is a more general trait of the data as opposed to a specific quality of the DMDEnKF’s modelling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' There is also a significant difference in the predictability from region to region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For example, region 1 was the most predictable region for both the DMDEnKF and historical baseline, which is consistent with the findings in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' However, the DMDEnKF improved on the historical baseline’s forecast for only two of the four age groups in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In [50] it was found that the most overall improvement gained by forecasting for a region using a model as opposed to the historical baseline prediction also occurred in region 1, so one would expect to see improvements by the DMDEnKF over the historical baseline in all four age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' As log score is heavily influenced by the amount of uncertainty in a forecast, it is possible that the covariance matrices used in the DMDEnKF were too large for this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, setting the magnitude of the DMDEnKF’s variance on a region by region basis could lead to better results and more accurate forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 6 was the worst forecast region by the DMDEnKF, and the historical baseline predictions were slightly more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Again, this is consistent with [50] where region 6 was the lowest scoring region for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In that work however, region 6 experienced the second most improvement by using a model over the historical baseline prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Hence, for this region a larger variance within the DMDEnKF may have been more appropriate to account for its extra unpredictability, further supporting the idea of varying the variance by region in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 28 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='35 5-24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='30 score Ages 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='25 s 25-64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20 Log 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15 65 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 1 2 3 4 5 6 7 8 9 10 Region Region Region Region egion egion Region Region Region Region Re e R R Regions-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='00 5-24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='05 core Ages 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10S 25-64 Log 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='15 65 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20 Region 1 Region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 3 Region 4 5 Region 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Region 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' 8 9 Region 10 Region 3 Region Region 8 Region 9 Regions4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 Varying the truncation rank Having analysed the DMDEnKF and Hankel-DMDEnKF’s ILI forecasting with 8 DMD modes, we now investigate the effect different truncation ranks (r) have on their performance in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' From Figure 14a, the subjective process of identifying an “elbow” in the data could lead an observer to determine a suitable rank for truncation as low as 4 or as high as 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Application of the algorithm of Gavish and Donoho for identifying the optimal truncation threshold [26] also finds the truncation rank to be 12, hence we will focus on investigating values of r in the interval from 4 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (a) % of total variance in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (b) DMDEnKF log score/mean squared errors for r = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (c) % of total variance in the delay-embedded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' (d) Hankel-DMDEnKF log score/mean squared errors for r = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=', 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figure 14: On the left, the % of the total variance in the data (top) and delay-embedded data (bottom), dependent on the number of singular values that are retained (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' An “elbow” in the data occurs around r = 8 where we choose to truncate, however determining the exact position of the “elbow” is subjective and could be considered anywhere from r = 4 to r = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' On the right, the log score and mean squared errors for 4-step ahead forecasts generated using the DMDEnKF (top) and Hankel-DMDEnKF (bottom) with differing values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' In both cases, log score is maximised and mean squared error minimised for r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' Figures 14b and 14d show how the metrics we use to measure the DMDEnKF and Hankel- DMDEnKF’s forecasting skill vary with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' An ideal forecast will have a high log score reflecting a relatively tight and accurate probability distribution, with a low mean squared error indicating a point estimate close to the true percentage of ILI consultations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For both methods, log score is maximised and mean squared error minimised by r = 8, indicating this is the optimal rank to truncate at for our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' For r = 4, we have the simplest model tested, hence it has a low degree of uncertainty resulting in a relatively high log score, however is too simple to properly model the system so receives a high mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' By increasing r, we allow the DMDEnKFs more freedom to capture complexity within the system, resulting in a more accurate representation of the true dynamics and hence a generally lower mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' When the number of eigen- 29 lained 100% 95% ex 90% iance 85% vari 80% 75% total 70% Rank cut off of 65% at r=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content=' % 0 5 10 15 20 25 30 35 40 Singular values retained (r)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='7 rror 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 Mean Squared 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf'} +page_content='28 Log Score