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+On Fairness of Medical Image Classification
+with Multiple Sensitive Attributes via
+Learning Orthogonal Representations
+Wenlong Deng1∗, Yuan Zhong2∗, Qi Dou2, and Xiaoxiao Li1
+1 Department of Electrical and Computer Engineering,
+The University of British Columbia, Vancouver, BC, Canada
+2 Department of Computer Science and Engineering,
+The Chinese University of Hong Kong, Hong Kong, China
+Abstract. Mitigating the discrimination of machine learning models
+has gained increasing attention in medical image analysis. However, rare
+works focus on fair treatments for patients with multiple sensitive demo-
+graphic ones, which is a crucial yet challenging problem for real-world
+clinical applications. In this paper, we propose a novel method for fair
+representation learning with respect to multi-sensitive attributes. We
+pursue the independence between target and multi-sensitive representa-
+tions by achieving orthogonality in the representation space. Concretely,
+we enforce the column space orthogonality by keeping target information
+on the complement of a low-rank sensitive space. Furthermore, in the row
+space, we encourage feature dimensions between target and sensitive rep-
+resentations to be orthogonal. The effectiveness of the proposed method
+is demonstrated with extensive experiments on the CheXpert dataset.
+To our best knowledge, this is the first work to mitigate unfairness with
+respect to multiple sensitive attributes in the field of medical imaging.
+The code will be available at https://github.com/vengdeng/FCRO.
+1
+Introduction
+With the increasing application of artificial intelligence systems for medical im-
+age diagnosis, it is notably important to ensure fairness of image classification
+models and investigate concealed model biases that are to-be-encountered in
+complex real-world situations. Unfortunately, sensitive attributes (e.g., race and
+gender) accompanied by medical images are prone to be inherently encoded by
+machine learning models [5], and affect the model’s discrimination property [20].
+Recently, fair representation learning has shown great potential as it acts as a
+group parity bottleneck that mitigates discrimination when generalized to down-
+stream tasks. Existing methods [1,4,15,16,21] have studied the parity between
+privileged and unprivileged groups upon just a single sensitive attribute, but
+neglecting the flexibility with respect to multiple sensitive attributes, in which
+the conjunctions of unprivileged attributes might also deteriorate discrimination.
+* These authors contributed equally to this work.
+arXiv:2301.01481v1  [cs.CV]  4 Jan 2023
+
+2
+W. Deng et al.
+Fig. 1. A t-SNE [10] visualization of (a) sensitive attribute and (b) target representa-
+tions learned from our proposed methods FCRO on the CheXpert dataset [7]. Sensitive
+embeddings capture subgroups’ variance. We claim FCRO enforces fair classification on
+the target task by learning orthogonal target representations that are invariant over
+different attributes.
+This is a crucial yet challenging problem hindering the applicability of machine
+learning models, especially for medical image classification where patients always
+have many demographic attributes.
+To date, it is still challenging to effectively learn target-related representa-
+tions which are both fair and flexible to multiple sensitive attributes, regardless of
+some promising investigations recently. For instance, adversarial methods [1,11]
+produce robust representations by formulating a min-max game between an en-
+coder that learns class-related representation and an adversary that removes sen-
+sitive information from it. Disentanglement-based methods [4,21] achieve separa-
+tion by minimizing the mutual information between target and sensitive attribute
+representations. These methods typically gain efficacy by means of carefully de-
+signing objectives. To extend them to the multi-attribute setting, additional loss
+functions have to be explored, which should handle gradient conflict or interfer-
+ence. Methods using variational autoencoder [3] decompose the latent distribu-
+tions of target and sensitive and penalize their correlation for disentanglement.
+However, aligning the distribution of the sensitive attributes is difficult or even
+intractable given the complex combination of multiple factors. Besides, there
+are some fairness methods based on causal inference [12] or bi-level optimization
+[16], which also learn debiased while multi-attributes inflexible representations.
+Recently, disentanglement is vigorously interpreted as the orthogonality of a de-
+composed target-sensitive latent representation pair by [15], where they predefine
+a pair of orthogonal subspaces for target and sensitive attribute representations.
+In a multi-sensitive attributes setting, the dimension of the target space would
+be continuously compressed and how to solve it is still an open problem.
+In this paper, we propose a new method to achieve Fairness via Column-
+Row space Orthogonality (called FCRO) by learning fair representations for med-
+ical image classification with multiple sensitive attributes. FCRO considers multi-
+sensitive attributes by encoding them into a unified attribute representation. It
+achieves a best trade-off for fairness and data utility (see illustrations in Fig. 1)
+
+Race-Sex-Age
+White, Male, 60-
+White, Male, 18-60
+White, Female, 60-
+福
+White, Female, 18-60
+non-White, Male, 60-
+non-White, Male, 18-60
+non-White, Female, 60.
+non-White, Female, 18-60
+(a) Sensitivie attribute representation
+(b) Target representationOn Fairness of Image Classification with Multi-Sensitive Attributes
+3
+Classi%ier
+ℎ!!
+𝑆!
+𝑆!
+"
+𝑧̃#
+$
+⋯
+Sensitive Encoder
+𝜙!
+𝑍!
+Classifier
+ℎ!"
+	𝑋
+	𝑌
+		𝐴%
+		𝑍!
+		𝑍#
+		𝐴&
+	⋮
+𝜙!(𝑋)
+𝜙"(𝑋)
+×
+(a)
+∈ ℝ'×)
+𝑋
+𝑧̂#
+*
+𝑧̂!
++
+Row space
+(c)
+𝑧̃!
+𝐿!!
+𝐿!"
+𝑗
+Column space
+(b)
+𝐿#$%&' = 𝑐(𝑧̃"
+(, 𝑆!)
+𝐿)$)*+ = 𝑟(𝑧̂"
+, , 𝑧̂!
+-)
+Classi%ier	
+ℎ"
+Target Encoder
+𝜙#
+𝑍#
+∈ ℝ'×)
+𝑧̃#
+𝐿#
+𝑖
+𝑘
+𝑍# ⊥ 𝑍!
+frozen modules
+forward
+backward
+cause
+correlated
+Fig. 2. Overview of our proposed method FCRO. (a) The graphical model of orthogo-
+nal representation learning for fair medical image classification with multiple sensitive
+attributes. (b) The novel column-row space orthogonality. In the column space, we
+encourage the target model to learn representations in the complement of a low-rank
+sensitive space. In the row space, we enforce each row vector (feature dimension) of the
+target and sensitive attribute representations to be orthogonal to each other (c) The
+overall training pipeline. We use a pre-trained multi-sensitive branch, and propagate
+orthogonal gradients to target encoder φT .
+via orthogonality in both column and row spaces. Our contributions are summa-
+rized as follows: (1) We tackle the practical and challenging problem of fairness
+given multiple sensitive attributes for medical image classification. To the best
+of our knowledge, this is the first work to study fairness with respect to multi-
+sensitive attributes in the field of medical imaging. (2) We relax the independence
+of target and sensitive attribute representations by orthogonality which can be
+achieved by our proposed novel column and row losses. (3) We conduct exten-
+sive experiments on the CheXpert [7] dataset with over 80,000 chest X-rays.
+FCRO achieves a superior fairness-utility trade-off over state-of-the-art methods
+regarding multiple sensitive attributes race, sex, and age.
+2
+Methodology
+2.1
+Problem Formulation
+Notations. We consider group fairness in this work, group fairness articulates
+the equality of some statistics like predictive rate between certain groups. Con-
+sidering a binary classification problem with column vector inputs x ∈ X,
+
+AP PORT UPRICHT4
+W. Deng et al.
+labels y ∈ Y = {0, 1}. Multi-sensitive attributes a ∈ A is vector of m at-
+tributes sampled from the conjunction, i.e., Cartesian product, of sensitive at-
+tributes A = �
+i∈[m] Ai * where the i-th sensitive attribute Ai ∈ {0, 1}. Our
+training data consist of tuples D = {(x, y, a)}. We denote the classification
+model �y = f(x) = hT (φT (x)) that predicts a class label given an input x,
+where φT : X �→ Rd is a feature encoder for target embeddings, and hT :
+Rd �→ R is a scoring function. Similarly, we consider a sensitive attribute model
+g(x) = {hA1(φA(x)), ..., hAm(φA(x))} that predicts sensitive attributes associ-
+ated with input x. Given the number of samples n, the input data representa-
+tion is X = [x1, . . . , xn] and we denote the feature representation ZT = φT (X),
+ZA = φA(X) ∈ Rd×n.
+Fair classifier on multiple sensitive attributes. A classifier predicts y given
+an input x by estimating the posterior probability p(y|x). When inputs that
+are affected by their associated attributes (i.e., {A1, . . . , Am} → X) are fed
+into the network, the posterior probability is written as p(y|x, a). Since biased
+information from A is encoded, this can lead to an unfair prediction by the
+classifier. For example, in the diagnosis of a disease with sensitive attributes age,
+sex, and race, a biased classifier will result in p(�y|A = male, old, black) ̸= p(�y|A =
+female, young, white). In this work, we focus on equalized odds (ED), which is a
+commonly used and crucial criterion of fair classification in the medical domain
+[19]. In our case, ED regarding multiple sensitive attributes can be formulated
+as follows:
+P(�Y = y|A = π1, Y = y) = P(�Y = y|A = π2, Y = y),
+∀π1, π2 ∈ A.
+(1)
+Recent methods [16] suggest achieving ED for a classifier by enforcing �Y ⊥ A|Y .
+In other words, a fair classifier is expected to be independent of multi-sensitive
+information: p(y|x) = p(y|x, a).
+Fair representation. To enforce our aforementioned conditions, we follow [15]
+and introduce target embedding zT and multi-attribute embedding zAi that is
+generated from x. As in the causal structure graph for the classifier depicted
+in Fig. 2 (a), if zT and zAi are independent, the probability of a fair classifier
+p(y|x, a) is written as:
+p(y, a|x) = p(y|x, a)p(x|a)p(a)
+p(x)
+= p(y|x)p(a|x)
+(2)
+= p(y|zT )p(zT |x)
+�
+i∈[m]
+p(ai|zAi)p(zAi|x),
+(3)
+and we call zT fair representation for the target task (e.g., disease diagnosis).
+To this end, we aim to maximize Eq. (3) with the conditional independence
+constraint to train a fair classifier. It is noteworthy that in the multisensitive
+attributes setting, forcing zT to be independent on all zAi, ∀i ∈ [m] is challenging
+and even intractable when m is large. Therefore, we propose to encode multi-
+sensitive attributes into a single compact encoding zA that is still predictive for
+* [m] = {0, 1, .., m}
+
+On Fairness of Image Classification with Multi-Sensitive Attributes
+5
+classifying attributes (i.e., zA → {a1, . . . , am}). Then we can rewrite Eq. (3) as
+maximizing the likelihood with the independence constraint on zT and zA:
+p(y, a|x) = p(y|zT )p(zT |x)p(a|zA)p(zA|x).
+(4)
+However, optimizing Eq. (4) brings two technical questions:
+Q1: How to satisfy the independence constraint for zT and zA?
+A1: We relax the independence by enforcing orthogonality. Different from pre-
+defined orthogonal space in [15], we enforce orthogonality in both column spaces
+(Sec. 2.2) and row spaces (Sec. 2.3) of ZT and ZA.
+Q2: How to estimate p(y|zT ), p(zT |x), p(a|zA), p(zA|x)?
+A2: We train two convolutional neural nets encoders zT = φT (x) and zA =
+φA(x) to approximate p(zT |x) and p(zA|x) respectively; we train two multi-
+layer perception classifier y = hT (zT ) and a = hA(zA) to approximate p(y|zT )
+and p(a|zA) respectively (Sec. 2.4).
+2.2
+Column Space Orthogonality
+First, we focus on the column space of the target and the sensitive attribute
+representations. Column space orthogonality aims to learn target representations
+ZT that fulfill the following two aims: 1) have the least projection onto the
+sensitive space SA and 2) preserve the representation power to predict Y .
+Denote the target representation ZT = [�z1
+T , �z2
+T , . . . , �zn
+T ] and the sensitive at-
+tribute representation ZA = [�z1
+A, �z2
+A, . . . , �zn
+A], where �zi ∈ Rd×1 is a column vector
+for i ∈ [n], we represent the column space for ZT and ZA as ST = span(ZT )
+and SA = span(ZA) respectively. Aim 1 can be achieved by forcing ST = S⊥
+A.
+Although both �zT , �zA ∈ Rd, their coordinates may not be aligned as they are
+generated from two separate encoders. As a result, if d ≪ ∞, then there is
+no straightforward way to achieve ST ⊥ SA by directly constraining �zi
+T , �zj
+A
+(e.g., forcing (�zi
+T )⊤�zj
+A = 0). Aim 2 can be achieved by seeking a low-rank rep-
+resentation �SA for SA, whose rank is k such that k ≪ d, because we have
+rank(ST ) + rank(SA) = d if ST = S⊥
+A holds. Then S⊥
+A would be a high-
+dimensional space with sufficient representation power for target embeddings.
+This is especially important when we face multiple sensitive attributes, as the
+total size of the space is d, and increasing the number of sensitive attributes
+would limit the capacity of ST to learn predictive �zT . To this end, we first pro-
+pose to find the low rank sensitive attribute representation space �SA, and then
+encourage ZT to be in �SA’s complement �S
+⊥
+A.
+Construct low-rank multi-sensitive space. We apply Singular Value De-
+composition (SVD) on ZA = UAΣAVA to construct the low-rank space �SA,
+where UA, VA ∈ Rd×d are orthogonal matrices with left and right singular vec-
+tors ui ∈ Rd and vi
+A ∈ Rn respectively. And ΣA ∈ Rd×n is a diagonal matrix with
+descending non-negative singular values {δi
+A}min{n,d}
+i=1
+. Then we extract the most
+important k left singular vectors to construct �SA = [u1
+A, ..., uk
+A], where k controls
+how much sensitive information to be captured in �SA. It is notable that �SA is
+
+6
+W. Deng et al.
+agnostic to the number of sensitive attributes because they share the same ZA.
+For situations that can not get the whole dataset at once, we follow [8] to select
+most important bases from both bases of old iterations and newly constructed
+ones. Thus providing an accumulative low-rank space construction variant to
+update �SA iteratively. As we do not observe significant performance differences
+between these two variants (see Fig. 4 (a)), we use and refer to the first one in
+this paper if there is no special clarification.
+Column orthogonal loss. With the low-rank space �SA for multiple sensitive
+attributes, we encourage φT to learn representations in its complement �S⊥
+A. No-
+tice that �S⊥
+A can also be interpreted as the kernel of the projection onto �SA,
+i.e., �S⊥
+A = Ker(proj �
+SA�zT ). Therefore, we achieve column orthogonal loss by
+minimizing the projection of ZT to �SA, which can be defined as:
+Lcorth = c(ZT , �SA) =
+n
+�
+i=1
+��� �S⊤
+A �zi
+T
+���
+2
+2
+���zi
+T
+��2
+2
+.
+(5)
+As �SA is a low-rank space, �S⊥
+A will have abundant freedom for φT to extract
+target information, thus reserving predictive ability.
+2.3
+Row Space Orthogonality
+Then, we study the row space of target and sensitive attribute representations.
+Row space orthogonality aims to learn target representations ZT that have the
+least projection onto the sensitive row space �SA. In other words, we want to
+ensure orthogonality on each feature dimension between ZT and ZA. Denote
+target representation ZT = [�z1
+T ; �z2
+T ; . . . ; �zd
+T ] and sensitive attribute representation
+ZA = [�z1
+A; �z2
+A; . . . ; �zd
+A], where �zi ∈ R1×n is a row vector for i ∈ [d]. We represent
+row space for target representations and sensitive attribute representations as
+�ST = span(Z⊤
+T ) and �SA = span(Z⊤
+A) correspondingly. Different from column
+space orthogonality, as the coordinates (i.e., the index of samples) of �zA and �zT
+are aligned, forcing �ST = �S
+⊥
+A can be directly applied by achieving ZT Z⊤
+A:
+{(ZT Z⊤
+A)i,j = �zi
+T (�zj
+A)⊤, i, j ∈ d} =
+n
+�
+t=1
+(�zi
+T )t(�zj
+A)t.
+(6)
+Unlike column space, the orthogonality here won’t affect the utility, as the row
+vector �zT is not directly correlated to the target y. To be specific, we let pair-wise
+row vectors ZT = [�z1
+T , �z2
+T , . . . , �zd
+T ] and ZA = [�z1
+A, �z2
+A, . . . , �zd
+A] have a small inner
+product. Then for any i, j ∈ [d], we try to minimize < �zi
+T , �zj
+A >. Here we slightly
+modify the orthogonality by extra subtracting the mean vector µA and µT from
+ZA and ZT respectively, where µ = Ei∈[d]�zi ∈ R1×n. Then orthogonality loss
+will naturally be integrated into a covariance loss:
+Lrorth = r(ZT , ZA) = 1
+d2
+d
+�
+i=1
+d
+�
+j=1
+�
+(�zi
+T − µT )(�zj
+A − µA)⊤�2
+.
+(7)
+
+On Fairness of Image Classification with Multi-Sensitive Attributes
+7
+Table 1. CheXpert dataset statistics and group positive rate p(y = 1|a) regarding
+pleural effusion with three sensitive attributes race, sex, and age.
+Dataset
+#Sample
+Group Positive Rate
+Race
+Sex
+Age
+(White/Non-white/gap) (Male/Female/gap) (>60/≤ 60/gap)
+Original
+127130
+.410/.393/.017
+.405/.408/.003
+.440/.359/.081
+Augmented 88215
+.264/.386/.122
+.254/.379/.125
+.264/.386/.122
+In this way, the resulting loss encourages each feature of ZT to be independent
+of features in ZA thus suppressing the sensitive-encoded covariances that cause
+the unfairness.
+2.4
+Overall Training
+In this section, we introduce the overall training schema as shown in Fig. 2 (c).
+For the sensitive branch, since we observe that using a shared encoder may
+threaten sensitive information leakage to classification [4] or obtain unsatisfied
+sensitive attribute representations [15], we pretrain {φA, hA1, ..., hAm} for mul-
+tiple sensitive attributes using the sensitive objective as Lsens = 1
+m
+�
+i∈[m] LAi.
+Here we use cross-entropy loss as LAi for the i-th sensitive attribute. Hence
+p(zA|x) and p(a|zA) in Eq. (4) can be obtained. Then, the multi-sensitive space
+SA is constructed as in Section 2.2 over the training data. For the target branch,
+we use cross-entropy loss as our classification objective LT to supervise the train-
+ing of φT and hT and estimate p(zT |x) and p(y|zT ) in Eq. (4) respectively. Here
+we do not make additional constraints to LT , which means it can be replaced
+by any other task-specific losses. At last, we apply our column and row orthog-
+onality losses Lcorth and Lrorth to representations as introduced in Section 2.2
+and Section 2.3 along with detached SA and ZA to approximate independence
+between p(zA|x) and p(zT |x). The overall target objective is given as:
+Ltarg = LT + λcLcorth + λrLrorth,
+(8)
+where λc and λr are hyper-parameters to weigh orthogonality and balance fair-
+ness and utility.
+3
+Experiments
+3.1
+Setup
+Dataset. We adopt CheXpert dataset [7] to predict Pleural Effusion in chest
+X-rays, as it’s crucial for chronic obstructive pulmonary disease diagnosis with
+high incidence. Subgroups are defined based on the following binarized sensitive
+attributes: self-reported race and ethnicity, sex, and age. Note that data bias
+
+8
+W. Deng et al.
+Table 2. Comparasion of predicting Pleural Effusion on CheXpert dataset. We report
+the mean and standard deviation of 5-fold models trained with multi-sensitive
+attributes. AUC is used as the utility metric, and fairness is evaluated using disparities
+among subgroups defined on multi-sensitive attributes jointly and individually.
+Methods
+AUC (↑)
+Subgroup Disparity (↓)
+Joint
+Race
+Sex
+Age
+∆AUC
+∆ED
+∆AUC
+∆ED
+∆AUC
+∆ED
+∆AUC
+∆ED
+ERM [17]
+0.863
+0.119
+0.224
+0.018
+0.055
+0.046
+0.142
+0.023
+0.038
+(.005)
+(.017)
+(.013)
+(.009)
+(.017)
+(.008)
+(.014)
+(.004)
+(.010)
+G-DRO [14]
+0.854
+0.101
+0.187
+0.015
+0.048
+0.034
+0.105
+0.035
+0.051
+(.004)
+(.012)
+(.034)
+(.003)
+(.014)
+(.010)
+(.025)
+(.002)
+(.010)
+JTT [9]
+0.834
+0.103
+0.166
+0.019
+0.056
+0.026
+0.079
+0.017
+0.030
+(.020)
+(.017)
+(.023)
+(.008)
+(.016)
+(.002)
+(.004)
+(.006)
+(.007)
+Adv [18]
+0.854
+0.089
+0.130
+0.017
+0.027
+0.022
+0.039
+0.016
+0.023
+(.002)
+(.009)
+(.018)
+(.004)
+(.009)
+(.003)
+(.008)
+(.004)
+(.004)
+BR-Net [1]
+0.849
+0.113
+0.200
+0.018
+0.051
+0.037
+0.109
+0.027
+0.039
+(.001)
+(.025)
+(.023)
+(.008)
+(.013)
+(.012)
+(.025)
+(.006)
+(.006)
+PARADE [4]
+0.857
+0.103
+0.193
+0.017
+0.052
+0.042
+0.104
+0.026
+0.031
+(.002)
+(.022)
+(.032)
+(.002)
+(.010)
+(.006)
+(.023)
+(.006)
+(.011)
+Orth [15]
+0.856
+0.084
+0.177
+0.011
+0.045
+0.022
+0.083
+0.025
+0.032
+(.007)
+(.022)
+(.016)
+(.005)
+(.012)
+(.009)
+(.012)
+(.006)
+(.005)
+FCRO (ours)
+0.858
+0.057 0.107 0.012
+0.033
+0.015 0.024 0.013 0.019
+(.001)
+(.022) (.013)
+(.003)
+(.008)
+(.004) (.008)
+(.004) (.006)
+(positive rate gap) is insignificant in the original dataset (see Table 1, row ’orig-
+inal’). To demonstrate the effectiveness of bias mitigation methods, we amplify
+the data bias by (1) firstly dividing the data into different groups according to the
+conjunction of multi-sensitive labels; (2) secondly calculating the positive rate of
+each subgroup; (3) sampling out patients and increase each subgroup’s positive
+rate gap to 0.12 (see Table 1, row ‘augmented’). We resize all images to 224×224
+and split the dataset into a 15% test set, and an 85% 5-fold cross-validation set.
+Evaluation metrics. We use the area under the ROC curve (AUC) to evaluate
+the utility of classifiers. To measure fairness, we follow [13] and compute subgroup
+disparity with respect to ED (denoted as ∆ED, which is based on true positive
+rate (TPR) and true negative rate (TNR)) in (1). We quantify ED disparity as:
+∆ED =
+max
+y∈Y,π1,π2∈A
+���P(�Y = y|A = π1, Y = y) − P(�Y = y|A = π2, Y = y)
+��� . (9)
+We also follow [20] and compare subgroup disparity regarding AUC (denoted
+as ∆AUC), which gives a threshold-free fairness metric. Note that we evaluate
+disparities both jointly and individually. The joint disparities are calculated with
+respect to the conjunction of multiple sensitive attributes A, and the individual
+disparities are calculated with respect to a specific sensitive attribute Ai.
+Implementation details. In our implementation, all methods use the same
+training protocol. We choose DenseNet-121 [6] as the backbone, but replace the
+
+On Fairness of Image Classification with Multi-Sensitive Attributes
+9
+(a)
+(c)
+(b)
+INPUT
+FCRO
+ERM
+(a)
+(b)
+(c)
+INPUT
+FCRO
+ERM
+(a)
+(b)
+Fig. 3. (a) Subgroup calibration curves. We report quantile calibration curves of the
+mean (the line) and standard deviation (the shadow around it) of different subgroups
+defined by the conjunction of race, sex, and age. Larger shadow areas correspond to
+more severe unfairness. (b) Class activation map [2] generated from vanilla ERM [17]
+and FCRO (ours).
+final layer with a linear layer to extract 128-dimensional representations. The
+optimizer is Adam with learning rate of 1e−4, and weight decay of 4e−4. We train
+for 40 epochs with a batch size of 128. We sweep a range of hyper-parameters
+for each method and empirically set λc = 80, λr = 500, and k = 3 for FCRO. We
+train models in 5-fold with different random seeds. In each fold, we sort all the
+validations according to AUC and select the best model with the lowest average
+∆ED regarding each sensitive attribute among the top 5 utilities.
+Baselines. We compare our method with (i) G-DRO [14] and (ii) JTT [9] – seek-
+ing low worst-group error by minimax optimization on group fairness and target
+task error, which can be naturally regarded as multi-sensitive fairness methods by
+defining subgroups with multi-sensitive attributes conjunctions. We also extend
+recent state-of-the-art fair representation learning methods on single sensitive
+attributes to multiple ones and compare our method with them, including (iii)
+Adv [18] and (iv) BR-Net [1] – achieve fair representation via disentanglement
+using adversarial training, (v) PARADE [4] – a state-of-the-art method that ad-
+versarially eliminates mutual information between target and sensitive attribute
+representations and (vi) Orth [15] hard codes the means of both sensitive and
+target prior distributions to orthogonal means and re-parameterize the encoder
+output on the orthogonal priors. Besides, we give the result of (vii) ERM [17] –
+vanilla classifier trained without any bias mitigation technique.
+3.2
+Comparsion with Baselines
+Quantitative results. We summarize quantitative comparisons in Table 2.
+It can be observed that all the bias mitigation methods can improve fairness
+compared to ERM [17] at the cost of utility. While ensuring considerable clas-
+sification accuracy, FCRO achieves significant fairness improvement both jointly
+
+1.0
+0.861
+ERM
+ERM
+PARADE
+0.25
+PARADE
+Adv
+0.860
+(个)
+0.8
+Adv
+FCRO (ours)
+FCRO (ours)
+/ Conjunctional 
+AUC (
+0.858
+0.15
+0.4
+★
+optimal
+0.857
+moving space
+0.10
+w/o column space
+0.2 
+0.856
+w/o row space
+sweep ^c
+0.855
+sweep 入,
+0.0 +
+0.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.09
+0.10
+0.11
+0.12
+0.13
+R
+s 
+A
+R×S
+RxASxA
+Mean Predicted Probability
+Fairness - Conjunctional AeD (↓)
+SensitiveAttributes0.8
+AUC(↑)
+0.7
+k=1
+k=30
+k=3
+k=50
+0.6
+k=5
+k=100
+k=10
+0.5
+500
+750
+100012501500 1750 2000 22502500
+Iterations
+%
+0.115
+△ED
+kept info
+0.110
+0.9
+0.105
+0.8
+0.100
+ik= 3
+0
+20
+40
+60
+80
+100
+Rank of SA (k)10
+W. Deng et al.
+Fig. 4. (a) Fairness-accuracy trade-off. The perfect point lies in the top left corner.
+We report ablations and Pareto fronts of the sweep of hyperparameters. (b) Fairness
+of models trained with various numbers and permutations of three sensitive
+attributes: race (R), sex (S), and age (A). (c) AUC convergence with different rank k
+of SA. (d) Fairness and total variance (the percentage of sensitive information captured
+by SA) under different k.
+and individually, demonstrating the effectiveness of our representation orthogo-
+nality motivation. To summarize, compared with the best performance in each
+metric, FCRO reduced classification disparity on subgroups with joint ∆AUC by
+2.7% and joint ∆ED by 2.3% respectively, and experienced 0.5% ∆AUC and 0.4%
+∆ED boosts regarding the average of three sensitive attributes.
+As medical applications are sensitive to classification thresholds, we further
+give calibration curves with the mean and standard deviation of subgroups de-
+fined on the conjunction of multiple sensitive attributes in Fig. 3 (a). It can be
+observed that the vanilla ERM [17] suffers from biased calibration among sub-
+groups. Fairness algorithms can help mitigate this, while FCRO shows the most
+harmonious deviation and the most trustworthy classification.
+Qualitative results. We present the class activation map [2] in Fig. 3 (b). We
+observe that the vanilla ERM [17] model tends to look for sensitive evidence
+outside the lung regions, e.g., breast, which threatens unfairness. FCRO focuses
+on the pathology-related part only for fair Pleural Effusion classification, which
+visually confirms the validity of our method.
+3.3
+Ablation Studies
+Loss modules and hyperparameters. We further investigate the key com-
+ponents of FCRO with reference to the fairness-utility trade-off. As shown in
+Fig. 4 (a), we present the ablation of key components and the Pareto fronts
+(i.e., the set of optimal points) curve of the sweep of a range of hyperparameters
+
+0.861
+ERM
+0.25
+0.860
+(个)
+PARADE
+Adv
+≤ 0.859
+FCRO(ours)
+optimal
+accumulative space
+w/ocolumnspace
+0.10
+w/orowspace
+0.856
+sweep 入c
+0.855
+sweep 入r
+0.00
+0.09
+0.10
+0.11
+0.12
+0.13
+R
+s
+A
+R×S
+RXA
+SxA
+Fairness -Joint AED（↓）
+Sensitive Attributes
+(a)
+(b)
+0.118
+1.00
+0.80
+0.116
+AED
+(%) 
+0.75
+total variance
+0.95
+nation
+0.114
+0.90 
+Infori
+0.60
+k=1
+k=30
+20.108
+k=3
+k=50
+0.55
+k=5
+k=100
+0.106
+0.80.0
+k=10
+0.104
+0.50
+500
+750
+1000
+1250
+1500
+1750
+2000
+2250
+2500
+0 3
+20
+40
+60
+80
+100
+Iterations
+Rank of Sa (k)
+(c)
+(d)On Fairness of Image Classification with Multi-Sensitive Attributes
+11
+λc and λr. We observe that removing either column or row space orthogonality
+shows a decrease in joint ∆ED of 2.4% and 1.8% respectively, but still being
+competitive. Besides, model utility is not sensitive to weights, which fulfills our
+motivation of handling a large number of sensitive attributes. We also observe
+that applying accumulative space introduced in Section 2.2 achieves a compara-
+ble performance.
+Training with different sensitive attributes. We present an in-depth abla-
+tion study on multiple sensitive attributes in Fig. 4 (b), where models are trained
+with various numbers and permutations of attributes. We show all methods per-
+form reasonably better than ERM when trained with a single sensitive attribute
+but FCRO brought significantly more benefits when trained with the union of
+discriminated attributes (e.g., Sex × Age), which consolidate FCRO’s ability for
+multi-sensitive attributes fairness. FCRO stand out among all methods.
+Different rank k for �SA. We show the effect of choosing different k for column
+space orthogonality. As shown in Fig. 4 (c), a lower rank k benefits convergence
+of the model thus improving accuracy, which validates our insights in Section. 2.2
+that lower sensitive space rank will improve the utility of target representations.
+In Fig. 4 (d), we show that k = 3 is enough to capture over 95% sensitive
+information and keep increasing it does not bring too much benefit for fairness,
+thus we choose k = 3 to achieve the best utility-fairness trade off.
+4
+Conclusion and Future Work
+This work studies an essential yet under-explored fairness problem in medical
+image classification where samples are with sets of sensitive attributes. We for-
+mulate this problem mathematically and propose a novel fair representation
+learning algorithm named FCRO, which pursues orthogonality between sensitive
+and target representations. Extensive experiments on a large public chest X-
+rays demonstrate that FCRO significantly boosts the fairness-utility trade-off both
+jointly and individually. Moreover, we show that FCRO performs stably under dif-
+ferent situations with in-depth ablation studies. For future work, we plan to test
+the scalability of FCRO on an extremely large number of sensitive attributes.
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+

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+MULTISCALE TRANSFORMS FOR SIGNALS ON SIMPLICIAL COMPLEXES
+NAOKI SAITO∗, STEFAN C. SCHONSHECK †, AND EUGENE SHVARTS ‡
+Abstract. Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-
+Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Pack-
+ets (NGWPs); and their relatives—were developed for analyzing data recorded on nodes of a given graph. In this
+article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more gen-
+erally κ-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to
+use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional simplices in
+a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demon-
+strate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial
+complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.
+Key words. Simplicial complexes, graph basis dictionaries, hierarchical partitioning, Fiedler vectors, Hodge
+Laplacians, Haar-Walsh wavelet packets
+1. Introduction. For conventional digital signals and images sampled on regular lat-
+tices, multiscale basis dictionaries, i.e., wavelet packet dictionaries including wavelet bases,
+local cosine dictionaries, and their variants (see, e.g., [50, Chap. 4, 7], [23, Chap. 6, 7], [30,
+Chap. 8]), have a proven track record of success: JPEG 2000 Image Compression Stan-
+dard [41, Sec. 15.9]; Modified Discrete Cosine Transform (MDCT) in MP3 [41, Sec. 16.3]; dis-
+criminant feature extraction for signal classification [37, 38, 39], just to name a few. Consid-
+ering the abundance of data measured on graphs and networks and the increasing impor-
+tance to analyze such data (see, e.g., [11, 31, 6, 29, 46]), it is quite natural to lift/generalize
+these dictionaries to the graph setting. Our group have developed the graph versions of the
+block/local cosine and wavelet packet dictionaries for analysis of graph signals sampled
+at nodes so far. These include the Generalized Haar-Walsh Transform (GHWT) [17], the
+Hierarchical Graph Laplacian Eigen Transform (HGLET) [18], the Natural Graph Wavelet
+Packets (NGWPs) [7], and their relatives [20, 45, 40]; see also [19, 21]. Some of these will be
+briefly reviewed in the later sections.
+In this article, we propose their generalization for analyzing data recorded on edges,
+faces (i.e., triangles), or more generally cells (i.e., polytopes) of a class of special graphs
+called simplicial complexes (e.g., a triangle mesh of a manifold). The key idea is to use the
+Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional
+simplices in a given simplicial complex, and then build localized basis functions on these
+partitioned subsets. We demonstrate their usefulness for data representation on both il-
+lustrative synthetic examples and real-world simplicial complexes generated from a co-
+authorship/citation dataset and an ocean current/flow dataset.
+1.1. Related work. Graph-based methods for analyzing data have been widely adopted
+in many domains, [2, 32, 10]. Often, these graphs are fully defined by data (such as a graph
+of social media “friends"), but they can also be induced through the persistence homology
+of generic point clouds [4]. In either case, the vast majority of these analytical techniques
+deal with signals which are defined on the nodes of a given graph. More recently, there
+has been a surge in interest in studying signals defined on edges, triangles, and higher-
+dimensional substructures within the graph [4, 47, 14, 1, 5]. The fundamental tool em-
+ployed for analyzing these signals, the Hodge Laplacian, has been studied in the context
+of differential geometry for over half a century but has only recently entered the toolbox
+∗Department of Mathematics, University of California, Davis ([email protected], ).
+†Department of Mathematics, University of California, Davis ([email protected]).
+‡Department of Mathematics, University of California, Davis ([email protected])
+1
+arXiv:2301.02136v1  [cs.SI]  28 Dec 2022
+
+of applied mathematics. This rise in popularity is largely due to the adaptation of discrete
+differential geometry [9] in applications in computer vision [28, 36], statistics [24], topo-
+logical data analysis [5, 43], and network analysis [42].
+One of the key challenges to applying wavelets and similar constructions to node-
+based graph signals is that graphs lack a natural translation operator, which prevents the
+construction of convolutional operators and traditional Littlewood-Paley theory [19, 25,
+44]. This challenge is also present for general κ-dimensional simplices. One method for
+overcoming this difficulty is to perform convolution solely in the “frequency” domain and
+define wavelet-like bases entirely in the coefficient space of the Laplacian (or in this case
+Hodge Laplacian) transform. Following this line of research, there have been several ap-
+proaches to defining wavelets [35] and convolutional neural networks [12] in which the
+input signal is transformed in a series of coefficients in the eigenspace of the Hodge Lapla-
+cian. Unfortunately, the atoms (or basis vectors) generated by these methods are not al-
+ways locally supported, and can be difficult to interpret their role in analyzing a given
+graph signal.
+An alternative path to the creation of wavelet-like dictionaries and transforms is to
+first develop a hierarchical block decomposition of the domain and then use this to de-
+velop multiscale transforms [18, 17, 40]. These techniques rely on recursively computing
+bipartitions of the domain and then generating localized bases on the subsets of the do-
+main. In this work, we propose a simplex analog to the Fielder vector [16] to solve a relaxed
+version of the simplex-normalized-cut problem, which we can apply iteratively to develop
+a hierarchical bipartition of the κ-dimensional simplices in a simplicial complex. From
+here, we are able to apply the general scheme of [18] and [17] to develop the Hierarchi-
+cal Graph Laplacian Eigen Transform and the Generalized Haar-Walsh Transform, respec-
+tively, for a given collection of simplices of an arbitrarily high order. As a result, we can
+also generate orthonormal Haar bases, orthonormal Walsh bases, as well as data-adaptive
+orthonormal bases using the best-basis selection method [8].
+1.2. Outline. This article is organized as follows: In Section 2 we formally describe
+simplicial complexes and how their geometry leads to notions of adjacency and orienta-
+tion. This allows us to define boundary operators, which admits a map between the κ and
+κ ± 1 degree faces of the complex as well discrete differential operators acting on signals
+defined on the complex. In Section 3 we use these boundary operators to describe the
+Hodge Laplacian and discuss several different variants, some analogous to different nor-
+malizations of the graph Laplacian and some more novel. In Section 4 we show how the
+eigenvectors of the Hodge Laplacian can be use to solve relaxed-cut-like problems to parti-
+tion a complex. We also develop hierarchical bipartitions, which decompose a given com-
+plex roughly in half at each level until we are left with a division into individual elements.
+In Section 5 we use these bipartitions to develop orthonormal Haar bases. In Section 6, we
+create overcomplete dictionaries based on given bipartitions and, as a consequence, are
+also able to define a canonical orthonormal Walsh Basis. In Section 7, we present numeri-
+cal experiments on both illustrative synthetic examples and real-world problems in signal
+approximation, clustering, and supervised classification. Finally, we conclude this article
+with Section 8 discussing our potential future work.
+2. Simplicial Complexes. In this section we review concepts from algebraic topology
+to formally define simplicial complexes and introduce some notions of how two simplices
+can be “adjacent", for a more thorough review see [4, 14]. Given a vertex set V = {v1,...,vn},
+a κ-simplex σ is a (κ+1)-subset of V . A face of σ is a κ-subset of σ, and so σ has κ+1 faces.
+A co-face of σ is a (κ+1)-simplex, of which σ is a face.
+Suppose σ = {vi1,...,viκ+1}, i1 < ··· < iκ+1, and α ⊂ σ is its face. Then, σ\α consists of a
+2
+
+Fig. 1: In this small 2-complex C, e1 ∼ e4 because they share the face v2, and e1 ∼ e2 be-
+cause they share the face v1. Further e1 ≃ e2 because their hull t1 ∈ C, but e1 � e4, so that
+e1 ∼
+1 e4. We have t1 ∼ t2 because they share the face e3, and also t1 ∼
+2 t2.
+single vertex; let viℓ∗ be that vertex where 1 ≤ ℓ∗ ≤ κ+1. Then the natural parity of σ with
+respect to its face α is defined as
+nat(σ,α) := (−1)ℓ∗ .
+When α is not a face of σ, nat(σ,α) = 0. The natural parity of κ-simplices with respect to
+their faces generalizes the idea of a directed edge having a head vertex and a tail vertex, and
+is “natural” because it disallows situations analogous to a directed edge with two heads or
+two tails.
+A simplicial complex C is a collection of simplices closed under subsets, where if σ ∈ C,
+then α ⊂ σ =⇒ α ∈ C. In particular, if σ ∈ C, so does each face of σ. Let κmax(C) �
+max
+�
+κ|σ ∈ C is a κ-simplex
+�
+, and let Cκ denote the set of κ-simplices in C for each κ =
+1,...,κmax. When κ > κmax, Cκ = �. We also refer toC as a κ-complex to note that κmax(C) =
+κ. Let a κ-region of C refer to any non-empty subset of Cκ.
+Let C be a simplicial complex, and σ,τ ∈ Cκ, for some κ > 0. When σ,τ share a
+face, they are weakly adjacent, denoted by σ ∼ τ. Their shared boundary face is denoted
+bd(σ,τ). When σ ∼ τ, additionally they both share a co-face, their hull, denoted by hl(σ,τ).
+If σ,τ ∈ C, σ ∼ τ, and hl(σ,τ) ∈ C, then σ,τ are strongly adjacent, denoted by σ ≃ τ. If σ ∼ τ,
+but σ � τ in C, then σ,τ are κ-adjacent, denoted σ ∼
+κ τ.
+2.1. Oriented Simplicial Complexes and Boundary Operators. An oriented simplex
+σ further has an orientation pσ ∈ {±1}, which indicates whether its parity with its faces
+is the same as, or opposite to, its natural parity. When pσ = +1, we say σ is in natural
+orientation. For example, a directed edge e = (vi,v j ) for i < j is in natural orientation,
+while if i > j, pe = −1. An oriented simplicial complex contains at most one orientation
+for any given simplex.
+Let Xκ be the space of real-valued functions on Cκ for each κ ∈ {0,1,...,κmax(C)}. In
+the case of graphs, X0 consists of functions taking values on vertices, or graph signals.
+X1 consists of functions on edges, or edge flows. A function in X1 is positive when the
+corresponding flow direction agrees with the edge orientation, and negative when the flow
+disagrees. X2 consists of functions on oriented triangles.
+Given an oriented simplicial complex C, for each κ ∈ {0,1,...,κmax}, the boundary op-
+erator is a linear operator Bκ : Xκ+1 �→ Xκ, where for σ ∈ Cκ+1, α ∈ Cκ, the corresponding
+matrix entries are [Bκ]ασ = pσpα nat(σ,α). Likewise, the coboundary operator for each
+κ ∈ {0,1,...,κmax} is just BκT : Xκ → Xκ+1, the adjoint to Bκ. The entries of Bκ express
+relative orientation between simplex and face, and they are a natural way to construct
+functions taking local signed averages, according to adjacency in the simplicial complex.
+3
+
+2
+e4
+e3
+t2
+t1
+V1
+V
+4
+e2
+e5Fig. 2: Pairs of κ-simplices demonstrating consistency at their boundary face, for κ = 1,2.
+The mixed-color pairs are consistent, and the same-color pairs are inconsistent.
+2.2. Data on Simplicial Complexes. Signal processing on simplicial complexes arises
+as a natural problem in the setting where richer structure is incorporated in data, than just
+scalar functions and pairwise relationships. In this article, we assume the input data is
+given on an existing simplicial complex.
+A simple directed graphG = (V,E) can always be represented as an oriented 1-complex
+˜G, with each directed edge e = (vi,v j ) inserted as a 1-simplex having orientation pe =
+sign(i − j). With this convention, natural orientation corresponds to the agreement of the
+edge direction with the global ordering of the vertices.
+3. Hodge Laplacian. The boundary operators just introduced represent discrete dif-
+ferential operators encoding the structure of κ-regions in a simplicial complex, and so can
+be building blocks towards a spectral analysis of functions on those regions. For analyzing
+functions on κ-simplices with κ > 0, we will construct operators based on the Hodge Lapla-
+cian, or κ-Laplacian. As in [28], the combinatorial κ-Laplacian is defined for κ-simplices
+as
+Lκ � BT
+κ−1Bκ−1 +BκBT
+κ .
+We refer to L ∨
+κ � BT
+κ−1Bκ−1 and L ∧
+κ � BκBT
+κ as the lower and upper κ-Laplacians, respec-
+tively.
+3.1. Simplex consistency. Let C be an oriented simplicial complex, and σ ∼ τ ∈ Cκ,
+with α = bd(σ,τ). Then we may write Lκ as diag(Lκ)−Sκ, where for κ > 0, Sκ is the signed
+adjacency matrix
+[Sκ]στ �
+�
+−pσpτ nat(σ,α)nat(τ,α)
+σ ∼
+κ τ
+0
+otherwise
+.
+When Sκ > 0, we say σ,τ are consistent, and otherwise they are inconsistent. A consistent
+pair of simplices view their shared boundary face in opposite ways; one as a head face,
+and the other as a tail face. An inconsistent pair of simplices view their shared boundary
+face identically. In the case of κ = 1, two directed edges are consistent when they flow
+into each other at their boundary vertex, and are inconsistent when they collide at the
+boundary vertex, either both pointing toward it, or both pointing away. Cases for κ = 1,2
+are demonstrated in Figure 2.
+The combinatorial κ-Laplacian represents signed adjacency between κ-adjacent sim-
+plices via their consistency. In particular, this means that Lκ depends only on the ori-
+entations of simplices in Cκ. Naively, constructing the boundary matrices Bκ−1,Bκ then
+additionally requires superfluous sign information – the orientation of each member of
+both Cκ−1 and Cκ+1. This situation exactly mirrors that of the graph Laplacian L0: in order
+to construct L0 for an undirected graph via the product B0BT
+0 , one must assign an arbi-
+trary direction to each edge, and the resulting Laplacian is independent of that choice of
+directions.
+4
+
+Swt
+1 = 1
+4
+�
+�����
+0
+2
+−1
+1
+0
+2
+0
+2
+0
+1
+−2
+2
+0
+2
+−2
+1
+0
+2
+0
+2
+0
+1
+−2
+2
+0
+�
+�����
+Fig. 3: The complex from Figure 1 on the left, with natural orientation displayed as directed
+edges, together with its weighted, unnormalized signed adjacency matrix Swt
+1 , with D2 = I.
+Notice that weights differ depending on consistency and presence or lack of hull, and that
+the presence of a hull can switch the expected sign.
+3.2. Weighted and Normalized Hodge Laplacian. In order to introduce a weighted
+simplicial complex, consider the symmetrically normalized graph Laplacian
+Lsym
+0
+� D−1/2
+0
+B0D1BT
+0 D−1/2
+0
+=
+�
+D−1/2
+0
+B0D1/2
+1
+��
+D−1/2
+0
+B0D1/2
+1
+�T ,
+where D0 = diag(|B0|1), the diagonal matrix of node degrees, and D1 is the diagonal ma-
+trix of edge weights. Letting Dκ generally refer to a diagonal matrix containing κ-simplex
+weights, we proceed as in [5] and define the symmetrically normalized κ-Laplacian as
+Lsym
+κ
+� BT
+κ−1Bκ−1 +BκBT
+κ ,
+where Bκ � D−1/2
+κ
+BκD1/2
+κ+1. Here Dℓ = diag(|Bℓ|1) for ℓ = κ−1,κ, and Dκ+1 is the diagonal
+matrix of (κ+1)-hull weights.
+From Lsym
+κ
+we may define the usual weighted unnormalized, and random-walk nor-
+malized κ-Laplacians Lwt
+κ and Lrw
+κ , whose eigenvectors will be the basis for our bipartition-
+ing:
+Lwt
+κ � D1/2
+κ Lsym
+κ
+D1/2
+κ
+and
+Lrw
+κ � D−1
+κ Lwt
+κ
+.
+While in the combinatorial case, Lκ vanishes for pairs σ ≃ τ, each of the weighted
+Laplacians may be nonzero whenever σ ∼ τ. Finally, we define the weighted analogues of
+the signed adjacency matrices, Swt
+κ ,Ssym
+κ
+,Srw
+κ , as the off-diagonal parts of their respective
+Laplacians.
+4. Cuts, Fielder Vectors, and Hierarchical Bipartitions.
+4.1. Fielder Vector. Let C be a simplicial complex, such that G = (C0,C1) is a con-
+nected graph. For a given κ, let p be a vector of orientations over Cκ, with each [p]σ =
+pσ ∈ ±1, and let P = diag(p). Let Lwt
+κ , ˜Lwt
+κ denote the weighted κ-Laplacian of Cκ with nat-
+ural orientations, and with orientations given by p, respectively. Let λ0 ≤ ··· ≤ λn−1 be the
+eigenvalues of Lwt
+κ and φ0,φ1,...,φn−1 be the corresponding eigenvectors where n = |C0|.
+Then, let ( ˜λi, ˜φi) be the eigenpairs for ˜Lwt
+κ . Because ˜Lwt
+κ = PLwt
+κ P, ˜λi = λi and ˜φi = Pφi for
+0 ≤ i < n.
+For κ = 0, with the vertices of G in natural orientation, we have that λ0 = 0, λ1 > 0,
+φ0 = 1 and in particular is non-oscillatory, and that φ1 acts as a single global oscillation,
+appropriate to partition the vertices of G with. Considering ˜Lwt
+0 for nontrivial p � ±1, ˜φ0 is
+oscillatory, and ˜φ1 is no longer appropriate for clustering; this is one reason that oriented
+0-simplices are always considered to be in natural orientation.
+5
+
+e4
+t2
+e3
+ti
+V1
+V
+4
+e2
+e5
+3For κ > 0 however, it is no longer true that φ0 will be non-oscillatory. Let p∗ be a vector
+of orientations such that where [φ0]σ � 0, [p∗]σ = sign([φ0]σ). Then the corresponding
+˜φ0 is non-oscillatory, and acts as a DC component. This motivates taking sign(φ0) · φ1
+(element-wise) as the Fiedler vector of Lwt
+κ , with which to partition Cκ.
+We will aim to bipartition κ-regions by following a standard strategy in spectral clus-
+tering, of minimizing a relaxation of a combinatorial cut function over possible partitions.
+Just as a graph cut is typically defined as the volume of edge weight which crosses a parti-
+tion of the nodes, we can define the consistency cut of Cκ into subregions A,B as
+Ccut(A,B) �
+�
+σ∈A,τ∈B
+σ∼τ
+[Swt
+κ ]στ .
+Because of the signs introduced by consistency, we consider Swt
+κ as the signed, weighted
+adjacency matrix for a signed graph over Cκ, and so can utilize the framework of signed
+Laplacians [26]. Let [S+
+κ]στ � max(0,[Swt
+κ ]στ) and [S−
+κ]στ � min(0,−[Swt
+κ ]στ), i.e., indicator
+functions for consistent/inconsistent pairs, respectively. Then, we can define the consis-
+tency volume Cvol±(A) � Ccut±(A, A) and the signed κ-cut
+κCut(A,B) � 2Ccut+(A,B)+Cvol−(A)+Cvol−(B) .
+In the κ = 0 case, with all vertices in natural orientation, Swt
+0
+is just the usual adjacency
+matrix, and so S−
+0 = 0; hence κCut = 2Ccut, yielding the traditional cut objective. For
+κ > 0, κCut increases with the number of consistent pairs of κ-adjacent simplices across
+the partition, and with the number of inconsistent pairs within each κ-region. Equiva-
+lently, minimizing κCut requires maximizing consistent pairs within each κ-region, and
+maximizing inconsistent pairs across the partition.
+Let Lκ be the signed Laplacian with signed adjacency Swt
+κ . Let A be a κ-region, r A �
+1A − 1Cκ\A, and define RA(L) � r T
+ALr A. Then because Lκ differs from Lwt
+κ only on the di-
+agonal, RA(Lκ) differs from RA(Lwt
+κ ) by a constant independent of A. From [26], we know
+that RA(Lκ) ∝ κCut(A,Cκ \ A). Hence, minA⊂Cκ RA(Lwt
+κ ) = minA⊂Cκ κCut(A,Cκ \ A), and we
+obtain φ0 as a relaxed solution to κ-cut minimization.
+Now, notice that if the orientations of Cκ were changed according to some p, this
+would be equivalent to a different choice of A; namely, if [p]σ = −1, then σ moves to the
+other side of the partition, either into or out of A. As all orientations are available to us,
+this includes one for which ˜φ0 is non-oscillatory, so that its sign does not partition Cκ. We
+then instead take ˜φ1 as our relaxed solution, which we may compute via sign(φ0)·φ1.
+An improved cut objective is the signed Ratio Cut, which encourages more balanced
+partitions:
+SignedRatioCut(A) �
+� 1
+|A| +
+1
+|Cκ \ A|
+�
+κCut(A,Cκ \ A) .
+From [26], we know that with rA above scaled by a factor of cA � �|A|/|Cκ \ A|, the analo-
+gous result holds, that the eigenvectors of Lκ yield a relaxed solution to minA⊂Cκ SignedRatioCut(A).
+However, the new dependence on A means the resulting objective is slightly different for
+Lκ, so the relaxation is only approximate.
+Finally, the signed Normalized Cut balances the partitions by degree rather than sim-
+plex count:
+SignedNormalizedCut(A) �
+�
+1
+Cvol(A) +
+1
+Cvol(Cκ \ A)
+�
+κCut(A,Cκ \ A).
+Here, the eigenvectors of diag(Lκ)−1Lκ yield a relaxed solution to minA⊂Cκ SignedNormalizedCut(A),
+and an approximate relaxed solution is given by the eigenvectors of Lrw
+κ . In our numeri-
+6
+
+Fig. 4: One possible hierarchical bipartitioning of a simple 2-complex, from j = 0 with no
+partition on the left, to j = 5 on the right, where each of the 27 triangles form their own
+subregion. Colors indicate distinct subregions.
+cal experiments, we use the random-walk κ-Laplacian for bipartitioning simplicial com-
+plexes.
+4.2. Hierarchical Bipartitions. The foundation upon which our multiscale transforms
+on a κ-simplices Cκ of a given simplicial complex C are constructed is a hierarchical bi-
+partition tree (also known as a binary partition tree) of Cκ, a set of tree-structured κ-
+subregions of Cκ constructed by recursively bipartitioning Cκ. This bipartitioning opera-
+tion ideally splits each κ-subregion into two smaller κ-subregions that are roughly equal in
+size while keeping tightly-connected κ-simplices grouped together. More specifically, let
+C j
+k denote the kth κ-subregion on level j of the binary partition tree of Cκ and n j
+k �
+���C j
+k
+���,
+where j,k ∈ Z≥0. Note C 0
+0 = Cκ, n0
+0 = n, i.e., level j = 0 represents the root node of this
+tree. Then the two children of C j
+k in the tree, C j+1
+k′
+and C j+1
+k′+1, are obtained through parti-
+tioning C j
+k using the Fiedler vector of Lrw
+κ (C j
+k). This partitioning is recursively performed
+until each subregion corresponding to the leaf contains only a simplex singleton. Note
+that k′ = 2k if the resulting binary partition tree is a perfect binary tree. We note that even
+other (non-spectral) partitioning methods can be used to form the binary partition tree,
+but in this article, we stick with the spectral clustering using the Fielder vectors. For more
+details see on hierarchical partitioning, (specifically for the κ = 0 case), see [22, Chap. 3]
+and [40]. Figure 4 demonstrates such a hierarchical bipartition tree for a simple 2-complex
+consisting of triangles.
+5. Orthonormal κ-Haar Bases. The classical Haar basis [15] was introduced in 1909
+as a piecewise-constant compactly-supported multiscale orthonormal basis (ONB) for square-
+integrable functions but has since been recognized as a wavelet family and adapted to
+many domains. In one dimension, the family of Haar wavelets on the interval [0,1] can be
+generated by the following mother and scaling (or father) functions:
+ψ(x) =
+�
+�
+�
+�
+�
+1,
+0 ≤ x < 1
+2;
+−1,
+1
+2 ≤ x < 1;
+0,
+otherwise.
+φ(x) =
+�
+1,
+0 ≤ x < 1;
+0,
+otherwise.
+Unfortunately, these definitions do not generalize to non-homogeneous domains due to
+the lack of appropriate translation operators and dilation operators [44]. Instead, several
+methods have been proposed to generate similar bases, and overcomplete dictionaries to
+apply more abstract domains such as graphs and discretized manifolds [17, 45, 40]. Here,
+we describe a method to compute similar, piecewise-constant locally supported bases for
+κ-simplex valued functional spaces, which we call the (orthonormal) κ-Haar bases.
+Rather than basing our construction on some kind of translation or transportation
+schemes, we instead employ the hierarchical bipartition, as we discussed in Section 4.2, to
+7
+
+Fig. 5: The 2-Haar basis vectors on the same simple 2-complex shown in Figure 4. The
+yellow, dark green, violet regions in each vector indicate its positive, zero, and negative
+components.
+divide the domain, i.e., the κ-simplicesCκ of a given simplicial complexC into appropriate
+locally-supported κ-regions. For each κ-region in the bipartition tree, if that region has two
+children in the tree, then we create a vector that is positive on one child, negative on the
+other, and zero elsewhere. To avoid sign ambiguity, we dictate that the positive portion is
+on the region whose region index is smaller among these two.
+Several remarks on this basis are in order. First, since the division is not symmetri-
+cally dyadic, we need to compute the scaling factor for each region separately. For each
+given basis vector ξ except the scaling vector, we break it into positive and negative parts
+ξ+ and ξ− and ensure that �
+i([ξ+]i + [ξ−]i) = 0 and ∥ξ∥ = 1. If the members of κ-region
+are weighted, then this sum and norm can be computed with respect to those weights. Fi-
+nally, we note that different hierarchical bipartition schemes may arise from the different
+weighting of the Hodge Laplacian, which will correspond to bases with different supports.
+Figure 5 demonstrates the 2-Haar basis on the simple 2-complex used in Figure 4, which
+has a hole in the center.
+6. Overcomplete Dictionaries. In this section, we introduce two overcomplete dic-
+tionaries for analyzing real-valued functions defined on κ-simplices in a given simplicial
+complex: the κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET), based on the
+Hierarchical Graph Laplacian Eigen Transform (HGLET) [18] and the κ-Generalized Haar-
+Walsh Transform (κ-GHWT), based on the Generalized Haar-Walsh Transform (GHWT) [17]
+for graph signals.
+6.1. κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET). The first over-
+complete transform we describe can be viewed as a generalization of the Hierarchical
+Block Discrete Cosine Transform (HBDCT). The classical HBDCT is generated by creat-
+ing a hierarchical bipartition of the signal domain and computing the DCT of the local
+signal supported on each subdomain. We note that a specific version of the HBDCT (i.e., a
+homogeneous split of an input image into a set of blocks of size 8×8 pixels) has been used
+in the JPEG image compression standard [34]. This process was generalized to the graph
+case in [18], i.e., the Hierarchical Graph Laplacian Eigen Transform (HGLET), from which
+we base our algorithm and notation. The basis given by the set {φj
+k,l} where j denotes the
+level of the partition (with j = 0 being the root), k indicates the partition within the level,
+and l indexes the elements within each partition in increasing frequency.
+To compute the transform, we first compute the complete set of eigenvectors {φ0
+0,l}l=1:n
+of the Hodge Laplacian of the entire κ-simplices Cκ of a given simplicial complex C and or-
+8
+
+Fig. 6: 2-HGLET dictionary on the 2-complex shown in Figure 4. Here, the color scale is
+consistent across each row (which corresponds to the level) to better visualize the smooth-
+ness of the elements
+der them by nondecreasing eigenvalues. We then partition Cκ into two disjoint κ-regions
+C 1
+0 and C 1
+1 as described in Section 4. We then compute the complete set of eigenvectors of
+the Hodge Laplacian on C 1
+0 and C 1
+1. We again order each set by nondecreasing frequency
+(i.e., eigenvalue) and label these {φ1
+0,l}l=1:n1
+0 and {φ1
+1,l}l=1:n1
+1 Note that n1
+0 + n1
+1 = n0
+0 = n,
+and that all of the elements in {φ1
+0,l} are orthogonal to those in {φ1
+1,l} since their supports
+are disjoint. Then the set {φ1
+0,l}l=1:n1
+0 ∪ {φ1
+1,l}l=1:n1
+1 form an orthonormal basis for vectors
+on Cκ. From here, we apply this process recursively, generating an orthonormal basis for
+each level in the given hierarchical bipartition tree.
+If the hierarchical bipartition tree terminates at every region containing only a κ-
+simplex singleton, then the final level will simply be the standard basis of Rn. Each level
+of the dictionary contains an ONB whose vectors have the support of roughly half the size
+of the previous level. There are roughly (1.5)n possible ONBs formed by selecting differ-
+ent covering sets of regions from the hierarchical bipartition tree; see, e.g., [49, 40] for more
+about the number of possible ONBs in such a hierarchical bipartition tree. Finally, we note
+that the computational cost of generating the entire dictionary is O(n3) and that any valid
+hierarchical bipartition tree can be used to create a similar dictionary. Figure 6 shows the
+2-HGLET constructed on the same 2-complex shown in Figure 4.
+6.2. κ-Generalized Haar-Walsh Transform (κ-GHWT). The second transform we present
+here is based on the Generalized Haar-Walsh Transform (GHWT) [17], which can itself be
+viewed as a generalization of the Wash-Hadamard transform. This basis is formed by first
+generating a hierarchical bipartition tree of Cκ. We then work in a bottom-up manner, be-
+ginning with the finest level in which each region only contains a single element. We call
+these functions scaling vectors and label them {ψjmax
+k,0 }k=0:n−1. For the next level, we first
+assign a constant scaling vector for support on each region. Then, for each region that con-
+tains two children in the partition tree, we form a Haar-like basis element by subtracting
+the scaling function associated with the child element with a higher index from that child
+element with a lower index. This procedure will form an ONB {ψjmax−1
+k,l
+}k=0:k′−1,l=0:l(k)−1
+(where k′ is the number of κ-subregions at level jmax − 1 and l(k) = 1 or 2 depending on
+the partition k) whose vectors have support of at most 2. For the next level, we begin by
+computing the scaling and Haar-like vectors as before. Next, for any region that contains
+three or more elements, we also compute Walsh-like vectors by adding and subtracting the
+Haar-like vectors in the children’s regions. From here, we form the rest of the dictionary
+recursively. A full description of this algorithm (for the κ = 0 case) is given in [18]. Figure 7
+9
+
+Fig. 7: Course-to-Fine (C2F) 2-GHWT dictionary. The yellow, dark green, and violet regions
+in each vector indicate its positive, zero, and negative components, respectively.
+displays the 2-GHWT dictionary on the same 2-complex used in Figures 5 and 7. We make
+several observations about this dictionary. First, like the κ-HGLET, each level of the dic-
+tionary forms an ONB, and at each level, basis vectors have the support of roughly half the
+size of the previous level. These basis vectors also have the same support as the κ-HGLET
+basis vectors (that is, supp(φj
+k,l) = supp(ψj
+k,l) for all j,k,l). However, the computational
+cost of computing the κ-GHWT is only O(n logn) compared to the O(n3) of the κ-HGLET.
+Finally, we note that at the coarsest level (j = 0) the κ-GHWT dictionary contains
+globally-supported piecewise-constant basis vectors, which are ordered by increasing os-
+cillation (or “sequency”). This forms an ONB analogous to the classical Walsh Basis. This
+allows us to define an associated Walsh transform and conduct Walsh analysis on signals
+defined on simplicial complexes. Although not the primary focus of this article, we con-
+duct some numerical experiments using the Walsh bases explicitly in Section 7.
+6.3. Organizing the Dictionaries. For many downstream applications, it is impor-
+tant to organize the order of these bases. In general, the κ-HGLET dictionary is naturally
+ordered in a Coarse-to-Fine (C2F) fashion. In each region, the basis vectors are ordered
+by frequency (i.e., eigenvalue). Similarly, the GHWT dictionary is also naturally ordered
+in a C2F fashion, with increasing “sequency” within each subgraph. Another useful way
+to order the GHWT is in a Fine-to-Coarse (F2C) ordering, which approximates “sequency”
+domain partitioning. See, e.g., Figure 8, which shows the F2C 2-GHWT dictionary on the
+triangle graph. We also note that the F2C ordering is not possible for the κ-HGLET dictio-
+nary because some parent subspaces and the direct sum of their children subspaces are
+not equivalent; see, e.g., [22, Eq. (5.6)] for the details. Other relabeling schemes, such as
+those proposed in [45, 40] may also be useful but are outside the scope of this article and
+will be explored further in our future work. .
+6.4. Basis and Frame Selection. Once we have established these arrangements of ba-
+sis vectors, we can efficiently apply the best-basis algorithm [8] to select an ONB that is op-
+timal for a task at hand for a given input signal or a class of input signals; see also our previ-
+ous work of applying the best-basis algorithm in the graph setting [18, 17, 19, 21, 45, 7, 40].
+Given some cost function F and signal x, we traverse the partition tree and select the basis
+that minimizes F restricted to each region. For the C2F dictionary, we initialize the best
+basis as the finest (j = jmax) level of the GHWT dictionary. We then proceed upward one
+level at a time and compute the cost of each subspace at that level and compare it to the
+10
+
+Fig. 8: Fine-to-Coarse (F2C) 2-GHWT dictionary. Note that this dictionary is not generated
+by simply reversing the row indices of the C2F dictionary, but instead by arranging each
+level (row) by “sequency”.
+cost of the union of its children subspaces. If the latter cost is lower, the basis is updated;
+if not, the children subspaces (and their basis vectors) are propagated to the current level.
+This algorithm yields the C2F best basis. The F2C best basis is performed similarly, i.e., we
+begin with the globally-supported basis (j = 0) at the bottom of the rearranged tree and
+proceed in the same bottom-up direction. As for the HGLET dictionary, it has only a C2F
+basis as we discussed earlier.
+In some contexts, it is not necessary to generate a complete ONB, but rather some
+sparse set of vectors in the dictionary (also known as atoms) that most accurately approx-
+imate a given signal or class of signals. In this case, we can directly apply the orthogo-
+nal matching pursuit of [3] to find the best m-dimensional orthogonal subframe (m ≤ n)
+selected from the dictionary. Additionally, for some downstream tasks, such as sparse ap-
+proximation or sparse feature selection, generating orthogonal sets of atoms is not critical.
+In these cases, we can employ a greedy algorithm to generate efficient approximation. This
+algorithm simply selects the atoms in the dictionary with the largest coefficient, removes
+it, then computes the transform of the residual and proceeds so forth. These basis and
+subframe algorithms are studied intensively in the subsequent section.
+7. Numerical Experiments. We demonstrate the efficacy of our proposed partition-
+ing techniques and basis constructions by conducting a series of experiments. In Sec-
+tion 7.1 we show how our multiscale bases and overcomplete dictionaries can be used
+to sparsely approximate signals defined on κ-simplices. In Section 7.2 we show how these
+representations can be used in supervised classification and unsupervised clustering prob-
+lems.
+7.1. Approximation and Signal Compression. We begin with an illustrative example
+by creating some synthetic data for 1- and 2-simplices by triangulating a digital image. We
+start with a 512 × 512 “peppers” image and map it to a Cartesian grid on the unit square
+[0,1]2. We then randomly sample 1028 points within this square (not necessarily on a grid).
+We then create a triangular mesh from these points using Delaunay triangulation. Next,
+we interpolate the image from the Cartesian grid to the sampled vertices by computing the
+barycentric coordinate of each vertex from the square inside the Cartesian grid. Finally,
+we interpolate the signal to the edges and triangles of the triangulation by averaging the
+values of the vertices that they contain. The result, for our random seed, is a signal defined
+11
+
+Fig. 9: Nonlinear approximation of the peppers image for κ = 2
+on the 3050 edges of the triangulation and another on the 2067 triangles. We now consider
+the sparse representation of these signals. Figure 9 shows the nonlinear approximation
+(i.e., using the largest expansion coefficients in magnitude) of the triangle-based signals
+in the Hodge Laplacian eigenbasis (Fourier), the orthonormal Haar basis, orthonormal
+Walsh basis as well as the approximation prescribed by applying the best-basis and greedy
+algorithms to the HGLET and GHWT dictionaries. Figure 10 shows the approximation
+error vs the number of terms used for both the edge-based and triangle-based functions.
+A number of observations are in order. First, the multiscale dictionary-based meth-
+ods consistently outperformed the generic orthonormal bases. The greedy approximation
+algorithm achieved the best approximation results, but it is also more costly to compute
+than any of the other methods, and the set of atoms used in the approximation may not
+be orthogonal. This may be detrimental to downstream tasks. Overall the GHWT-based
+method performed best, with the F2C best basis performing much better than the C2F
+best basis, which suggests that the fine-scale features of this signal are the most impor-
+tant. Similarly, the Walsh basis achieved much better results than the Haar basis, again
+emphasizing the necessity of capturing details at the fine scale.
+Next, we apply our approach to real-world data for higher degree signals for κ = 0,...,5.
+The citation complex [33, 12] is a simplicial complex derived from the Cora citation com-
+plex [48], which models the interactions between multiple authors of scientific papers. A
+paper with κ authors is represented by a (κ − 1)-simplex. We first build a graph whose
+vertices represent the authors in this Cora database. Then, the vertices are connected by
+edges that represent co-authored papers. Note that if two authors co-authored multiple
+12
+
+1%
+5%
+10%
+25%
+50%
+75%
+90%
+Delta
+Fourier
+Haar
+Walsh
+HGLET (BB)
+GHWT (BB)Fig. 10: Nonlinear approximation errors of the peppers image, Left: L2 error, Right: log(L2
+error) for up to 50% of the terms retained. Top κ = 1, Bottom: κ = 2.
+papers, these two vertices are connected by a single edge. Next, we assign each edge the
+sum of the citation numbers of all the co-authored papers by the authors, forming this
+edge as its weight (or value). Finally, we assign each higher-order simplex the sum of the
+values of its lower-order simplices as its value. See [12] for a more thorough description
+of the construction of this complex. Table 1 reports some basic information about the
+number of simplices of different degrees in this citation complex. Figure 11 shows the
+approximation of this signal(i.e., a vector of citation numbers) for κ = 0,1,...,5 with the
+Delta, Fourier, Haar, HGLET, and GHWT bases. Figure 12 shows the log error. The HGLET
+and GHWT bases were selected with the best-basis algorithm using the C2F ordering for
+the GHWT dictionary.
+In these experiments, we observe that the best bases (GHWT and HGLET) outper-
+formed the canonical bases, with the GHWT being the most efficient basis for each κ. Ad-
+ditionally, for κ > 0, the orthonormal Haar basis performed best in the semi-sparse regime
+(1 and 10% of terms retrained). This suggests that the signals on each degree of the citation
+complex are similar in that they are all close to being piecewise constant. However, when
+13
+
+K=l, Approximation Error
+K=1, Log Approximation Error
+1.0
+Delta Basis
+0.0
+Frourier Basis
+Orthogonal Haar
+Orthogonal Walsh
+-0.5
+BB HGLET
+0.8
+HGLET (Greedy)
+BB GHWT C2F
+Log L2 Approximation Error 
+ Approximation Error
+BB GHWT F2C
+-1.0
+GHWT (Greedy)
+0.6
+1.5 
+0.4
+Delta Basis
+Frourier Basis
+Orthogonal Haar
+-2.0-
+Orthogonal Walsh
+0.2
+BB HGLET
+HGLET (Greedy)
+-2.5 -
+BB GHWT C2F
+BB GHWT F2C
+0.0 
+GHWT (Greedy)
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+# of Terms
+# of TermsK=2, Approximation Error
+K=2, Log Approximation Error
+Delta Basis
+1.0
+0.0
+Frourier Basis
+Orthogonal Haar
+Orthogonal Walsh
+-0.5
+BB HGLET
+0.8
+HGLET (Greedy)
+BB GHWT C2F
+Log L2 Approximation Error
+Approximation Error
+BB GHWT F2C
+-1.0
+GHWT (Greedy)
+0.6
+-1.5 
+0.4
+Delta Basis
+-2.0
+Frourier Basis
+Orthogonal Haar
+Orthogonal Walsh
+0.2
+BB HGLET
+-2.5
+HGLET (Greedy)
+BB GHWT C2F
+BB GHWT F2C
+-3.0 -
+0.0 -
+GHWT (Greedy)
+750
+0
+200
+400
+600
+800
+0
+250
+500
+100012501500 1750 2000
+1000
+# of Terms
+# of Termsκ
+0
+1
+2
+3
+4
+5
+# of elements
+1126
+5059
+11840
+18822
+21472
+17896
+Table 1: The number of element in the κ-simplices in the Cora complex for κ = 0,1,...,5
+Fig. 11: Approximation of the Citation Complex for κ = 0,...,5.
+more terms are considered, the HGLET best basis achieved a lower approximation error
+than the orthonormal Haar basis achieved.
+7.2. Signal Clustering and Classification. Since the basis (and dictionary) vectors we
+present are both multiscale and built from the Hodge Laplacians that are aware of both
+topological and geometric properties of the domain [5], they can function as very powerful
+feature extractors for general data science applications. In this section, we present two
+clustering-type applications—one supervised and one unsupervised. For baselines, we
+compare our proposed dictionaries with Fourier and Delta (indicator function) bases and
+with the Hodgelets proposed in [35] for cases when κ = 1.
+7.2.1. Supervised Classification. First, we present our study in supervised classifi-
+cation. We begin by computing edge-valued signals for 1000 handwritten digits from the
+MNIST dataset [27] by sampling 500 points in the unit square and following the interpola-
+tion method presented for the peppers image in Section 7.1. We then compute the features
+of these images using the proposed orthogonal transforms and best bases from the over-
+complete dictionaries. Next, we train a support vector machine (SVM) to classify the digits
+for each of the transformed representations using the 1000 training examples. Finally, we
+test these SVMs on the rest of the whole MNIST dataset.
+We repeat this experiment for the FMNIST dataset [51], again using only 1000 exam-
+ples for training data. Results are presented in Table 2. We remark that these tests are not
+meant to achieve state-of-the-art results for image classification but rather to showcase
+the effectiveness of these representations for downstream tasks. Unsurprisingly, the dic-
+tionary methods outperformed the basis methods. Again, the piecewise constant meth-
+ods (GHWT, Haar) achieved better approximations than the smoother methods (Fourier,
+14
+
+Approximation k=0
+1.0
+Delta
+Fourier
+Haar
+HGLET
+0.8
+GHWT
+0.6
+ux
+0.4
+0.2
+0.0
+0
+50
+100
+150
+200
+250
+300
+350
+# of elementsApproximation k=1
+1.0
+Delta
+Fourier
+Haar
+0.8
+HGLET
+GHWT
+0.6
+x
+-
+X
+0.4
+0.2
+0.0
+0
+200
+400
+600
+800
+1000
+1200
+1400
+# of elementsApproximation k=2
+1.0
+Delta
+Fourier
+Haar
+0.8
+HGLET
+GHWT
+0.6
+x
+-
+0.4
+0.2
+0.0
+0
+500
+1000
+1500
+2000
+2500
+3000
+# of elementsApproximation k=3
+1.0
+Delta
+Fourier
+Haar
+HGLET
+0.8
+GHWT
+0.6
+x
+-
+0.4
+0.2
+0.0
+0
+1000
+2000
+3000
+4000
+5000
+# of elementsApproximation k=4
+1.0
+Delta
+Fourier
+Haar
+HGLET
+0.8
+GHWT
+0.6
+x
+-
+0.4
+0.2
+0.0
+0
+1000
+2000
+3000
+4000
+5000
+# of elementsApproximation k=5
+1.0
+Delta
+Fourier
+Haar
+HGLET
+0.8
+GHWT
+0.6
+x
+-
+0.4
+0.2
+0.0
+0
+1000
+2000
+3000
+4000
+# of elementsFig. 12: Top: Approximation of the Citation Complex for κ = 0,...,5. Bottom: Log of the
+error for up to 50% of the terms retained.
+Basis Methods
+Dictionary Methods
+Delta
+Fourier
+Haar
+Walsh
+HGLET
+(BB)
+GHWT
+(BB C2F)
+GHWT
+(BB F2C)
+Joint
+Separate
+HGLET
+GHWT
+# of terms
+661
+661
+661
+661
+661
+661
+661
+5288
+5288
+9254
+9254
+MNIST
+68.675
+77.053
+75.388
+77.011
+77.991
+78.779
+77.156
+79.202
+80.038
+80.001
+81.089
+FMNIST
+64.370
+76.753
+76.779
+75.230
+76.117
+76.991
+76.121
+78.761
+78.738
+79.739
+80.789
+Table 2: Test Accuracy for SVMs trained on transforms of MNIST signals interpolated to a
+random triangulation
+HGLET, Joint, and Separate Hodgelets). This is likely due to the near-binary nature of im-
+ages in both datasets.
+7.2.2. Unsupervised Clustering. A natural setting for studying κ = 1 valued signals is
+the analysis of trajectories [5, 36, 35]. Of particular interest is the case where the domain
+has nontrivial topological features. Such is the case of the Global Drifter Program dataset,
+which tracks the positions of 334 buoys dropped into the ocean at various points around
+the island of Madagascar [35].
+We begin by dividing the dataset into three subsets, train (|Xtr| = 176), test (|Xte| =
+83) and validation (|Xvl| = 84). We then use orthogonal matching pursuit [3] (OMP) to
+compute the m significant features of the training set. Next, we extract these features for
+the test set and use them to compute the centroids {c j }d
+j=1 for each cluster. To evaluate
+these clusters K -score (i.e. the standard k-means objective) on the transformed features
+of the validation set:
+K −score := 1
+N
+N
+�
+i=1
+min
+1≤j≤d ∥f (xi)−c j ∥2,
+xi ∈ Xvl.
+where f (·) represents the feature extraction prescribed by applying OMP to the test set. We
+repeat this experiment for m = 5,10,15,20,25 (number of features) and d = 2,...,7 (num-
+ber of clusters). Figure 13 summarizes the results of this test, while Table 3 shows the full
+15
+
+Approximationk=0
+0.0
+Delta
+Fourier
+0.5
+Haar
+HGLET
+GHWT
+1.0
+(llux
+-1.5
+-
+)601
+-2.0
+2.5
+3.0
+3.5
+0
+25
+50
+75
+100
+125
+150
+175
+#ofelementsApproximation k=1
+0.0
+Delta
+Fourier
+Haar
+-0.5 
+HGLET
+GHWT
+-1.0
+-1.5
+-2.0
+-2.5 
+-3.0 
+-3.5
+0
+100
+200
+300
+400
+500
+600
+700
+# of elementsApproximation k=2
+0.0
+Delta
+Fourier
+Haar
+-0.5
+HGLET
+GHWT
+-1.0
+-1.5
+-2.0
+-2.5
+-3.0 
+0
+250
+500
+750
+1000
+1250
+1500
+# of elementsApproximation k=3
+0.0
+Delta
+Fourier
+Haar
+-0.5
+HGLET
+GHWT
+-1.0 :
+- xII)60|
+-1.5
+-2.0
+-2.5 
+-3.0
+0
+500
+1000
+1500
+2000
+2500
+# of elementsApproximation k=4
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+Delta
+-2.5
+Fourier
+Haar
+HGLET
+-3.0
+GHWT
+0
+500
+1000
+1500
+2000
+2500
+# of elementsApproximation k=5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+Delta
+-2.5 
+Fourier
+Haar
+-3.0
+HGLET
+GHWT
+0
+500
+1000
+1500
+2000
+# of elementsFig. 13: Extensive results for buoy cluster test. Leftmost figure shows which method pre-
+formed best, the second to the left shows the second best and so on. The x-axis in each
+subplot indicates the number of coefficients used and the y-axis is the number of clusters.
+Full numerical results are presented in Table 3.
+numerical results. In this experiment, the GHWT outperformed all other bases because
+the trajectories are roughly constant and locally supported. The orthogonal matching pur-
+suit scheme can select elements with the correct support size, and the piecewise constant
+nature of the GHWT atoms can capture the action of the trajectory with very few elements.
+8. Conclusions and Future work. In this article, we have developed several general-
+izations of orthonormal bases and overcomplete transforms/dictionaries for signals de-
+fined on κ-simplices, and demonstrated their usefulness for data representation on both
+illustrative synthetic examples and real-world simplicial complexes generated from a co-
+authorship/citation dataset and an ocean current/flow dataset. However, there are many
+more tools from harmonic analysis that we have not addressed in this article. From a
+theoretical standpoint, future work may involve 1) defining additional families of mul-
+tiscale transforms such as the extended Generalized Haar-Walsh Transform(eGHWT) [45]
+and Natural Graph Wavelet Packets (NGWPs) [7]; 2) exploring different best-basis selection
+criteria tailored for classification and regression problems such as the Local Discriminant
+Basis [37, 39] and the Local Regression Basis [38] on simplicial complexes; and 3) inves-
+tigating nonlinear feature extraction techniques such as the Geometric Scattering Trans-
+form [13]. From an application standpoint, we look forward to applying the techniques
+presented here to data science problems in computational chemistry, weather forecasting,
+and genetic analysis, all of which have elements that are naturally modeled with simplicial
+complexes.
+Acknowledgments. This research was partially supported by the US National Science
+Foundation grants DMS-1418779, DMS-1912747, CCF-1934568; the US Office of Naval Re-
+search grant N00014-20-1-2381.
+16
+
+Validation set Best
+Validation set Second
+Validation set Third
+6
+6
+5 
+5+
+num_clusters
+4
+GHWT
+3 -
+3 
+ HGLET
+2 -
+2
+5
+5
+5
+num coefs
+num coefs
+ Haar
+num coefs
+Validation set Fourth
+Validation set Fifth
+Validation set Sixth
+Separate
+6 
+-Joint
+4
+- Fourier
+3
+3 .
+2
+num coefs
+num coefs
+9
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+
+Appendix A. Full Results for Buoy Clustering.
+Clusters
+# Feat.
+Fourier
+Joint
+Separate
+Haar
+HGLET
+GHWT
+5
+0.174
+0.183
+0.122
+0.115
+0.154
+0.024
+10
+0.150
+0.151
+0.109
+0.110
+0.124
+0.023
+2
+15
+0.129
+0.129
+0.120
+0.093
+0.119
+0.021
+20
+0.118
+0.113
+0.108
+0.084
+0.107
+0.023
+25
+0.104
+0.099
+0.096
+03073
+0.103
+0.024
+5
+0.174
+0.163
+0.110
+.0115
+0.126
+0.026
+10
+0.143
+0.137
+0.100
+0.108
+0.103
+0.023
+3
+15
+0.126
+0.112
+0.113
+0.095
+0.118
+0.021
+20
+0.114
+0.104
+0.100
+0.081
+0.095
+0.019
+25
+0.099
+0.092
+0.089
+0.069
+0.093
+0.021
+5
+0.139
+0.135
+0.096
+0.091
+0.101
+0.023
+10
+0.137
+0.120
+0.090
+0.096
+0.082
+0.019
+4
+15
+0.116
+0.099
+0.083
+0.079
+0.097
+0.018
+20
+0.111
+0.094
+0.084
+0.072
+0.090
+0.021
+25
+0.094
+0.083
+0.076
+0.062
+0.087
+0.022
+5
+0.135
+0.116
+0.087
+0.081
+0.074
+0.014
+10
+0.118
+0.109
+0.083
+0.090
+0.062
+0.018
+5
+15
+0.110
+0.090
+0.078
+0.074
+0.083
+0.017
+20
+0.103
+0.090
+0.075
+0.068
+0.079
+0.020
+25
+0.083
+0.079
+0.069
+0.058
+0.083
+0.019
+5
+0.135
+0.116
+0.087
+0.081
+0.074
+0.014
+10
+0.118
+0.109
+0.083
+0.090
+0.062
+0.018
+6
+15
+0.110
+0.090
+0.078
+0.074
+0.083
+0.017
+20
+0.103
+0.092
+0.075
+0.068
+0.073
+0.020
+25
+0.083
+0.073
+0.069
+0.058
+0.083
+0.019
+5
+0.116
+0.137
+0.084
+0.082
+0.065
+0.014
+10
+0.115
+0.106
+0.089
+0.092
+0.055
+0.013
+7
+15
+0.097
+0.088
+0.069
+0.074
+0.067
+0.013
+20
+0.095
+0.080
+0.055
+0.068
+0.067
+0.014
+25
+0.087
+0.070
+0.051
+0.058
+0.076
+0.013
+Table 3: K -score for buoys tests, smaller is better
+19
+

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@@ -0,0 +1,1010 @@
+THE PREDICTIVE FORWARD-FORWARD ALGORITHM
+Alexander Ororbia
+Rochester Institute of Technology
+[email protected]
+Ankur Mali
+University of South Florida
+[email protected]
+ABSTRACT
+In this work, we propose a generalization of the forward-forward (FF) algorithm that we call the
+predictive forward-forward (PFF) algorithm. Specifically, we design a dynamic, recurrent neural
+system that learns a directed generative circuit jointly and simultaneously with a representation
+circuit, combining elements of predictive coding, an emerging and viable neurobiological process
+theory of cortical function, with the forward-forward adaptation scheme. Furthermore, PFF
+efficiently learns to propagate learning signals and updates synapses with forward passes only,
+eliminating some of the key structural and computational constraints imposed by a backprop-
+based scheme. Besides computational advantages, the PFF process could be further useful for
+understanding the learning mechanisms behind biological neurons that make use of local (and
+global) signals despite missing feedback connections [11]. We run several experiments on image
+data and demonstrate that the PFF procedure works as well as backprop, offering a promising
+brain-inspired algorithm for classifying, reconstructing, and synthesizing data patterns. As a
+result, our approach presents further evidence of the promise afforded by backprop-alternative
+credit assignment algorithms within the context of brain-inspired computing.
+Keywords Brain-inspired computing · Self-supervised learning · Neuromorphic · Forward learning
+1
+Introduction
+The algorithm known as backpropagation of errors [38], or “backprop” for short, has long faced criticism concerning
+its neurobiological plausibility [8, 27, 12]. Despite having powered the tremendous progress and success behind
+deep learning and its every-growing myriad of promising applications [44, 9], it is improbable that backprop is
+a good, viable model of learning in the brain, such as in cortical regions. Notably, there are both practical and
+biophysical issues [12, 27], and among these issues are the following:
+• there is a lack of evidence that neural activities are explicitly stored to be used later for synaptic adjustment,
+• error derivatives are backpropagated along a global feedback pathway to generate teaching signals,
+• the error signals move back along the same neural pathways used to forward propagate information, and,
+• inference and learning are locked to be largely sequential (instead of massively/easily parallel).
+Furthermore, in processing temporal data, it is certainly not the case that the neural circuitry of the brain is unfolded
+backwards through time in order to calculate and adjust synapses [33] (as it is for backprop through time).
+Recently, there has been a growing interest in the research domain of brain-inspired computing, which focuses
+on developing algorithms and computational models that attempt to circumvent or resolve the critical issues
+such as those highlighted above. Among the most powerful and promising ones is predictive coding (PC)
+[15, 37, 10, 3, 40, 32], and among the most recent ones is the forward-forward (FF) algorithm [16]. These
+alternatives offer powerful, different means of conducting credit assignments that have shown similar performance
+as backprop, but to the contrary, are more likely consistent with and similar to real biological neuron learning
+(see Figure 1 for some representative credit assignment depictions). This paper will propose a novel model and
+learning/inference process, the predictive forward-forward (PFF) process, that generalizes and combines FF and
+PC into a robust (stochastic) neural system that simultaneously learns a representation and generative model in a
+biologically-plausible fashion. Like the FF algorithm, the PFF procedure offers a promising, potentially helpful
+model of biological neural circuits, a potential candidate system for low-power analog hardware and neuromorphic
+circuits, and a potential backprop-alternative worthy of future investigation and study.
+arXiv:2301.01452v1  [cs.LG]  4 Jan 2023
+
+Preprint
+h3
+h2
+h1
+W1
+W2
+W3
+WT
+3
+WT
+2
+WT
+1
+h3
+h2
+h1
+W1
+W2
+W3
+B3
+B2
+B1
+L, Y
+L, Y
+h3
+h2
+h1
+W1
+W2
+W3
+WT
+3
+WT
+2
+WT
+1
+L3, Y
+L2
+L,
+h3
+h2
+h1
+W1
+W2
+W3
+E3
+E2
+E1
+L3, Y
+L2
+L1
+X
+X
+X
+BP
+h3
+h2
+h1
+W1
+W2
+W3
+L3, Ypos
+L2
+L1
+X_pos
+hNg
+3
+hNg
+2
+hNg
+1
+X_neg
+FF
+h3
+h2
+h1
+W1
+W2
+W3
+L3, Ypos
+L2
+L1
+X_pos
+hNg
+3
+hNg
+2
+hNg
+1
+X_neg
+zG
+3
+zG
+2
+zG
+1
+GM
+L = Global loss
+LN = Local loss
+WN = Forward weights
+BN = Fixed backward weights (random 
+weights)
+EN = Learnable recurrent error weights
+WT
+N = Transpose of forward activity
+GN = Generative weights
+GM = Generative model
+HN = Hidden States
+HNG
+N = Hidden States obtained by doing 
+2nd forward pass on negative data
+ZN = Error Corrected State
+ZG
+N = Generative Model State
+X = Input
+Y = Output
+X_pos = Positive input data
+X_neg = Negative input data
+Yneg
+Yneg
+G3
+G2
+G1
+FA
+PC
+LRA
+h3
+h2
+h1
+W1
+W2
+W3
+E3
+E2
+E1
+L3, Y
+L2
+L1
+[X, Y]
+z3
+z2
+z1
+G3
+G2
+G1
+NGC
+PFF
+X
+Figure 1: Comparison of learning algorithms that relax constraints imposed by backpropagation of errors (BP).
+Algorithms visually depicted include feedback alignment (FA) [26], predictive coding (PC) [37, 41], local repre-
+sentation alignment (LRA) [35], neural generative coding (NGC) [34, 32], the forward-forward procedure (FF)
+[16], and predictive forward-forward algorithm (PFF).
+2
+The Predictive Forward-Forward Learning Process
+The brain-inspired neural process that we will design and study is called the predictive forward-forward (PFF)
+algorithm, which builds on and generalizes aspects of the FF algorithm [16]. At a high level, the PFF process
+consists of two neural structures or circuits, i.e., a representation circuit (parameterized by Θr) that focuses on
+acquiring distributed representations of data samples and a top-down generative circuit (parameterized by Θg) that
+focuses on learning to synthesize data given the activity values of the representation circuit. Thus, the PFF process
+can be characterized as a complementary system with the aim of jointly learning a classifier and generative model.
+We will first define the notation used throughout this paper, then proceed to describe the inference and learning
+mechanics of the representation circuit followed by those of the generative circuit.
+Notation:
+We use ⊙ to indicate a Hadamard product and · to denote a matrix/vector multiplication. (v)T is the
+transpose of v. Matrices/vectors are depicted in bold font, e.g., matrix M or vector v (scalars shown in italic). zj
+will refer to extracting jth scalar from vector z. Finally, ||v||2 denotes the Euclidean norm of vector v. The sensory
+input has shape x ∈ RJ0×1 (J0 is the number of input features, e.g., pixels), label has shape y ∈ RC×1 (where C
+is the number of classes), and any neural layer has shape zℓ ∈ RJℓ×1 (Jℓ is the number of neurons in layer ℓ).
+2.1
+The Forward-Forward Learning Rule
+The PPF process, like the FF algorithm when it is applied to a recurrent network, involves adjusting the synaptic
+efficacies of a group of neurons by measuring their “goodness”, or, in other words, the probability that their activity
+indicates that an incoming signal comes from the target training data distribution (or the “positive class”). Formally,
+for any single layer ℓ in an L-layered neural system, we calculate the goodness as the sum of the squared activities
+for a given neural activity vector zℓ and compare it to particular threshold value θz in one of two ways:
+p(c = 1)ℓ =
+1
+1 + exp
+�
+− (�Jℓ
+j (zℓ
+j)2 − θz)
+�, or, p(c = 1)ℓ =
+1
+1 + exp
+�
+− (θz − �Jℓ
+j (zℓ
+j)2)
+�
+(1)
+where p(c = 1)ℓ indicates the probability that the data comes from the data distribution (i.e., positive data, where
+the positive class is labeled c = 1) while the probability that the data does not come from the training data
+distribution is p(c = 0)ℓ = 1 − p(c = 1)ℓ. Note that p(c)ℓ indicates the probability that is assigned by a layer ℓ of
+neurons in a system/network. This means the cost function that any layer is trying to solve/optimize is akin to a
+binary class logistic regression problem formulated as follows:
+L(Θℓ) = − 1
+N
+N
+�
+i=1
+ci log p(ci = 1)ℓ + (1 − ci) log p(ci = 0)ℓ
+(2)
+2
+
+Preprint
+where the binary label ci (the label for the ith datapoint xi) can be generated correctly and automatically if one
+formulates a generative process for producing negative data samples. Data patterns sampled from the training set
+xj ∼ Dtrain can be labeled as cj = 1 and patterns sampled outside of Dtrain (from the negative data generating
+process) can be automatically labeled as cj = 0. Crucial to the success of the FF procedure is the design of a useful
+negative data distribution, much as is the case for noise contrastive estimation [13].
+It is important to notice that the FF learning rule is local in nature – this means that the synapses of any particular
+layer of neurons can be adjusted independently of the others. The rule’s form is furthermore different from a
+classical Hebbian update [14] (which produces a weight change by a product of incoming and outgoing neural
+activities), given that this synaptic adjustment requires knowledge across a group of neurons (goodness depends on
+the sum of squares of the activities of a group rather than an individual unit) and integrates contrastive learning
+into the dynamics. Synaptic updates are specifically calculated by taking the gradient of Equation 2, i.e., ∂L(Θℓ)
+∂Θℓ .
+In effect, a neural layer optimizes Equation 2 by either maximizing the squared activities of a layer (to be above
+threshold θz) (left form of the probability presented in Equation 1) or, alternatively, minimizing the squared
+activities (right form of the probability presented in Equation 1).
+2.2
+The Representation Circuit
+In order to take advantage of the above FF learning rule (and to model contextual prediction via top-down and
+bottom-up influences), a recurrent network was proposed in [16], where, at each layer, a set of top-down and
+bottom-up forces are combined to compute the activity of any layer ℓ, much akin to the inference process of a deep
+Boltzmann machine [39]. The core parameters of this model are housed in the construct Θr = {W1, W2, ..., WL}
+(later referred to as the representation parameters). Note that no additional classification-specific parameters are
+included in our model (in contrast to the model of [16]), although incorporating these is straightforward.1 Note that
+the representation circuit of the the PFF process will take the form of a recurrent network.
+To compute any layer’s activity within the representation circuit, top-down and bottom-up messages are combined
+with an interpolation of the layer’s activity at the previous time step. Specifically, in PFF, this is done as follows:
+zℓ(t) = β
+�
+φℓ�
+Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1))
+�
++ ϵℓ
+r
+�
++ (1 − β)zℓ(t − 1)
+(3)
+where ϵℓ
+r ∼ N(0, σ) is injected, centered Gaussian noise and z0(t − 1) = x. As in [16], we set the activation
+function φℓ() for each layer ℓ to be the linear rectifier, i.e., φℓ(v) = max(0, v). Notice the introduction of an
+interpolation coefficient β, which allows integration of the state zℓ over time (the new activity state at time t is a
+convex combination of the newly proposed state and the previous value of the state at t − 1). Furthermore, notice
+that this interpolation is similar to that of the “regression” factor introduced into the recirculation algorithm [19], a
+classical local learning algorithm that made use of carefully crafted autoencoders to generate the signals needed
+for computing synaptic adjustments. LN(z) is a layer normalization function applied to the activity vector, i.e.,
+LN(zℓ) = zℓ/(||zℓ||2 + ϵ) (ϵ is a small numerical stability factor for preventing division by zero). Note that the
+topmost layer of the representation circuit is clamped to a context vector y (which could be provided by another
+neural circuit or be set to be a data point’s label/target vector), i.e., zL+1 = y2, while the bottom layer is clamped
+to sensory input, i.e., z0(t) = x(t) (where x(t) could be the frame of video or a repeated copy of a static image x).
+Equation 3 depicts a synchronous update of all layer-wise activities, but, as noted in [16], the recurrent model could
+alternatively be implemented by cycling between even and odd-number layers, i.e., first updating all even-numbered
+layers given the activities of the odd-numbered layers followed by updating the values of the odd-numbered layers
+given the new values of the even layers, much like the generative stochastic networks of [5].
+To create the negative data needed to train this system, we disregard the current class indicated by the label y of
+the positive data xp and create an incorrect “negative label” yn by randomly (uniformly) sampling an incorrect
+class index, excluding the correct one.3 A final mini-batch of samples is dynamically created by concatenating
+positive and negative samples, i.e., x =< x, x > and y =< y, yn > (notice that positive image pixels are reused
+1If classification-specific parameters are desired, one could include an additional set of synaptic weights Θd = {W, b}
+that take in as input the top-most (normalized) activity LN(zL) of the recurrent representation circuit in order to make a rough
+prediction of the label distribution over y, i.e, p(y = i|LN(zL)) = exp(W · LN(zL) + b)i/
+� �
+c exp(W · LN(zL) + b)c
+�
+.
+This would make the recurrent model of this work much more similar to that of [16]. Softmax parameters W and b would then
+be adjusted by taking the relevant gradients of the objective Ly(W, b) = − log p(y = i|LN(zL)).
+2It is important to scale the label/context vector by a factor of about 5, i.e., the topmost layer activity would be zL+1 = y ∗ 5
+(Geoffrey Hinton, personal communication, Dec 12, 2022).
+3This deviates from how the negative label was made in [16], which chose an incorrect class index in proportion to the
+probabilities produced by a forward pass of the classification-specific parameters. This was not needed for the PFF algorithm.
+3
+
+Preprint
+and paired with the negative labels in order to create the negative samples). The PFF process then involves running
+the combined mini-batch through the neural system and calculating the relevant synaptic updates.
+Equation 3 is typically run several times (8 to 10 times as in this study and [16]), similar to the stimulus processing
+window that is simulated for predictive coding systems [37, 32]. Each time Equation 3 is run, the (bottom-up and
+top-down) synapses for layer ℓ are adjusted according to the following local update:
+∆Wℓ =
+�
+2
+∂L(Θℓ)
+∂ �Jℓ
+j (zℓ
+j)2 ⊙ zℓ�
+·
+�
+LN(zℓ−1)
+�T , and, ∆Vℓ =
+�
+2
+∂L(Θℓ)
+∂ �Jℓ
+j (zℓ
+j)2 ⊙ zℓ��
+·
+�
+LN(zℓ+1)
+�T
+(4)
+which can then be applied to the relevant parameters, i.e., Wℓ and Vℓ, via methods such as stochastic gradient
+descent (SGD) with momentum or Adam [22]. In principle, the neural layers of the representation circuit are
+globally optimizing the objective L(Θr) = �L
+ℓ=1 Lℓ(Θℓ = Wℓ) (the summation of local goodness functions).
+On Classifying Sensory Patterns:
+One might observe that our representation circuit does not include discrimi-
+natory parameters that classify inputs directly. Nevertheless, given that the supervised target y is used as context to
+mediate the top-most latent representations of the recurrent circuit above, the representation system should (positive
+data samples) acquire distributed representations that implicitly encode label information. To take advantage of
+the discriminative information encoded in PFF’s representations, as was also done in the FF algorithm, we may
+still classify by executing an inference process similar to that of early hybrid Boltzmann machine models [23, 36].
+Specifically, to classify an input x, we iterate over all possible (one-hot) values that y could be, starting with the
+first class index. Specifically, for any chosen y (such as the one-hot encoding of class index i), we run Equation
+3 for the representation circuit for T steps and then record the goodness across the layers in the middle three
+iterations (from T/2 − 1 to T/2 + 1), i.e., Gy=i = 1
+3
+�T/2+1
+T/2−1
+1
+L
+�L
+ℓ=1 θz − �Jℓ
+j (zℓ
+j
+2). This goodness calculation
+is made for all class indices y = 1, 2, ..., C, resulting in {Gy=1, Gy=2, ..., Gy=C} over which the argmax is applied
+in order to obtain the index of the class with the highest average goodness value. Note that, as mentioned in [16], if
+classification-specific parameters are included in PFF’s representation circuit, then a single feedforward pass could
+be used to obtain initial class probabilities. Then the above search could instead be simplified by conducting it
+over only the top M highest probabilities (and thus avoid an expensive search over a massive number of classes).
+To estimate the label probability distribution under the representation circuit, as we do in this work, we run the
+goodness (logits) through the softmax, i.e., p(y = i|x) ∼ exp(Gi)/(�
+c exp(Gc)).
+2.3
+The Generative Circuit
+As mentioned before, the PFF algorithm incorporates the joint adaptation of a top-down directed generative model.
+This aspect of the PFF process is motivated by the generative nature of predictive processing (PP) models [37, 10],
+particularly those that focus on learning a top-down generative model as in the framework of neural generative
+coding [32]. Crucially, we remark that jointly learning (in a biologically-plausible fashion) a generative feedback
+system could favorably provide a means of inspecting the content of the representations acquired by an FF-centric
+process as well as provide a plausible, alternative means for(internally) synthesizing negative data.
+The generative circuit, which is comprised of the set of synaptic parameters Θg = {G0, G1, ..., GL}, attempts to
+learn how to predict, at each layer, a local region of neural activity, which, as we will see by design, facilitates
+simple error Hebbian updates (much like those calculated in a PP system). Formally, the objective that this
+generative circuit will attempt to optimize (for a single data point) is:
+L(Θg) =
+L
+�
+ℓ=0
+Lℓ
+g(Gℓ) =
+L
+�
+ℓ=0
+Jℓ
+�
+j=1
+(¯zℓ
+j − zℓ
+j(t))2
+(5)
+where z0 = x (the bottom layer target is clamped to the data point being processed). Each layer of the generative
+circuit conducts the following computation:
+¯zℓ = gℓ(Gℓ · LN(�zℓ+1)), where, �zℓ+1 = φℓ+1(zℓ+1(t) + ϵℓ+1
+z
+) and, eℓ = ¯zℓ − zℓ(t)
+(6)
+¯zL = gL(GL · LN(zs)), where, zs ← zs − γ ∂LL
+g (Gℓ)
+∂zs
+// Topmost latent layer activity zs
+(7)
+where ϵℓ
+z ∼ N(0, σz) is controlled (additive) activity noise injected into layer ℓ (with a small scale, such as
+σz = 0.025). gℓ() is the elementwise activation function applied to any generative layer’s prediction and, in this
+work, we set the activation functions for layers ℓ >= 1 to be the linear rectifier while the bottom one is specifically
+set to be the clipped identity, i.e., g0(v) = HardClip(v, 0, 1). At each step of the inference process that in Section
+4
+
+Preprint
+y
+x
+Representation 
+Circuit
+Generative 
+Circuit
+z1
+z2
+z3
+e1
+e2
+e3
+e0
+𝛍1
+𝛍2
+𝛍3
+𝛍0
+zs
+Figure 2: The PFF algorithmic process depicted over three-time steps for a three hidden layer network system
+coupled to a four-layer generative system (topmost layer is the sampled latent variable zs). Solid arrows represent
+synaptic weights, dashed blue arrows depict interpolation between left and right states, and dash-dotted arrows
+depict state carry-over/direct copying. The dashed diamond curve represents a feedback pathway, gray circles
+represent neural units, and red diamonds represent error neurons. Note that since all elements of the system are
+adjusted dynamically, the generative circuit is run/updated each time the representation circuit is run/updated.
+2.2, the synaptic weights of the generative model (at each layer) are adjusted via the following Hebbian rule:
+∆Gℓ = eℓ ·
+�
+LN(zℓ+1(t))
+�T , and, ∆GL = eℓ ·
+�
+LN(zs)
+�T .
+(8)
+Notice that the topmost layer of the generative circuit (i.e., layer L + 1) is treated a bit differently from the rest, i.e.,
+the highest latent generative layer zs predicts the topmost neural activity of the representation circuit zL and is
+then adjusted by an iterative inference feedback scheme, much akin to that of sparse/predictive coding [31, 37, 32].
+Once trained, synthesizing data from the generative circuit can be done using ancestral sampling:
+¯zL+1 = zs ∼ P(zs)
+(9)
+¯zℓ = gℓ(Gℓ · LN(¯zℓ+1)), ℓ = L, (L − 1), ..., 0
+(10)
+where we choose the prior P(zs) to be a Gaussian mixture model (GMM) with 10 components, which, in this
+study, was retro-fit to samples of the trained system’s topmost activity values (acquired by running the training
+dataset Dtrain through the model), as was done for the top-down directed generative PP models of [32]. Note
+that for all circuits in PFF (both the representation and generative circuits), we treat the derivative of the linear
+rectifier activation function as a vector of ones with the same shape as the layer activity zℓ (as was done in [16]).
+The learning process of the PFF procedure is shown in Algorithm 1 and its neural circuits are depicted in Figure 2.
+Relationship to Contrastive Hebbian Learning:
+When designing a network much as we do above, one might
+notice that the inference process is quite similar to that of a neural system learned under contrastive Hebbian
+learning (CHL) [28], although there are several significant differences. Layer activities in a CHL-based system are
+updated as follows:
+zℓ(t) = zℓ(t − 1) + β
+�
+− zℓ(t − 1) + φℓ�
+Wℓ · zℓ−1(t − 1) + (Wℓ+1)T · zℓ+1(t − 1)
+��
+(11)
+where we notice that dynamics do not involve any normalization and the values for any layer ℓ are integrated a bit
+differently than in Equation 3, i.e., neural values change as a function of a form of a leaky Euler integration, where
+the top-down and bottom-up transmissions are combined to produce a perturbation to the layer rather than propose
+a new value of the state itself.
+Like CHL, FF and PFF require two phases (or modes of computation) where the signals propagated through the
+neural system will be used in contrast with one another. Given data sample (x, y), CHL specifically entails running
+the neural system first in an un-clamped phase (negative phase), where only the input image x is clamped to the
+sensory input/bottom layer, followed by a clamped phase, where both x and its target y are clamped, i.e., y is
+clamped to the output layer (positive phase). At the end of each phase (or inference cycle), the layer-wise activities
+are recorded and then used in a subtractive/contrastive Hebbian rule to calculate the updates for each matrix of
+5
+
+Preprint
+Algorithm 1 The predictive forward-forward (PFF) credit assignment algorithm. red denotes representation circuit
+computation and blue denotes generative circuit computation.
+1: Input: sample (yi, xi), data label ci (binary label: 1 = “positive”, 0 = “negative”), PFF parameters Θr and Θg
+2: Hyperparameters: State interpolation β, SGD step size η, noise scales σr and σz, stimulus time T
+3: // Note that LN(zℓ) = zℓ/(||zℓ||2 + 1e−8)
+4: function SIMULATE((yi, xi, ci), Θr, Θg)
+5:
+// Run forward pass to get initial activities
+6:
+z0 = xi,
+zℓ = φℓ(Wℓ · zℓ−1), for ℓ = 1, 2, ..., L,
+zL+1 = yi, �zL+1 = 0 (same as zs)
+7:
+for t = 1 to T do
+8:
+// Run representation circuit
+9:
+for ℓ = 1 to L do
+▷ Compute representation activities with layer-wise parameters Θℓ
+r = {Wℓ, Vℓ}
+10:
+Θℓ
+r = Θr[ℓ],
+Wℓ, Vℓ ← Θℓ
+r
+▷ Extract relevant parameters
+11:
+ϵℓ
+r ∼ N(0, σr),
+zℓ(t) = β
+�
+φℓ�
+Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1))�
++ ϵℓ
+r
+�
++ (1 − β)zℓ(t − 1)
+12:
+Calculate local goodness loss L(Θℓ
+r) (Equation 1 using data label ci)
+13:
+∆Wℓ =
+�
+2
+∂L(Θℓ
+r)
+∂ �Jℓ
+j
+(zℓ
+j)2 ⊙ zℓ�
+· �LN(zℓ−1)�T ,
+∆Vℓ =
+�
+2
+∂L(Θℓ
+r)
+∂ �Jℓ
+j
+(zℓ
+j)2 ⊙ zℓ�
+· �LN(zℓ+1)�T
+14:
+Wℓ ← Wℓ − η∆Wℓ,
+Vℓ ← Vℓ − η∆Vℓ
+▷ SGD update with step size η shown (could use Adam [22] instead)
+15:
+// Run generative circuit
+16:
+for ℓ = L to 1 do
+▷ Compute generative predictions with layer-wise parameters Θℓ
+g = {Gℓ}
+17:
+Θℓ
+g = Θg[ℓ],
+Gℓ ← Θℓ
+r
+▷ Extract relevant parameters
+18:
+ϵℓ ∼ N(0, σz), �zℓ+1 = φℓ+1(zℓ+1 + ϵℓ+1), ¯zℓ = φℓ(Gℓ · LN(�zℓ+1))
+19:
+Calculate local generative loss Lℓ
+g(Gℓ) = 1
+2
+�
+j(¯zℓ
+j − zℓ
+j(t))2
+20:
+eℓ = ¯zℓ − zℓ,
+∆Gℓ = eℓ · �LN(zℓ+1(t))�T
+▷ Note that eℓ =
+∂Lℓ
+g(Gℓ)
+∂¯zℓ
+21:
+Gℓ ← Gℓ − η∆Gℓ
+22:
+zL+1 ← zL+1 − γ
+∂LL
+g (GL)
+∂zL+1
+▷ Update latent variable zs (one step of iterative inference)
+23:
+Return Θg, Θr
+▷ Output newly updated PFF parameters
+synapses. Note that the positive phase of CHL depends on first running the negative phase. FF and PFF, in contrast,
+essentially amount to running the positive and negative phases in parallel (with each phase conditioned on different
+data), resulting in an overall faster pattern processing time (instead of one inference cycle being conditioned on the
+statistics of another, the same cycles are now run on either positive or negative data with opposite objectives [16]).
+Relationship to Predictive Coding:
+The PFF algorithm integrates the local hypothesis generation component
+of predictive coding (PC) into the inference process by leveraging the representations acquired within the recurrent
+representation network’s iterative processing window. Specifically, each layer of the representation circuit, at each
+time step, becomes the prediction target for each layer of the generative circuit. In contrast, PC generative models
+must leverage a set of feedback synapses to progressively modify their layerwise neural activities before finally
+adjusting synaptic values. Furthermore, PFF iteratively/dynamically modifies the synapses within each processing
+time step, whereas; typically, most PC circuits implement a form of expectation-maximization that, as a result,
+generally requires longer stimulus processing windows in order to learn effective generative models [32] given
+that Euler integration is being simulated (in this work, the PFF generative circuit learns a good-quality generative
+model in only 8 steps whereas the models of [32] required at least 50 steps).
+Relationship to Local Learning:
+It has been strongly argued that the synapses in the brain are likely to be
+adjusted according to a local scheme, i.e., only information closest spatially and in time to a target synapse is
+involved in computing its change in efficacy. Methods that adhere to this biological constraint/setup are referred to
+as local learning procedures [35, 25, 29, 30, 4, 21], offering a potential replacement for backprop for training deep
+neural networks, relaxing one or more of its core constraints (see Figure 3 for details related to some of the key
+ones). Desirably, it has even been shown that, empirically, updates from a local scheme can result in improved
+model generalization [25, 35]. There have been many efforts in designing biologically-plausible local learning
+algorithms, such as contrastive Hebbian learning (mentioned above) [28], contrastive divergence for learning
+harmoniums (or restricted Boltzmann machines) [17], the wake-sleep algorithm for learning Helmholtz machines
+[18], and algorithms such as equilibrium propagation [43]. Other efforts that directly integrate local learning into
+the deep learning pipeline include kickback [1] and decoupled neural interfaces [20]. It is worth pointing out that
+PFF does bear some similarity to the wake-sleep algorithm, which itself entails learning a generative model jointly
+with an inference (recognition) model. However, the wake-sleep algorithm suffers from instability, given that the
+recognition network could be damaged by random fantasies produced by the generative network and the generative
+network could itself be hampered by the low-quality representation capability of the inference network (motivating
+6
+
+Preprint
+Learning 
+Algorithms
+BP
+FA
+PC
+LRA
+NGC
+FF
+PFF
+Fwd locked
+Global
+Global
+Local
+Local
+Local
+None
+None
+Fwd error
+  ✅
+    ✅
+Fwd target
+  ✅
+    ✅
+Bwd locked
+Global
+Global
+None
+None
+None
+None
+None
+Bwd error
+   ✅
+   ✅
+    ✅
+  
+Bwd target
+   ✅
+   ✅
+    ✅
+Local loss
+   ✅
+   ✅
+    ✅
+  ✅
+    ✅
+Error Synapses
+Fixed
+Learned
+Learned
+Global signal
+     ✅
+   ✅
+   
+   ✅
+    
+  
+    
+Local Signal
+   ✅
+   ✅
+    ✅
+  ✅
+    ✅
+Generative 
+capabilities
+    ✅
+    ✅
+Generative 
+Weights
+    ✅
+    ✅
+Figure 3: Properties of different learning algorithms, i.e., backprop (BP), feedback alignment (FA), predictive
+coding (PC), local representation alignment (LRA), neural generative coding (NGC), the forward-forward algorithm
+(FF), and the predictive forward-forward process (PFF).
+the design of improvements, such as reweighted wake-sleep [6]). PFF, in contrast, aims to learn the generative
+model given the representation circuit, using the locally-adapted distributed neural activities as a guide for the
+synthesization process rather than randomly sampling the generative model to generate teaching signals for the
+recognition network (potentially distracting its optimization with nonsensical noisy signals).
+3
+Experiments
+This section describes the simulations/experiments that were run to test the proposed PFF procedure. We leverage
+several benchmark image datasets to quantitatively evaluate PFF’s classification ability (in terms of test-set error)
+and qualitatively evaluate its generative capability (in terms of visual inspection of sample reconstruction and
+pattern synthesization). The PFF process (PFF-RNN) is compared with the FF algorithm (FF) as well as several
+baselines, including the K-nearest neighbors algorithm (with K = 4, or 4-KNN), the recurrent network trained
+with the original FF algorithm [16], and two backprop-based models, i.e., a feedforward network that uses backprop
+to adjust all of its internal synapses (BP-FNN) and the same network but one that only adjusts the top-most
+softmax/output layer parameters and fixes the hidden layer synaptic parameters (Rnd-FNN). Both backprop-based
+networks are trained to minimize the categorical cross-entropy of each dataset’s provided labels. The partially-
+trained model, i.e., the Rnd-FNN, serves as a sort of lower bound on the generalization ability of a neural system,
+given that it is possible to obtain respectable classification performance with only random hidden feature detectors
+(a neural credit assignment algorithm should not perform worse than this).
+Datasets:
+In this study, we experiment with two (gray-scale) image collections, i.e., the MNIST and the Kuzushiji-
+MNIST databases. The MNIST dataset [24] specifically contains 28 × 28 images containing handwritten digits
+across 10 different categories. Kuzushiji-MNIST (KMNIST) is a challenging drop-in replacement for MNIST,
+containing 28 × 28 images depicting hand-drawn Japanese Kanji characters [7] (each class corresponding to the
+character’s modern hiragana counterpart, with 10 classes in total).
+7
+
+Preprint
+Table 1: Classification generalization results for neural systems trained under different learning algorithms (except
+for 4-KNN, which is a non-parametric learning baseline model). Measurements of mean and standard deviation are
+made across five experimental trial runs.
+MNIST
+K-MNIST
+Model
+Test Error (%)
+Test Error (%)
+4-KNN
+2.860 ± 0.000
+7.900 ± 0.000
+Rnd-FNN
+3.070 ± 0.018
+14.070 ± 0.189
+BP-FNN
+1.300 ± 0.023
+6.340 ± 0.202
+FF-RNN [16]
+1.320 ± 0.100
+6.590 ± 0.420
+PFF-RNN
+1.360 ± 0.030
+6.460 ± 0.120
+(a) MNIST recon.
+(b) MNIST synthesis.
+(c) K-MNIST recon.
+(d) K-MNIST synthesis..
+(e) MNIST rep. fields.
+(f) MNIST gen. fields.
+(g) K-MNIST rep. fields.
+(h) K-MNIST gen. fields.
+Figure 4: Model reconstruction (Left) and generated (Right) samples for MNIST and K-MNIST. In the bottom row,
+the receptive fields of the bottom-most layer of the representation (rep.) circuit (Left) and those of the generative
+(gen.) circuit (Right) are displayed.
+Simulation Setup:
+All models simulated in this study were constrained to use similar architectures in order to
+ensure a more fair comparison. All networks for all neural-based learning algorithms contained two hidden layers
+of 2000 neurons (which was also done for the FF models in [16]), with initial synaptic weight values selected
+according to the random orthogonal initialization scheme [42] (using singular value decomposition). Once any
+given learning algorithm calculated adjustment values for the synapses, parameters were adjusted, using the Adam
+adaptive learning rate [22] with mini-batches containing 500 samples. Both FF and PFF were set to use a threshold
+value of θz = 10.0 and PFF was set to use 20 latent variables (i.e., zs ∈ R20×1), representation noise ϵℓ = 0.05,
+and generative noise ϵz = 0.025.
+3.1
+Discussion
+Observe in Table 1 that the PFF procedure performs well in the context of the models simulated in this study,
+reaching a top/good-quality classification error of about 1.36% on MNIST, nearly reaching that of the well-tuned
+backprop-based classifier BP-FNN. Notably, the PFF-RNN model outperforms BP-FNN slightly on K-MNIST,
+arguably a more difficult benchmark. Both FF and PFF outperform the lower-bound baselines, i.e., 4-KNN and
+Rnd-FNN, indicating that they acquire hidden feature detectors that facilitate good discriminative performance.
+Qualitatively, in Figure 4 (Top Row), observe that PFF learns a good-quality reconstruction model and generative
+model of the image inputs. The reconstructed digits and Kanji characters are excellent and the image samples
+for both cases exhibit variety/diversity across the categories (albeit a bit blurry). Note that to sample from the
+PFF’s directed generative model, as mentioned earlier in Section 2.3, we retro-fit a GMM to samples of its latent
+8
+
+Acquired Filters3
+7455131
+0
+2
+72417172
+8
+2/5
+012
+Sb
+7375
+231
+02
+994
+038
+9b6b
+7
+3
+32.3
+22Z
+14
+3
+44
+4
+hhbth
+hb
+B0000000
+55555660
+5
+6
+5
+7777小Acquired FiltersAcquired FiltersPreprint
+variable zs, specifically optimizing a GMM via expectation-maximization with 10 components. In addition, as
+shown in the bottom row of Figure 4, the receptive fields (of the synapses of the layer closest to the sensory input
+layer) acquired by the fully-connected representation circuits of both the representation and generative circuits
+appear to extract useful/interesting structure related to digit or Kanji character strokes, often, as is expected for
+fully-connected neural structures, acquiring representative full object templates (if one desired each receptive field
+to acquire only single strokes/component features specifically, then an additional prior would need to be imposed,
+such as convolution or the locally-connected receptive field structure employed in [2, 16]).
+4
+Conclusion
+In this work, we proposed the predictive forward-forward (PFF) process for dynamically adjusting the synaptic
+efficacies of a recurrent neural system that jointly learns how to classify, reconstruct, and synthesize data samples
+without backpropagation of errors. Our model and credit assignment procedure integrates elements of the forward-
+forward algorithm, such as its local synaptic adaptation rule based on goodness and contrastive learning, with
+aspects of predictive coding, such as its local error Hebbian manner of adjusting generative synaptic weights,
+resulting in a promising brain-inspired, forward-only and backprop-free form of machine learning.
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+with deep neural networks and tree search. nature 529, 7587 (2016), 484–489.
+11
+
+Preprint
+Appendix
+Visualized Samples (Expanded)
+This appendix section presents the reconstruction and synthesized samples from the PFF models in the main paper
+at a larger image scale/size.
+(a) PFF reconstructed images.
+(b) PFF sampled images.
+(c) PFF representation receptive fields.
+(d) PFF generative receptive fields.
+Figure 5: MNIST model reconstruction (Left) and generated (Right) samples. In the bottom row, the receptive
+fields of the bottom-most layer of the representation circuit (Left) and those of the generative circuit (Right).
+12
+
+3
+7455131
+0
+2
+72417172
+8
+2/5
+012
+Sb
+7375
+231
+02
+994
+038
+9b6b
+7
+3
+32.3
+22Z
+14
+3
+44
+4
+hhbth
+hb
+B0000000
+55555660
+5
+6
+5
+7777Acquired FiltersAcquired FiltersPreprint
+(a) PFF reconstructed images.
+(b) PFF sampled images.
+(c) PFF representation receptive fields.
+(d) PFF generative receptive fields.
+Figure 6: In the top row, Kuzushiji-MNIST model reconstruction (Left) and generated (Right) samples. In the
+bottom row, the receptive fields of the bottom-most layer of the representation circuit (Left) and those of the
+generative circuit (Right).
+13
+
+Acquired Filters小Acquired Filters

6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/2301.04208v1.pdf.txt

@@ -0,0 +1,1277 @@
+Sequential Structure and Control Co-Design of
+Lightweight Precision Stages with Active Control of
+Flexible Modes
+Jingjie Wu
+Walker Department of Mechanical Engineering
+The University of Texas at Austin
+Austin, TX, 78712
+[email protected]
+Lei Zhou
+Walker Department of Mechanical Engineering
+The University of Texas at Austin
+Austin, TX, 78712
+[email protected]
+Abstract—Precision motion stages are playing a prominent role
+in various manufacturing equipment. The drastically increasing
+demand for higher throughput in integrated circuit (IC) man-
+ufacturing and inspection calls for the next-generation preci-
+sion stages that have light weight and high control bandwidth
+simultaneously. In today’s design techniques, the stage’s first
+flexible mode is limiting its achievable control bandwidth, which
+enforces a trade-off between the stage’s acceleration and closed-
+loop stiffness and thus limits the system’s overall performance.
+To overcome this challenge, this paper proposes a new hardware
+design and control framework for lightweight precision motion
+stages with the stage’s low-frequency flexible modes actively
+controlled. Our method proposes to minimize the resonance
+frequency of the controlled mode to reduce the stage’s weight, and
+to maximize that of the uncontrolled mode to enable high control
+bandwidth. In addition, the proposed framework determines
+the placement of the actuators and sensors to maximize the
+controllability/observability of the stage’s controlled flexible mode
+while minimizing that of the uncontrolled mode, which effectively
+simplifies the controller designs. Two case studies are used to
+evaluate the effectiveness of the proposed framework. Simulation
+results show that the stage designed using the proposed method
+has a weight reduction of more than 55% compared to a
+baseline stage design. Improvement in control bandwidth was
+also achieved. These results demonstrate the effectiveness of the
+proposed method in achieving lightweight precision positioning
+stages with high acceleration, bandwidth, and precision.
+Index Terms—Precision positioning systems, control co-design,
+structure control
+I. INTRODUCTION
+High-precision positioning stages are playing a critical role
+in a wide range of manufacturing and inspection tools such as
+photolithography scanners [2] and MEMS inspection systems
+[1]. The drastically growing demand for higher throughput in
+semiconductor manufacturing necessitates the next-generation
+precision motion stages with higher acceleration capability
+while maintaining excellent positioning accuracy and high
+control bandwidth [9]. Creating new lightweight precision
+positioning stages is critical to achieve this goal. However, as
+the stage’s weight reduces, its structural resonance frequencies
+will decrease to near or even within the control bandwidth
+(Fig. 1), which limits the stage’s control bandwidth and
+Control
+Bandwidth
+Frequency
+Loop gain
+Flexible dynamics 
+- Well above bandwidth
+- no stability challenges
+Control
+Bandwidth
+Frequency
+Loop gain
+Flexible dynamics 
+- Close or within bandwidth
+- Cause stability challenges
+How to design?
+Conventional Stages
+Lightweight Stages
+Fig. 1. Design challenge of lightweight precision positioning stages.
+Stage Acceleration
+✓ High Acceleration,
+✓ Low power consumption
+X Low control bandwidth 
+X Deformation during acceleration
+Closed-loop Stiffness/Control Bandwidth
+Conventional Rigid Stage
+✓ Rigid structure, high precision
+✓ High control bandwidth
+X Low acceleration 
+X High power consumption
+Today’s 
+stages
+Feasible Range
+New Feasible Range
+Lightweight stage 
+w/ Flexible Mode 
+Control
+Proposed Approach:
+Lightweight Stage w/o 
+Flexible Mode Control
+Acceleration
+Fig. 2. Illustration of acceleration and bandwidth trade-off in today’s precision
+positioning systems and motivation for the proposed lightweight stage with
+flexible mode control.
+positioning accuracy, and can even cause stability challenges
+[8].
+In the past decade, a number of research and engineering
+efforts have studied the design and control for lightweight
+precision positioning stages. For example, Laro et al. [7]
+presented an over-actuation approach to place actuators/sensors
+at the stage’s nodal locations to prevent the flexible dynamics
+from being excited by the feedback loops. Oomen et al. [9]
+proposed a system identification and robust control framework
+for wafer stages, which provides a systematic approach to
+create controller designs for stages exhibiting low-frequency
+flexible dynamics. Although effective, these studies mostly
+investigate the motion control for flexible stages, and the
+arXiv:2301.04208v1  [eess.SY]  10 Jan 2023
+
+Control
+Bandwidth
+Frequency
+Loop gain
+Flexible dynamics 
+- Well above bandwidth
+- no stability challenges
+Control
+Bandwidth
+Frequency
+Loop gain
+Flexible dynamics 
+- Close or within bandwidth
+- Cause stability challenges
+How to design?
+Conventional Stages
+Lightweight Stages
+Control
+Bandwidth
+Frequency
+Loop gain 
+Uncontrolled 
+flexible dynamics 
+- well above bandwidth
+- no stability challenges
+Actively controlled  
+flexible dynamics 
+- highly compliant
+- well within bandwidth
+baseline
+Design challenge 
+proposed
+Fig. 3. Illustration of the proposed lightweight stage design with active control
+for flexible modes.
+synergy between the structure design and controller design
+is not fully exploited. In recent years, the hardware-control
+co-design, or control co-design (CCD) [6], has been studied for
+the lightweight precision positioning stages, aiming at enabling
+a synergistic structure-control design method for precision
+positioning stages. For example, Van der Veen et al. [13]
+studied the integrated topology and controller optimization
+for a simple 2D motion stage structure. Delissen et al. [3]
+presented a topology-optimized wafer stage fabricated via
+metal additive manufacturing. In a recent study, Wu et al.
+[15] presented a nested CCD formulation of for lightweight
+precision stages with controller design constraints explicitly
+considered. Despite these advances, we make a key observation
+that in these prior lightweight precision stages designs, the
+first resonance frequency of the stage structure sets an upper
+limit for the achievable control bandwidth. This fact enforces
+a fundamental trade-off between the stage’s bandwidth and
+acceleration as illustrated in Fig. 2. Fundamental advances in
+the stage’s mechatronic design must be made to break this trade-
+off and thus enable stages with improved overall performance.
+Aiming at overcoming the aforementioned trade-off and thus
+creating new lightweight stages that can simultaneously have
+high acceleration and high closed-loop stiffness, this paper
+presents a sequential structure and control design framework
+where the low-frequency flexible modes of the stage are
+under active control. This approach has been explored in
+van Herpen et al. [14] where additional actuators and sensors
+are introduced for a lightweight stage to enhance the control
+bandwidth. However, in [14], the control for flexible dynamics
+is not considered in the stage’s structural design phase, which
+limits the achievable performance. In our work, to facilitate
+the controller design, we propose to minimize the resonance
+frequency of the stage’s mode being controlled and to maximize
+the resonance frequency of the uncontrolled mode. The target
+control bandwidth of the stage is in between the resonance fre-
+quencies, as shown in Fig. 3. We envision that this formulation
+will remove material in the stage’s structure to allow compliance
+in the actively-controlled modes thereby breaking the trade-off
+in lightweight stages, as shown in Fig. 2. With the stage’s
+structure designed, we further propose to use an optimization
+method to compute the best actuator/sensor placement. Our
+hypothesis is that maximizing the controllability/observability
+of the actively-controlled flexible modes while minimizing that
+of the uncontrolled modes will deliver the best positioning
+performance with reasonable control signal magnitude. Two
+case studies are simulated to evaluate the effectiveness of the
+proposed approach, where a stage weight reduction of > 55%
+is demonstrated compared to a baseline case. These results
+demonstrate the potential of the proposed lightweight precision
+stage design framework.
+The rest of the paper is organized as follows. Section II de-
+scribes the problem statement. Section III presents the proposed
+design framework for the lightweight precision positioning
+stage. Section IV shows the simulation evaluations with two
+case studies. Conclusion and future work are summarized in
+Section V.
+II. PROBLEM STATEMENT
+The dynamics of a precision positioning stage considering
+its flexible structural behaviors can be described by
+M(θp)¨x + D(θp) ˙x + K(θp)x = B(θp, θa)u,
+y = C(θp, θs)x,
+(1)
+where x is a vector of state variables of both rigid-body displace-
+ments and flexible displacements in the modal coordinate, M,
+D, K are the mass, damping and stiffness matrices, respectively,
+u is the vector of control signals, y is a vector of measurement
+signals, B is the input matrix which maps the control input u to
+corresponding states, C is the output matrix which maps state
+variables to measurements, θp is a vector of stage’s geometric
+design parameters, and θa, θs are the vectors of actuator and
+sensor locations, respectively.
+The design optimization problem for a lightweight precision
+stage described by (1) aims at finding a set of hardware design
+parameters θp, θa, and θs and a controller design that can
+minimize the stage’s weight while maximizing the control
+bandwidth, meanwhile satisfying certain robustness criteria.
+III. SEQUENTIAL HARDWARE AND CONTROL
+OPTIMIZATION FRAMEWORK
+This section presents a sequential framework of designing the
+hardware and controller for lightweight stages with their low-
+frequency flexible modes actively controlled. In the first step,
+an optimization problem that determines the stage’s geometric
+parameter is formulated to facilitate the active control for the
+stage’s low-frequency flexible modes. In the second step, an
+optimization is performed to determine the location of actuators
+and sensors. Finally, feedback controllers are synthesized for
+the designed stage to control the stage’s motion as well as the
+low-frequency flexible modes. The three steps are introduced
+in detail in the following sections.
+A. Stage Geometry Design Optimization
+In a lightweight precision stage with active control for
+low-frequency flexible modes, the stage’s geometry design
+
+ALoop-gain
+ANoop-gain
+How to design?
+Flexible
+vnamic
+Frequency
+Frequency
+Control
+Flexible
+Control
+bandwidth
+dynamics
+bandwidthoptimization is formulated as
+min
+θp
+Jm(θp),
+s.t.
+ωi ≤ ωlow,
+i = 1, ..., n
+ωj ≥ ωhigh,
+j = n + 1, ..., m
+θp,min ≤ θp ≤ θp,max.
+(2)
+Here, the objective function Jm represents the stage’s weight,
+θp is a vector for the stage’s geometric parameters, ωi is the i-th
+modal frequency with its corresponding vibration mode actively
+controlled, and ωj is the j-th resonance frequency where
+the corresponding mode shape is not controlled. ωlow is the
+upper bound for the actively-controlled resonance frequencies,
+and ωhigh is the lower bound for the uncontrolled resonance
+frequencies. θp,min and θp,max are the lower and upper bounds
+for the stage’s geometric parameter, respectively.
+With the stage structure design optimization formulation (2),
+the stage’s flexible modes under active control are having
+resonance frequencies below ωlow, and that of the uncontrolled
+modes are beyond ωhigh. Such an optimization process can
+enforce material removal in the stage’s structure to allow for
+compliance in the actively-controlled flexible modes, and add
+material to stiffen the uncontrolled modes.
+Remark 3.1: The selection of ωlow and ωhigh are highly
+important and determine the system’s dynamic behavior. The
+system’s target control bandwidth must be between ωlow and
+ωhigh, and ωhigh sets the new upper bound for the achievable
+control bandwidth for the lightweight precision stage with
+actively controlled flexible modes, as illustrated in Fig. 2.
+To facilitate controller design while maintaining design
+feasibility, the values of ωlow and ωhigh need to be se-
+lected according to the target control bandwidth, for example
+ωlow ∼
+1
+2 × ωbw and ωhigh ∼ 5 × ωbw, where ωbw is the
+target bandwidth. This method, although robust, may lead to
+a relatively conservative stage design. To fully evaluate the
+feasible design range in Fig. 2, the value of ωhigh needs to be
+swept while considering the actuator/sensor positioning, which
+will be introduced in Section IV-B.
+B. Actuator and Sensor Placement
+The actuator and sensor placement optimization problem
+for the proposed lightweight stage with active flexible mode
+controlled can be formulated as
+max
+θa∈Da Ja(θa) =
+�
+i=1,...,n
+Wpi(θa) − γ
+�
+i=n+1,...,m
+Wpi(θa),
+(3)
+max
+θs∈Ds Jo(θs) =
+�
+i=1,...,n
+Woi(θs) − γ
+�
+i=n+1,...,m
+Woi(θs),
+(4)
+where θa and θs are vectors of actuator and sensor placement
+parameters, respectively; Da and Ds are the design domains
+for actuator/sensor locations, and γ is a positive user-defined
+weighting constant. Wpi and Woi are the controllability and
++
++
+𝐶𝜃𝑥
+𝐶𝜃𝑦
+u𝜃𝑦
+u𝜃𝑥
+𝐶𝑧
+𝐶𝑚1
+Control Diagram
+Lightweight 
+Stage 
+Dynamics
+Actuation 
+Recoupling 
+Transformation
+𝑢𝑧
+𝑢𝑚1
+Measurement
+Decoupling
+Transformation
+𝑢1
+𝑢2
+𝑢3
+𝑢4
+𝑦1
+𝑦2
+𝑦3
+𝑦4
+𝑟𝑧
+𝑟𝜃𝑥
+𝑟𝜃𝑦
+𝑟𝑞1
+𝑥𝑧
+𝜃𝑥
+𝜃𝑦
+𝑞1
++
+− +
+−
+−
+−
+Fig. 4. Control block diagram for the lightweight precision positioning stage
+with model decoupling.
+TABLE I
+CONTROLLER PARAMETERS [2].
+Parameter
+Description
+Typical
+Value
+ωbw
+Desired bandwidth [rad/s]
+–
+α
+PID frequency ratio
+0.3
+Kp
+Proportional gain
+–
+ωi
+Integrator frequency
+ωbw/α2
+ωd
+Differentiator frequency
+ωbw/α
+ωlp
+Lowpass filter frequency
+αωbw
+zlp
+Lowpass filter damping ratio
+0.7
+observability grammians of i-th flexible mode, respectively,
+which can be calculated as
+Wpi = ∥φi(θa)⊤Ba(θa)∥2
+2
+4ζiωi
+, Woi = ∥Cs(θs)⊤φi(θs)∥2
+2
+4ζiωi
+, (5)
+where φi is the mass-normalized mode shape of i-th flexible
+mode, Ba and Cs are the force and measurement assembling
+matrices, ζi is the modal damping ratio, and ωi is the i-th
+resonance natural frequency. The controllability/observability
+grammians Wpi and Woi quantitatively evaluate the control-
+lability/observability of the corresponding flexible mode in
+the control system, which will reflect on the peak resonance
+magnitude in the system’s frequency response.
+With actuator/sensor placement optimization formulation
+in (3) and (4), our goal is to maximize the controllabil-
+ity/observability for the actively-controlled modes to reduce
+the required controller gain, and to minimize those of the
+uncontrolled modes to reduce their coupling with the control
+systems. The value of γ provides a trade-off between the two
+design goals: a low value in γ emphasizes reducing the needed
+controller gain for actively-controlled modes, and a high value
+in γ emphasizes reducing the cross-talk between uncontrolled
+modes and controlled modes.
+C. Feedback Control Design
+With the stage’s structure and actuator/sensor locations
+determined, the plant dynamics of the stage can be found.
+Feedback controllers can be designed for each degree of
+freedom (DOF) to enable precision positioning and disturbance
+rejection. Figure 4 shows a block diagram for the control loop
+for a lightweight stage with three rigid-body DOFs and one
+flexible mode under active control. Here, the lightweight stage
+plant dynamics P : u → y can be obtained from solving (2),
+(3), and (4). The sensor measurements y are transformed to
+individual DOFs via a measurement decoupling transformation.
+Four single-input, single-output (SISO) feedback controllers
+can then be designed for four decoupled channels assuming the
+
+Rib width: 
+4 mm
+Rib distance: 
+30 mm
+𝑥
+𝑦
+𝑧
+Rib height: 
+25 mm
+Base height: 
+3 mm
+Rib width 2: 
+𝜃𝑝2
+Rib width 1: 
+𝜃𝑝1
+Rib distance: 
+𝜃𝑝3
+Rib Height: 𝜃𝑝5
+Base Height: 𝜃𝑝4
+𝑥
+𝑦
+𝑧
+𝑎1
+𝑎2
+𝑎3
+𝑎4
+𝑥𝑎
+𝑦𝑎
+𝑥𝑠
+𝑦𝑠
+𝑠1
+𝑠2
+𝑠3
+𝑠4
+𝑎1
+𝑠1
+𝑎2
+𝑠3
+𝑎3
+𝑠2
+1st: 38 Hz
+2nd: 500 Hz
+3rd: 500 Hz
+4th: 553 Hz
+Proposed: Lightweight stage w/ 1st flexible mode controlled
+Baseline: Precision stage w/o flexible mode control
+1st: 250 Hz
+3rd: 1394 Hz
+4th: 1415 Hz
+Thought maybe colorful one is better for proposed case. Rainbow.
+Flexible Modes:
+2nd: 1260 Hz
+Flexible Modes:
+Fig. 5. Case study #1: proposed and baseline stage parameter definition and resultant dynamics.
+cross-coupling between different DOFs is negligible. For each
+DOF, a fixed-structure SISO controller is selected following
+reference [5] as
+Ck(s) = Kp
+�s + ωi
+s
+�� s
+ωd
++ 1
+��
+ω2
+lp
+s2 + 2zlpωlps + ω2
+lp
+�
+,
+(6)
+where the controller parameters are described in Table I. This
+controller design follows reference [2], [4] where all the con-
+troller parameters except the controller gain can be determined
+by a target control bandwidth ωbw. This approach effectively
+simplifies the parameter tuning process. The proportional gain
+Kp and the target bandwidth are determined such that the
+control bandwidth is maximized while satisfying a robustness
+criteria[10] of
+∥Sk(s)∥∞ ≤ 2, k = 1, ..., n,
+(7)
+where Sk(s) is the closed-loop sensitivity function of the k-
+th channel as Sk = (I − GkCk)−1. With the control effort
+signals uk for each channel computed, an actuation recoupling
+transformation is used to map the control signals to individual
+actuators.
+IV. SIMULATION EVALUATION
+Two case studies are simulated to evaluate the potential and
+effectiveness of the proposed lightweight precision stage design
+method. Case study #1 considers a simple rib-enhanced stage
+structure with arbitrary sensor/actuator placements, aiming at
+demonstrating the impact of the selection of the weighting
+variable γ on controller design. Case study #2 implements
+the proposed framework for a practical lightweight planar
+motor stage with the actuator’s weight and location constraints
+considered. The performance of both case studies compared to
+that of a baseline stage design without flexible mode control
+for evaluation.
+A. Case study #1
+Figure 5 shows the diagrams of the stage structure being
+considered, which shows a rib-reinforced structure made of
+6061-T6 aluminum alloy of 300 mm × 300 mm in size. The
+coordinate system being used is also shown in Fig. 5. Herein,
+the rigid-body motion of the stage in three DOFs, including
+vertical translation (z), roll (θx), and pitch (θy) are actively
+controlled. In addition, the proposed stage also actively controls
+its first vibration mode, and the baseline stage has no control
+for flexible modes. Therefore, three actuators and three sensors
+are used for the baseline stages for exact constraint, while
+the proposed case uses four actuators and four sensors. The
+geometric parameters θp ∈ R5 and the actuator/sensor location
+parameters θa = [xa, ya]⊤ and θs = [xs, ys]⊤ are also shown
+in Fig. 5.
+Due to the geometric complexity of the ribbed stage structure,
+analytical models are not sufficient to capture its structural dy-
+namics accurately. In this work, finite element (FE) simulation
+(with COMSOL Multiphysics) is used to simulate the stage’s
+spatial-temporal behavior. In the stage geometry optimization
+problem (2) formulation for the proposed stage in Fig. 5, to
+facilitate controller design with a target control bandwidth
+of ∼ 100 Hz, the values of ωlow and ωhigh are selected as
+50 Hz and 500 Hz, respectively. In addition, the rib width and
+base height are constrained to be larger than 1 mm for the
+sake of manufactuability. With the stage geometry optimization
+problem (2) fully formulated, the Optimization Module in
+COMSOL Multiphysics is selected to solve the problem, where
+an iterative method for derivative-free constrained optimization
+COBYLA [11] is employed. The resultant stage resonance
+frequencies and mode shapes are illustrated in Fig. 5.
+The actuator/sensor placement optimization problems (3)-(4)
+are then solved for the optimized structure. In case study #1, the
+actuator/sensor location range is over the entire top surface of
+the stage, i.e., Da = Ds = {(x, y, z)
+�� ∥x∥, ∥y∥ ≤ 0.15 m, z =
+0}. The normalized mode shapes over all mesh nodes φi(x, y, z)
+for the stage and their corresponding natural frequencies ωi can
+be obtained from the FE simulations. For each node location
+within placement domain, let θa or θs = (x, y, z) ∈ Da or
+Ds and thus the actuation/sensing matrices Ba(θa) or Cs(θs)
+can be found. A modal damping of ζ = 0.01 is assumed
+for all modes, and the grammians (5) for each mode can be
+computed. A direct search algorithm is utilized to find the
+optimal actuator/sensor locations.
+Remark 4.1: When γ and the placement domain of actuators
+and sensors being identical, i.e., Da = Ds, the optimal solution
+for both (3) and (4) will be identical too. Therefore, the optimal
+configuration is a “collocated” case with the actuator and
+sensors configured at the same location [12]. In addition, the
+
+stage structure being considered is symmetrical about the x
+and y axes. Therefore, the optimal actuator/sensor location
+will be also symmetrical as shown in Fig. 5. These two facts
+significantly simplify the numerical computation required for
+the actuator/sensor placement optimization problems.
+With the stage’s geometric design and the placement of
+actuator/sensor decided, we are able to extract the state-
+space models for the proposed lightweight stage from the FE
+simulations. The system’s undamped dynamics can be written
+as
+MF E ¨xF E + KF ExF E = BF Eu,
+y = CF ExF E,
+(8)
+where xF E ∈ RnF E is the vector of displacement of all nodes
+in the FE simulation, nF E is the number of nodes from mesh
+setting, MF E, KF E ∈ RnF E×nF E are the mass and stiffness
+matrices, respectively, and BF E and CF E are the input and
+output matrices determined by the actuator and sensor locations.
+Note that the dimension of the FE-computed system dynamics
+(8) is typically very large (nF E ∼ 104) especially when a fine
+mesh is used in the simulation. To overcome this problem, the
+system dynamics (8) is transformed into the modal coordinate
+as
+¨q + Kq = B(θa)u,
+y = C(θs)q,
+(9)
+where q = Φ−1xF E is the decoupled modal state vector,
+Φ = [φ1, · · · , φn] is an n × n matrix where φi represents
+the vector of corresponding i-th mode shape with mass matrix
+normalized, i.e. Φ⊤MF EΦ = I, K = Φ⊤KF EΦ is the diago-
+nal stiffness matrix, and B(θa) and C(θs) are decoupled input
+and output matrix, respectively. In this decoupled coordinate,
+we can reduce the model order by truncating high-frequency
+vibration modes. We keep only the 3 rigid-body modes and
+first 10 flexible modes in this paper. Such model is able to
+capture the system dynamics accurately up to 1200 Hz, which
+is sufficient for controller design. Then, a modal damping term
+is introduced into the (9), and a reduced-order model in the
+form of (1) can be derived. Finally, the actuation signals u
+and measurement signals y are transformed into the decouple
+DOFs as shown in Fig. 4.
+As is stated in Section III-B, the weighting parameter γ
+in (3)-(4) provides a balance between the need to have small
+control gains and the need to decouple controlled modes and
+uncontrolled modes. In this case study, (3)-(4) are solved for
+the stage structure with a varying value of γ, and the resultant
+actuator/sensor locations are shown in Fig. 6. Here in Fig. 6,
+the red crosses represent the optimal actuator/sensor locations
+with different γ values (note that the actuators and sensors
+are collocated), and the blue lines represent the nodal lines
+of the stage’s second to fourth vibration modes. Figure 7
+shows the decoupled plant frequency responses of the proposed
+lightweight stage with actuator/sensor location optimized under
+different values of γ. Several selections of the value γ are
+discussed as below.
+Gamma Gamma
+𝛾 = 0
+𝛾 = 5
+𝛾 = 5.5
+𝛾 = 6
+𝛾 = 50
+𝛾 = 10
+𝑥
+𝑦
+𝛾
+𝑥
+𝑦
+0
+5
+5.5
+6
+10
+50
+Fig. 6.
+Optimal actuator/sensor placements under varying γ. Blue: nodal
+points of uncontrolled modes.
+Phase [deg]
+Phase [deg]
+Magnitude 
+[𝐦/𝐍]
+Phase [deg]
+Magnitude
+[𝐫𝐚𝐝/(𝐍 ∙ 𝐦)]
+Phase [deg]
+Magnitude
+[𝐫𝐚𝐝/(𝐍 ∙ 𝐦)]
+Magnitude
+[𝐦/𝐍]
+Frequency [Hz]
+Frequency [Hz]
+Fig. 7. Open-loop plant with different γ.
+(a): γ = 0: With γ = 0, the optimal actuator/sensor locations
+are at the corners of the stage (Fig. 6), where the first vibration
+mode’s modal displacement is maximized. This is because
+with γ = 0 we are only considering the need to maximize the
+controllability/observability of the actively-controlled modes,
+and not considering the effects of high-frequency uncontrolled
+modes. This is confirmed by the plant frequency response
+shown in Fig. 7 with γ = 0 (blue dashed line), where the
+last channel of the plant dynamics (the stage’s first flexible
+mode) is having high magnitude. However, this design results in
+strong coupling between the stage’s rigid body motion and the
+uncontrolled flexible modes (e.g. the second mode at 500 Hz).
+(b): γ = 50. As γ increases, the actuator/sensor locations move
+towards the the nodal location of the stage’s uncontrolled
+flexible modes, as shown in Fig. 6. This is also confirmed by
+the plant frequency responses shown in Fig. 7: as γ increases,
+the peak of uncontrolled flexible modes decreases, while the
+magnitude of the last channel in the plant dynamics (the stage’s
+first flexible mode) reduces as well.
+From the discussions above, it can be concluded that a large
+value in γ is beneficial for obtaining high control bandwidth
+at the cost of needing a higher controller gain in the flexible
+mode control. Therefore, the value of γ should be selected
+as its maximum allowed value to produce an acceptable plant
+magnitude in the actively controlled flexible mode. In this
+case study, γ = 50 (i.e. the plant as red solid lines in Fig. 7)
+is selected to enable a high control bandwidth. The resultant
+
+×0.2
+0.15
+0.1
+XX
+X
+0.05
+0
+0.05
+X X
+-0.1
++
+-0.15
+-0.2
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2G1:vertical translation
+10~3
+!! ! !
+10~5
+10-7
+101
+102
+103
+0
+7=0
+=5
+-90
+=6
+=50
+-180
+101
+102
+103G2 : tip
+10~3
+10~5
+10-7
+107
+102
+103
+0
+=0
+/=5
+-90
+=6
+=50
+-180
+101
+102
+103G3 : tilt
+10~3
+10~5
+10~7
+107
+102
+103
+0
+2=0
+1=5
+-90
+=6
+=50
+-180
+101
+102
+103=0
+=5010~1
+104
+10-7
+107
+102
+103
+0
+06-
+180
+101
+102
+103𝑎
+𝑏
+𝝎𝒃 = 𝟏𝟎𝟎 𝐇𝐳
+Frequency [Hz]
+Frequency [Hz]
+𝝓𝒎 = 𝟑𝟕°
+𝝎𝒃 = 𝟏𝟎𝟎 𝑯𝒛
+𝝎𝒃 = 𝟐𝟔 𝑯𝒛
+𝝓𝒎 = 𝟑𝟖°
+𝝓𝒎 = 𝟑𝟕°
+𝟓𝟎𝟎 𝐇𝐳
+Magnitude [abs]
+𝟏𝟐𝟔𝟎 𝐇𝐳
+𝟐𝟓𝟎 𝐇𝐳
+𝟓𝟓𝟑 𝐇𝐳
+Phase [deg]
+Fig. 8. Case study #1: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed). (a) z-DOF (translation in the
+vertical direction). (b) θx-DOF (pitch).
+TABLE II
+CASE STUDY #1 PERFORMANCE COMPARISON.
+Baseline Design
+Proposed Design
+Stage weight
+2.31 kg
+0.34 kg
+1st res. freq.
+250 Hz
+38 Hz
+2nd res. freq.
+1260 Hz
+500 Hz
+z bandwidth
+100 Hz
+100 Hz
+θx/θy bandwidth
+26 Hz
+100 Hz
+Max sensitivity
+1.89
+1.84
+optimal actuator/sensor locations are close to the nodal positions
+of the uncontrolled flexible modes, see Fig. 6. Finally, four
+SISO controllers in the form of (6) are designed for each
+actively-controlled DOFs, with a target control bandwidth of
+ωbw = 100 Hz.
+To evaluate the effectiveness of our proposed design method,
+a baseline lightweight precision stage as illustrated in Fig. 5 is
+used for comparison. This baseline stage lightweight stage does
+not have active control for its flexible modes, and only has the
+rigid body motions under feedback control. Three actuators and
+three sensors are used to achieve exact constraint in the stage
+actuation and control. In such a design, the first resonance
+frequency of the stage structure places an upper limit to the
+achievable control bandwidth. With a target control bandwidth
+of 50 Hz, the geometric parameters of the baseline stage are
+designed such that the first resonance frequency of the stage
+structure is above 250 Hz (i.e. 5× of the target bandwidth).
+Similarly, SISO controllers in the form of (6) are designed
+for all decoupled DOFs under active control such that the
+robustness criteria 7 is satisfied.
+Table. II summarizes the performance of the proposed
+lightweight stage in case study #1 and that of the baseline stage,
+and Fig. 8 shows the loop gains of both proposed and baseline
+designs in the z-DOF (translation in the vertical direction) and
+the θx-DOF (pitch direction). Comparing the loop frequency
+responses shown in Fig. 8a, it can be observed both stages
+can reach a high control bandwidth of 100 Hz with sufficient
+stability margins in the z-DOF, and the 250 Hz resonance
+in the baseline stage is not shown in its z-DOF frequency
+response. This is because the baseline’s first flexible mode is
+not controllable or not excitable by the z-axis control loop,
+and thus this resonance does not limit the stage’s control
+bandwidth in this axis. However, the 250 Hz resonance of
+the baseline stage can couple in the stage’s z-axis dynamics
+under imperfect actuator or position placement, and stability
+issue can arise in the control under such situations. In addition,
+the lightly-damped resonance at 250 Hz in the baseline stage
+is not actively controlled and thus can be easily excited by
+disturbances, which can impair the stage’s positioning accuracy.
+Comparing the loop frequency responses shown in Fig. 8b,
+it can be observed that the bandwidth of the baseline stage is
+only 26 Hz. This is primarily due to the 250 Hz resonance peak
+in the stage dynamics is coupled into the stage’s control in
+the θx direction with the current actuator/sensor configuration,
+and thus limits the achievable control bandwidth. In contrast,
+the proposed design can robustly achieve a control bandwidth
+of 100 Hz since the stage’s first resonance mode at 50Hz is
+actively controlled.
+Finally, comparing the performance shown in Table II, it can
+be seen that the weight of the proposed stage design is reduced
+by 85% compared to baseline design. To our understanding,
+this significant gain in weight reduction is due to the proposed
+stage is allowing compliance in the first flexible mode, which
+effectively removes material in the stage structure needed to
+reinforce the stage. This result shows the tremendous potential
+of the proposed approach in stage acceleration improvement and
+the power consumption reduction. In addition, comparing the
+closed-loop damping performance of the stage’s first resonance
+mode, it can be seen that the baseline stage’s resonance at
+250 Hz is only having a low damping ratio of 0.01, which can
+be excited by external disturbances. In contrast, the first flexible
+mode of the proposed stage is under closed-loop control, which
+has a bandwidth of 100 Hz and has a closed-loop damping ratio
+of 0.37. This improvement in the structural damping shows
+the potential of the proposed approach to improve the stage’s
+positioning accuracy under external disturbances.
+B. Case study #2
+Case study #2 considers a magnetically-levitated planar
+motion stage as illustrated in Fig. 9, where four neodymium
+
+100
+360
+Proposed
+180
+Baseline
+0
+180
+360
+10
+102
+10310°
+06-
+-180
+-270
+-360
+Proposed
+Baseline
+450
+10
+102
+103permanent magnet arrays of 60mm × 60 mm × 6 mm are
+arranged at the corner of the stage to provide both thrust forces
+for planar motion and the levitation forces. The inclusion of the
+actuator magnets enhances the practical relevance of the case
+study for wafer positioning application. The vertical-directional
+levitation forces are assumed to be located at the center of the
+permanent magnet arrays. All other stage geometry parameters
+are defined in the same way with case study #1.
+As stated in Remark 3.1, the value of ωhigh sets an upper
+bound for the achievable control bandwidth for the proposed
+positioning stage. However, using a high value of ωhigh can
+enforce the stage design to increase materials to stiffen the
+corresponding resonance mode, and thus increase the stage’s
+weight. Therefore, to fully explore the feasible designs set
+as illustrated in Fig. 2 and thus to remove possible design
+conservatism, the value of ωhigh needs to be swept. It is
+worth pointing out that the stage geometry design (2) and
+the actuator/sensor placement design (3)-(4) collaboratively
+determine the plant dynamics of the positioning stage. When
+conducting a parameter sweep for ωhigh, the actuator/sensor
+placement problems must also be solved for each stage
+geometry design for effective design optimization.
+To reduce possible design conservatism and thus fully exploit
+the advantages brought by the flexible mode control, the feasible
+stage design set for case study #2 is explored as follows:
+First, a target control bandwidth is selected to be 120 Hz for
+the positioning stage. Next, the stage geometry optimization
+problem (2) is solved with ωhigh = 600 Hz, i.e. 5× of the
+target bandwidth. Then, the sensor positioning optimization
+problem (4) is solved with γ = 50. Note that the actuator’s
+locations are fixed due to the inclusion of magnet arrays. With
+one feasible stage and sensor positioning design provided by
+the previous steps, we then decrease the value of ωhigh by a
+constant step δω = 10 Hz and resolve (2) and (4). Assuming δω
+is sufficiently small, the change in optimal geometric parameters
+can be assumed continuous, which allows us to use the optimal
+solution from the previous run as the initial parameters when
+resolving (2). This method effectively reduces the required
+computation time. The previous steps are repeated until ωhigh
+is sufficiently low such that it may be excited by external
+disturbances. In this case study, the lowest value of ωhigh is
+selected to be at 300 Hz.
+In the stage geometry optimization problem, the optimal
+solutions always have the stage’s second resonance frequency
+match ωhigh. Fig. 11 shows the stage geometric parameters
+and the resultant stage weight and actuator/sensor placement
+objectives under varying ωhigh. It can be observed that the
+stage’s weight is reducing as the value of ωhigh decreases,
+and the value of Jp + Jo (i.e. sum of objectives of (3)-(4))
+is also decreasing along with the reduction of ωhigh. These
+observations reveal new trade-off between the stage’s achievable
+control bandwidth and acceleration (assuming constant thrust
+force generation), which is illustrated by the orange line in
+Fig 2.
+The stage hardware design can be manually made among
+the optimal designs based on the results shown in Fig. 11.
+TABLE III
+CASE #2 OPTIMAL PARAMETERS
+Baseline Design
+Proposed Design
+Stage weight
+2.67 kg
+1.20 kg
+First res. freq.
+251 Hz
+50 Hz
+2nd res. freq.
+1080 Hz
+540 Hz
+z motion bandwidth
+25 Hz
+120 Hz
+θx/θy bandwidth
+120 Hz
+120 Hz
+Max sensitivity
+1.80
+1.94
+In this case study, ωhigh = 540 Hz is selected to provide
+sufficiently high Jp + Jo values while reducing the stage’s
+weight. Compared to the initial stage design using ωhigh =
+600 Hz, the stage’s weight is reduced by 4.5%. Although the
+improvement is not significant, it is worth pointing out that the
+geometry optimization of the stage is relatively limited in the
+current formulation with only five parameters that can be varied.
+A more significant improvement in the stage’s performance
+may be expected given increased design flexibility is allowed
+in the stage structure. The resultant stage’s flexible modes are
+illustrated in the bottom left in Fig. 9. The state-space dynamic
+model of the stage can be derived for this stage in the same
+way as discussed in case study #1, and controllers are designed
+for the decoupled motions.
+To evaluate the effectiveness of our proposed framework con-
+sidering actuator weight and constraints, a baseline lightweight
+stage with same magnet array is simulated for comparison.
+In the baseline stage, only the rigid-body motions are under
+active control, and all flexible modes are uncontrolled. With a
+target bandwidth of 50 Hz, the stage’s geometric parameters
+are designed to constrain the first resonance frequency above
+250 Hz. Fig. 9 show the baseline stage design parameters
+and actuator/sensor location. Three SISO controllers as (6) are
+designed for all decoupled DOFs in the same way with case
+study #1.
+Table. III summarizes the performance and comparison of
+the proposed and baseline stage design in case #2, Fig. 10
+illustrates the loop gains of both proposed and baseline designs
+in z- and θx-DOFs. Comparing the loop frequency responses in
+Fig. 10a, it can be observed that the bandwidth of the baseline
+design is limited to 25 Hz due to the 251 Hz resonance peak.
+In contrast, the proposed design can reach a bandwidth of
+120 Hz with sufficient stability margin. Fig. 10b shows that
+both designs can reach a bandwidth of 120 Hz in the θx-DOF.
+This is because the 251 Hz resonance peak in the baseline
+stage is not excitable by the θx feedback loop. However, similar
+to the z-DOF in case study #1, stability issue can be caused
+if the actuator/sensor placement is imperfect. Moreover, the
+lightly-damped 251 Hz resonance mode can be easily excited
+by external disturbance and thus impair the stage’s positioning
+precision.
+Finally, Table III shows that the weight of the proposed
+stage design is reduced by 55% compared to baseline design.
+The significant improvement for a stage considering the weight
+of magnet array shows the effectiveness and generality of our
+proposed approach. In addition, comparing the closed-loop
+damping performance of stage’s first resonance mode, it can be
+
+𝑥
+𝑦
+𝑧
+60 mm
+𝑎1
+𝑎4
+𝑎2
+𝑎3
+Rib width 1: 
+𝜃1
+Rib width 2: 𝜃2
+𝑥
+𝑦
+Rib distance: 
+𝜃3
+Rib Height: 
+𝜃5
+Base 
+Height: 𝜃4
+6 mm
+𝑠1
+𝑠2
+𝑠3
+𝑠4
+60 mm
+Rib distance: 
+30 mm
+Rib width: 3 
+mm
+6 mm
+𝑥
+𝑦
+𝑥
+𝑦
+𝑧
+𝑎1
+𝑎4
+𝑎2
+𝑎3
+Rib height: 
+25 mm
+Base height: 
+3 mm
+𝑠1
+𝑠2
+𝑠3
+𝑠4
+Proposed: Practical lightweight stage w/ 1st flexible mode controlled
+Baseline: Practical precision stage w/o flexible mode control
+1st: 50 Hz
+2nd: 540 Hz
+3rd: 540 Hz
+4th: 547 Hz
+1st: 251Hz
+2nd: 1080 Hz
+3rd: 1183 Hz
+4th: 1241 Hz
+Flexible Modes:
+Flexible Modes:
+Fig. 9. Case study #2 proposed and baseline stages. Both stages consider a permanent magnet array with 60 mm × 60 mm × 6 mm for planar motor force
+generation.
+𝑎
+𝑏
+Frequency [Hz]
+Frequency [Hz]
+Magnitude [abs]
+Phase [deg]
+𝟐𝟓𝟏 𝐇𝐳
+𝟏𝟐𝟒𝟎 𝐇𝐳
+𝟓𝟒𝟕 𝐇𝐳
+𝟓𝟒𝟎 𝐇𝐳
+𝝓𝒎 = 𝟑𝟕°
+𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛
+𝝎𝒃 = 𝟐𝟓 𝑯𝒛
+𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛
+𝝓𝒎 = 𝟑𝟕°
+Fig. 10. Case study #2: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed). (a) z-DOF (translation in
+the vertical direction). (b) θx-DOF (pitch).
+Rib Distance
+Rib Height
+Base Height, Rib Width 1&2
+2nd Resonance Frequency [Hz]
+Length [mm]
+2nd Resonance Frequency [Hz]
+Mass [kg]
+Jp+Jo [𝐚𝐛𝐬]
+𝑎
+𝑏
+2nd Resonance Frequency [Hz]
+Fig. 11.
+(a) Geometric parameter history. (b) Stage weight and grammian
+history.
+stated that the proposed design is more robust against external
+disturbances with the first lightly-damped mode at 547 Hz,
+while that of the baseline stage is at 251 Hz. The comparison
+indicates the huge potential of our framework to improve both
+the stage’s acceleration capability and positioning accuracy
+simultaneously.
+V. CONCLUSION AND FUTURE WORK
+In this work, we proposed and evaluated a sequential
+hardware and controller co-design framework for lightweight
+precision stages, aiming at enabling designs that can achieve
+high control bandwidth and high acceleration simultaneously.
+The algorithm of the framework is presented, and the effec-
+tiveness of the proposed method is demonstrated by numerical
+simulations using two case studies. The significant weight
+reduction (>55%) and improvement in control bandwidth
+show the potential. Future work will consider the experimental
+evaluations for the proposed method. A fully integrated
+controller and hardware co-optimization that can better exploit
+the synergy between hardware and control designs will also
+be studied.
+REFERENCES
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+type mirau interferometer for parallel inspection of mems,” Journal of
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+in IECON 2020 The 46th Annual Conference of the IEEE Industrial
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+350
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+Proposed
+Baseline
+0
+-90
+-180
+270
+10 1
+102
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+180
+270
+101
+102
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+

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+SYNERGY BETWEEN NP AND HEP RESEARCH GOALS
+AND EFFORTS IN FUNDAMENTAL SYMMETRIES AND
+INTERACTIONS
+Tanmoy Bhattacharya and Rajan Gupta
+Los Alamos National Laboratory, T-2, Los Alamos, NM 87545, USA
+Kate Scholberg
+Department of Physics, Duke University, Durham, NC, 27708, USA
+(Dated: January 10, 2023)
+The aim of this white paper is to highlight several areas for which the Department
+of Energy’s Office of Nuclear Physics has primary stewardship or significant invest-
+ment and expertise, and for which there is also significant interest and expertise
+within the HEP community. These areas of overlap offer exciting opportunities for
+collaboration.
+The 2021 Snowmass process brought to the fore a remarkable collaboration between nu-
+clear and high energy physicists to elucidate the potential for significant progress through
+joint experimental and theoretical efforts in four areas of great interest to the “Fundamental
+Symmetries” subprogram of the DOE Office of Science, Nuclear Physics. This collaboration
+is evident from the joint authorship of four contributions [1–4], including the associated top-
+ical group reports [5–7]. These four areas are: (i) neutrinoless double beta decay (0νββ), (ii)
+the neutron electric dipole moment (nEDM), (iii) tests of CKM unitarity through precision
+calculations for the extraction of the Vud matrix element, and (iv) lepton-nucleus scattering.
+In addition, there are ongoing searches for novel scalar and tensor interactions at the TeV
+scale, and NN oscillations for baryon number violation. Conclusive results in any of these
+areas could merit the Nobel prize, and will open new directions in beyond-the-standard-
+model (BSM) physics. In this short document we summarize the physics goals, the open
+challenges and why collaborative efforts by multiple communities would greatly accelerate
+progress.1
+Neutrinoless double beta decay [8]: A signal in experiments searching for 0νββ will
+be a clear evidence of lepton-number-violation (LNV) and will demonstrate the Majorana
+nature of neutrinos. An observation in the next-generation experiments will either identify
+the neutrino mass ordering or, if oscillation experiments and advances in cosmology will
+show that neutrinos are organized in the “normal ordering”, may provide decisive evidence
+of BSM physics, shedding light on the mechanism of neutrino mass generation. There are
+several experiments worldwide [9], with the US program stewarded by DOE NP pursuing a
+multi-experiment international strategy.
+1 We note that the areas highlighted here do not represent all possible opportunities for joint NP/HEP
+collaboration. For example, instrumentation development challenges are shared between the communities
+as well.
+arXiv:2301.03086v1  [hep-ph]  8 Jan 2023
+
+2
+0νββ experiments are sensitive to a variety of LNV mechanisms, from the “standard
+mechanism” of light-Majorana-neutrino exchange, to contributions mediated by new parti-
+cles at the TeV scale, or by weakly coupled light particles such as sterile neutrinos. Identi-
+fying the microscopic mechanism behind a signal demands a rich theoretical program over
+a wide range of energy scales [4]. At high energy, particle physics models with LNV need to
+be further developed, and the complementarity between 0νββ experiments, cosmology, and
+searches at present and future high-energy colliders needs to be further explored.
+The 0νββ rates induced by light-Majorana exchange or less minimal LNV models can
+be computed by using a tower of effective field theories (EFTs), systematically linking the
+electroweak to the nuclear scale. Because of the lack of experimental data, the couplings
+in the nuclear EFTs need to be determined directly from QCD. Lattice QCD is currently
+the only way to systematically and reliably compute the necessary matrix elements. Signif-
+icant progress has already been achieved in the calculation of LNV pion couplings [10–14].
+The determination of 0νββ transition operators requires, in addition, nucleon-nucleon LNV
+couplings [15], even for light-Majorana-neutrino exchange [16]. Progress on this front will re-
+quire further theoretical developments to relate lattice QCD results to physical two-nucleon
+matrix elements [17, 18], coupled with computational advances to obtain precise two-nucleon
+spectra and matrix elements.
+The results from Lattice QCD will then serve as input for many-body calculations of nu-
+clear matrix elements (NME) in experimentally relevant isotopes. Here ab initio methods are
+starting to appear alongside more traditional phenomenological approaches. If accompanied
+by more Lattice QCD and EFT work towards the construction of nuclear interactions and
+transition operators at the same order and in the same regularization scheme, these meth-
+ods will provide NMEs, and thus 0νββ rates, with a controlled estimate of the theoretical
+uncertainties.
+Neutron Electric Dipole Moment (nEDM) [3]: One of the profound mysteries of
+nature is the lack of matter-antimatter symmetry in the universe, i.e., the almost total
+absence of antibaryons. The symmetry between baryons and antibaryons is expected to
+have been broken during the evolution of the universe post inflation [19], and requires CP
+violation (��
+CP) [20]. If it is in the quark sector, then it has to be larger than that present
+in the CKM quark mixing matrix [20]. In that case, weak-scale Baryogenesis is the favored
+mechanism for creating the asymmetry [21]. If it is in the neutrino mixing matrix, then
+it would be through Leptogenesis [22]. Any ��
+CP interaction in the quark sector necessarily
+contributes to the nEDM, and most popular BSM models have additional ��
+CP that would
+give a dn > 10−28 e-cm [23].
+The DOE NP Flagship SNS EDM experiment being built in the US at Oak Ridge is
+designed to reach dn ∼ 3×10−28 e-cm [24], and there is a less ambitious effort at LANL [25]
+using already proven technology. A successful measurement will give credence to electroweak
+baryogenesis [26] as the mechanism for the baryon asymmetry. The value (or the lowering
+of the bound in case of a null result) for dn will provide stringent constraints on possible
+BSM theories, provided results for the matrix elements of low energy novel ��
+CP operators
+of dimension six or less can be calculated between the neutron ground state with O(20%)
+accuracy. Lattice QCD [3], with effective field theory methods [27] providing the connection
+between ��
+CP couplings in BSM theories and the low-energy effective ��
+CP operators [23, 28, 29],
+is attempting to reach this precision over the next decade—there are currently multiple col-
+laborations between nuclear and HEP physicists doing the lattice and the EFT calculations
+to achieve this. This combined effort is designed to elucidate fundamental symmetries and
+
+3
+interactions at far beyond the TeV scale, often complementary to the searches at the LHC.
+Lepton-Nucleus scattering [2]: The flagship of the HEP program in the US is the
+DUNE experiment at Fermilab [30]. It is designed to quantify ��
+CP in the neutrino sector.
+Since there is ��
+CP in the quark sector, it is important to quantify it in the neutrino sector.
+Reaching the design precision requires accurate measurements of the ν-nucleus cross-section.
+Essential, but the least constrained, ingredients for this are the nucleon axial vector form fac-
+tors and transition matrix elements over the range of a few hundred MeV to a couple of GeV
+incident neutrino energy, and corrections to these from nuclear effects [2]. This energy range
+covers the difficult-to-model quasi-elastic and resonant regions, making the cross-section cal-
+culations and Monte Carlo event generators challenging. The most promising approach to
+reach the required precision is to use lattice QCD to calculate the axial form factors of the
+nucleons and input them into nuclear many-body calculations of the cross-section.
+At lower energies (few to few-hundred MeV), neutrino-nucleus interactions are relevant
+for astrophysical neutrinos (e.g., solar, atmospheric and supernova neutrinos), and their
+understanding is important both for the interpretation of detected signals and for processes
+occurring in the sources. Thus, astrophysical signals provide information on both the sources
+and the properties of neutrinos themselves. Neutrino cross-section measurements in this
+regime are also relevant for the understanding of weak couplings and nuclear transitions, as
+well as for searches for BSM physics [31, 32]. Experimental data in this energy regime are
+sparse and theoretical understanding is also modest. Joint HEP-NP efforts for both theory
+and experiment are underway, for example in the context of experiments at stopped-pion
+sources [33–35].
+The planned electron-ion collider (EIC) is designed to provide a detailed 3D tomographic
+map of the structure of nucleons in terms of quarks and gluons [36]. Experiments at the
+EIC will significantly improve the measurements of electric and magnetic form factors that
+also enter the analysis of ν-nucleus interactions. Similarly, improvements in the extraction
+of parton distribution functions are of interest to both the NP and HEP communities [37].
+In all three areas, the ongoing collaborative efforts between HEP and NP physicists again
+demonstrate that the relevant communities are already working together.
+Test of CKM unitarity [38–41]: Understanding of nuclear β decays was instrumental
+in the discovery of the Standard Model. Even in the era of the LHC, β decay experiments
+can probe BSM physics at scales of ≳ 10 TeV, highly competitive with direct searches.
+Tests of unitarity of the first row of the Cabibbo-Kobayashi-Maskawa mixing matrix
+are particularly sensitive to these effects. Recently, a revaluation of the “inner radiative
+correction” [39, 41–43] has led to a reduction of the uncertainty in the extraction of Vud from
+superallowed 0+ → 0+ decays, while progress in lattice QCD resulted in permille accuracy
+on the form factor f+(0) and on the ratio fK+/fπ+, needed to extract Vus and Vus/Vud from
+kaon decays [44]. These advances revealed a ∼ 3σ tension with the SM [38–41]. Understand-
+ing the tension is limited by theoretical errors, with an uncertainty currently dominated by
+nuclear corrections in 0+ → 0+ decays [43]. In the near future, measurements of the neutron
+lifetime τn with uncertainty ∆τn ∼ 0.1 s, and of ratio λ = gA/gV of the neutron axial and
+vector coupling with uncertainty ∆λ/|λ| ∼ 0.03%, will allow for the extraction of Vud from
+neutron decay with accuracy comparable to superallowed β decay. Such an extraction will
+have the advantage of not being affected by nuclear corrections. Lattice QCD can play an
+important role in validating and reducing the error on the radiative corrections to meson
+and nucleon decays. The first calculations for pion and kaon decays have already appeared
+[45–49], and work on nucleon decay is ongoing. In addition to CKM unitarity, decay spectra
+
+4
+and correlations also provide tests of new charged-current interactions at scales of about
+10 TeV. Lattice QCD has provided precise calculations of the scalar and tensor charges
+[44, 50–53], which are needed to convert bounds on the Fierz interference terms onto bounds
+on quark-level operators (see below). Comparing experimental extractions and lattice QCD
+calculations of the nucleon axial charge gA can provide strong bounds on right-handed
+charged currents. With lattice QCD approaching the percent level precision [44, 50, 54, 55],
+these comparisons are now limited by electromagnetic corrections [56].
+Novel Scalar and Tensor Interactions at the TeV scale [57]: The two commu-
+nities are also working to search for novel scalar and tensor interactions at the TeV scale.
+The low-energy approach requires precision measurements of the neutron or nuclear decay
+distributions, the calculation of neutron matrix elements using lattice QCD [50, 58], and,
+in the case of nuclear decays, the ab initio calculation of nuclear matrix elements. At the
+moment, the best bounds on the neutron Fierz interference term, a probe of both scalar
+and tensor currents, come from the UCNA and Perkeo III experiments [59, 60], while the
+Nab experiment will provide bounds of a few per-mil [61]. The Fierz interference term in-
+duced by scalar interactions is sensitively probed in 0+ → 0+ superallowed β decays [43],
+while new experiments such as He6-CRES [62, 63] can investigate TeV-scale tensor cur-
+rents. At high energy, scalar and tensor interactions affect the high transverse mass tail
+of the charged-current Drell-Yan process at the LHC [64]. The latest high-transverse-mass
+Drell-Yan dataset from the ATLAS and CMS collaborations [65, 66], which use the full lu-
+minosity of the LHC Run II, provide constraints on scalar and tensor interactions that are
+very competitive with present and future β decay experiments [67].
+∆B = 2 baryon number violation in NN oscillations: The current limit on the free
+neutron oscillation time τNN ≳ 108 sec can be converted into new physics scales of 102 −103
+TeV, and upcoming experiments at the European Spallation Source will probe parameter
+space relevant to low-scale baryogenesis scenarios in which the baryon asymmetry is induced
+by the B violating decays of new particles that mediate NN oscillations [68, 69].
+Synergy in theoretical methods used: There is close synergy and often collaborations
+between NP and HEP physicists exploiting two theoretical tools needed to achieve physics
+goals: lattice QCD and effective field theory methods.
+Lattice QCD [1, 44, 55]: Large-scale simulations of lattice QCD is the most promising
+tool for many of the theoretical calculations of matrix elements needed in all physics drivers.
+Explicit examples are connecting the 0νββ, nEDM, neutron decay distributions, and N ¯N
+oscillation experiments to BSM physics, and obtaining crucial input in the the extraction
+of Vud, Vus and axial vector form factors for lepton-nucleus scattering. The US lattice QCD
+communities in both nuclear and high energy physics collaborate and work jointly, for exam-
+ple, to procure resources that are then allocated by the umbrella USQCD collaboration [70].
+Many of the teams that receive these awards have members from both communities work-
+ing collaboratively on the above six areas and have a history of producing state-of-the-art
+results. These efforts would benefit from an increase in computing resources.
+Effective Field Theory Methods [71] EFT is a systematic method to express interac-
+tions and their couplings arising in BSM theories in terms of low-energy effective operators
+composed of quark and gluon fields and organized by symmetries and dimension (roughly
+translating into importance). The renomalization group and QCD perturbation theory are
+used to run the associated couplings from the high scale to the hadronic scale of a few GeV,
+and in the process integrating out the heavy degrees of freedom systematically. Lattice QCD
+
+5
+can then be used to calculate, incorporating full non-perturbative QCD dynamics, the ma-
+trix elements of these effective operators between hadron states. These matrix elements then
+provide the connection between low energy experiments and possible fundamental theories,
+for example, between bound/value of neutron EDM and allowed values for ��
+CP couplings in
+BSM theories, i.e., constraining the space of possible theories.
+Acknowledgments
+T. Bhattacharya and R. Gupta, were partly supported by the U.S. DOE, Office of Sci-
+ence, HEP under Contract No. DE-AC52-06NA25396 and the LANL LDRD program. K.
+Scholberg is funded by the Department of Energy Office of Science, HEP and the National
+Science Foundation.
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+arXiv:2301.01957v1  [astro-ph.GA]  5 Jan 2023
+MNRAS 000, 1–6 (2022)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+A practicable estimation of opening angle of dust torus in Type-1.9 AGN
+with double-peaked broad H훼
+Xue-Guang Zhang1★
+1 School of Physical Science and Technology, GuangXi University, No. 100, Daxue Road, 530004, Nanning, P. R. China
+6 January 2023
+ABSTRACT
+In this manuscript, an independent method is proposed to estimate opening angle of dust torus in AGN, through unique properties
+of Type-1.9 AGN with double-peaked broad H훼 (Type-1.9 DPAGN) coming from central accretion disk. Type-1.9 AGN without
+broad H훽 can be expected by the commonly accepted unified model of AGN, considering central BLRs seriously obscured
+by dust torus with its upper boundary in the line of sight. For the unique Type-1.9 DPAGN, accretion disk originations of
+double-peaked broad H훼 can be applied to determine the inclination angle of the central accretion disk, which is well accepted
+as substitute of the half opening angle of the central dust torus. Then, among low redshift Type-1.9 DPAGN in SDSS, SDSS
+J1607+3319 at redshift 0.063 is collected, and the half opening angle of the central dust torus is determined to be around
+46±4◦, after considering disfavoured BBH system to explain the double-peaked broad H훼 through long-term none variabilities
+and disfavoured local physical conditions to explain disappearance of broad H훽 through virial BH mass properties. The results
+indicate that more detailed studying on dust torus of AGN can be appropriately done through Type-1.9 DPAGN in the near
+future.
+Key words: galaxies:active - galaxies:nuclei - quasars:emission lines - quasars: individual (SDSS J1607+3319)
+1 INTRODUCTION
+An unified model of Active Galactic Nuclei (AGN) is well known
+and widely accepted to explain different spectroscopic phenomena
+between Type-1 AGN with optical both broad and narrow emission
+lines and Type-2 AGN with only optical narrow emission lines, af-
+ter mainly considering obscurations on central Broad Line Regions
+(BLRs) by central dust torus. The unified model has been firstly dis-
+cussed in Antonucci (1993); Urry & Padovani (1995), and more re-
+cently reviewed and discussed in Netzer (2015); Kuraszkiewicz et al.
+(2021); Zhang (2022a). The Unified model has been strongly sup-
+ported by clearly detected polarized broad emission lines and/or
+clearly detected broad infrared emission lines in some Type-2 AGN
+(Tran 2003; Savic et al. 2018; Moran et al. 2020). Moreover, there
+are observational/theoretical evidence to support central dust torus
+as one fundamental structure in the unified model, such as the re-
+sults in NGC1068 in Rouan et al. (1998); Marco & Alloin (2000);
+Gratadour et al. (2015) through direct Near-IR images and polari-
+metric images, the resolved dust torus in the Circinus galaxy in
+Tristram et al. (2007), the reported diversity of dusty torus in AGN
+in Burtscher (2013), the estimated covering factors of central dust
+torus in local AGN in Ezhikode et al. (2017), the determined size of
+central dust torus in H0507+164 in Mandal et al. (2018), the well dis-
+cussed X-ray clumpy torus model in Ogawa et al. (2021) etc.. More
+recent review on dust torus can be found in Almeida & Ricci (2017).
+Under the framework of the unified model, considering different
+orientations of central dust torus in the line of sight, there is a spe-
+cial kind of AGN, Type-1.9 AGN (firstly discussed in Osterbrock
+★ Corresponding author Email: [email protected]
+(1981)), with broad H훼 emission lines but no broad H훽 indicat-
+ing central BLRs seriously obscured by dust torus with its upper
+boundary in the line of sight, besides the Type-1 and Type-2 AGN.
+Commonly, as a transition type, Type-1.9 AGN are considered as
+the best candidates on studying properties, especially properties of
+spatial structures, of the unified model expected central dust torus.
+Actually, there are some reports on the opening angles (covering
+factor) of the central dust torus in the literature. Arshakian (2005)
+have proposed a receding torus model, based on statistically signif-
+icant correlation between the half opening angle of the torus and
+[O iii] emission-line luminosity, and then followed and discussed in
+Simpson (2005); Alonso-Herrero et al. (2011); Marin et al. (2016);
+Matt et al. (2019). Zhuang et al. (2018) have reported that the half
+opening angle of the torus declines with increasing accretion rate
+until the Eddington ratio reaches 0.5, above which the trend reverses.
+Netzer et al. (2016); Stalevski et al. (2016) have found no evidence
+for a luminosity dependence of the torus covering factor in AGN not
+to support the receding torus model, similar conclusions can also be
+found in Mateos et al. (2017). More recent interesting discussions on
+central obscurations by dust torus can be found in Ricci et al. (2022)
+to support a radiation-regulated unification model in AGN.
+Until now, there are rare reports on the opening angles of the cen-
+tral dust torus in AGN through direct spatial resolved images. How
+to measure/determine the opening angle of the central dust torus in
+an individual AGN is still an interesting challenge. Here, based on
+unique properties of Type-1.9 AGN with BLRs being seriously ob-
+scured by the central dust torus, an independent method is proposed
+to estimate the opening angle of the central dust torus in a special
+kind of Type-1.9 AGN, the Type-1.9 AGN with double-peaked broad
+H훼 (Type-1.9 DPAGN). The manuscript is organized as follows.
+© 2022 The Authors
+
+2
+Zhang
+Section 2 presents our main hypothesis to estimate the half opening
+angle of the central dust torus in special Type-1.9 DPAGN. Sec-
+tion 3 shows the spectroscopic results of the Type-1.9 DPAGN SDSS
+J160714.40+331909.12 (=SDSS J1607+3319) at redshift 0.063. Sec-
+tion 4 gives the main discussions. Section 5 gives our final con-
+clusions. And the cosmological parameters have been adopted as
+퐻0 = 70km · s−1Mpc−1, ΩΛ = 0.7 and Ωm = 0.3.
+2 MAIN HYPOTHESIS
+Accretion disk originations have been well accepted to double-
+peaked broad emission lines, as well discussed in Chen & Halpern
+(1989); Eracleous et al. (1995); Storchi-Bergmann et al. (2003,
+2017). The inclination angle of the central accretion disk can be
+well estimated through double-peaked broad line emission features.
+Meanwhile, considering the serious obscurations from central dust
+torus in Type-1.9 DPAGN, the accretion disk origination determined
+inclination angle should be well accepted to trace the half opening
+angle of the central dust torus. Certainly, beside the accretion disk
+origination, commonly known binary black hole (BBH) system can
+also be applied to explain double-peaked broad emission lines, such
+as the results shown in Shen & Loeb (2010). However, assumed BBH
+system should lead to optical quasi-periodic oscillations (QPOs) with
+periodicities about hundreds to thousands of days, such as the results
+shown in Graham et al. (2015a,b); Zhang (2022), and will be dis-
+cussed to disfavor the BBH system in the target in the manuscript.
+Moreover, it should be confirmed that the seriously obscured broad
+H훽 are not due to local intrinsic physical conditions (such as the case
+in H1320+551 discussed in Barcons et al. (2003)), but due to serious
+obscurations by the central dust torus.
+It is exciting to check whether the method can be applied to esti-
+mate the opening angle of the central dust torus in Type-1.9 DPAGN,
+which is the main objective of the manuscript. And the following
+three criteria are accepted to collect targets of the manuscript. First,
+the targets are Type-1.9 DPAGN, with apparent double-peaked broad
+H훼 but no apparent broad H훽. Second, there are no signs for optical
+QPOs in the targets, indicating BBH systems not preferred to explain
+the double-peaked broad H훼. Third, after considering the BH mass
+properties which will be well discussed in the Section 4, serious ob-
+scurations by the central dust torus are well accepted to explain the
+seriously obscured broad H훽 in the targets.
+Among the low redshift (푧 < 0.35) broad line AGN listed
+in Shen et al. (2011) with SPECIAL_INTEREST_FLAG=1 and in
+Liu et al. (2019) with flag MULTI_PEAK=2, there are 561 low red-
+shift DPAGN with reliable broad H훼 emission lines (both reported
+line width and line luminosity at least five times larger than their re-
+ported uncertainties). Based on the main hypothesis and correspond-
+ing criteria above, Type-1.9 DPAGN SDSS J160714.40+331909.12
+(=SDSS J1607+3319) at redshift 0.063 is collected as the unique
+target of the manuscript, based on two main unique features through
+properties of its spectroscopic and long-term variabilities, well dis-
+cussed in the next section. On the one hand, among the Type-1.9
+DPAGN, the SDSS J1607+3319
+has the most apparent double-
+peaked features in broad H훼. On the other hand, there are no appar-
+ent variabilities in SDSS J1607+3319, which can be well applied to
+disfavour the BBH system in the SDSS J1607+3319, combining its
+double-peaked features in broad H훼.
+Figure 1. Top panel shows the SSP method determined descriptions (solid
+red line) to the SDSS spectrum (solid dark green line) with emission lines
+being masked out. In top panel, solid blue line and dashed blue line show
+the determined host galaxy contributions and power law AGN continuum
+emissions, respectively, solid cyan line shows the line spectrum calculated
+by the SDSS spectrum minus the sum of host galaxy contributions and AGN
+continuum emissions. Bottom panels show the best fitting results (solid red
+line) to absorption features (solid dark green line) of Ca ii H+K (left panel),
+Mg i (right panel). In each panel, the determined 휒2/푑표 푓 and stellar velocity
+dispersion are marked in red characters.
+3 SPECTROSCOPIC RESULTS OF THE TYPE-1.9 DPAGN
+SDSS J1607+3319
+SDSS J1607+3319 has its SDSS spectrum (plate-mjd-fiberid=1419-
+53144-0453) with signal-to-noise about 34 shown in Fig. 1. In or-
+der to measure emission lines, the commonly accepted SSP (Sim-
+ple Stellar Population) method is applied to determine host galaxy
+contributions. More detailed descriptions on the SSP method can
+be found in Bruzual & Charlot (2003); Kauffmann et al. (2003);
+Cid Fernandes et al. (2005); Cappellari (2017). And the SSP method
+has been applied in our previous papers Zhang (2021a,b,d, 2022a,b).
+Here, we show simple descriptions on SSP method as follows.
+The 39 simple stellar population templates from Bruzual & Charlot
+(2003); Kauffmann et al. (2003) have been exploited, combining with
+a power law component applied to describe intrinsic AGN continuum
+emissions. When the SSP method is applied, optical narrow emission
+lines are masked out by full width at zero intensity about 450 km/s,
+and the spectrum with wavelength range from 6250 to 6750Å are
+also masked out due to the strongly broad H훼. Then, through the
+Levenberg-Marquardt least-squares minimization technique, SDSS
+spectra with emission lines being masked out can be well described
+by combinations of broadened stellar population templates and the
+power law component. The best descriptions are shown in Fig. 1
+with 휒2/푑표 푓 ∼ 0.91 (the summed squared residuals divided by de-
+gree of freedom) and with determined stellar velocity dispersion (the
+broadening velocity) about 224±5 km/s.
+Moreover, in order to determine reliable stellar velocity dispersion,
+absorption features of around Ca ii H+K from 3750 to 4200Å and
+around Mg i from 5050 to 5250Å are applied to re-measure stel-
+lar velocity dispersions, through the same SSP method above. The
+best fitting results are shown in bottom panels of Fig. 1 with deter-
+mined stellar velocity dispersions in units of km/s about 222±11 and
+208±26 through the Ca ii H+K and Mg i, respectively. Therefore, in
+MNRAS 000, 1–6 (2022)
+
+Opening angle of Dust Torus
+3
+Figure 2. Top panels show the best fitting results (solid red line) to the emission lines (solid dark green line), and bottom panels show the corresponding
+residuals. In top left panel, solid blue line shows the determined narrow H훽, solid green lines show the determined [O iii] doublet. In top right panel, solid blue
+line shows the determined narrow H훼, solid cyan line shows the determined double-peaked broad H훼 described by the elliptical accretion disk model, solid
+green lines show the determined [O i], [N ii] and [S ii] doublets, dashed purple lines show the determined broad H훼 described by two broad Gaussian functions.
+In each bottom panel, horizontal dashed lines show residuals=0, ± 1, respectively.
+Figure 3. MCMC technique determined two-dimensional posterior distributions in contour of the model parameters in the elliptical accretion disk model
+applied to describe the double-peaked broad H훼. In each panel, sold circle plus error bars in red mark the positions of the accepted values and corresponding
+uncertainties of the model parameters. The number densities related to different colors are shown in color bar in top region of each panel.
+the manuscript, the inverse variance weighted mean stellar velocity
+dispersion 휎★ =222±26 km/s in SDSS J1607+3319 is accepted,
+which is consistent with the SDSS pipeline reported 230 km/s.
+After subtractions of host galaxy contributions and AGN contin-
+uum emissions, emission lines in the line spectrum can be well mea-
+sured. Similar as what we have previously done in Zhang (2021a,b,
+2022a,b,c), for the emission lines within rest wavelength range from
+4600 to 5150Å, there are one broad and one narrow Gaussian func-
+tions applied to describe probable broad and apparent narrow H훽,
+two Gaussian functions applied to describe [O iii]휆4959, 5007Å dou-
+blet. When the functions above are applied, each component has line
+intensity not smaller than zero, and the [O iii] components have the
+same redshift and the same line width and have flux ratio to be fixed
+to the theoretical value 3. Then, through the Levenberg-Marquardt
+least-squares minimization technique, the best fitting results to the
+emission lines and the corresponding residuals (line spectrum minus
+thebest fittingresultsandthendividedbyuncertaintiesofSDSS spec-
+trum) are shown in left panels of Fig. 2 with 휒2/푑표 푓 ∼ 0.70. Based
+on the fitting results, it is not necessary to consider broad Gaussian
+component in H훽, because the determined line width and line flux
+(around to zero) of the broad Gaussian component are smaller than
+their corresponding uncertainties, indicating there are no apparent
+broad H훽 in SDSS J1607+3319.
+Meanwhile, Gaussian functions can be applied to describe the
+narrow emission lines within rest wavelength range from 6200 to
+6850Å, the [O i], [N ii], [S ii] and narrow H훼. But the commonly ac-
+cepted elliptical accretion disk model with seven model parameters
+well discussed in Eracleous et al. (1995) is applied to describe the
+double-peaked broad H훼, because the model can be applied to ex-
+plain almost all observational double-peaked broad H훼 of the SDSS
+J1607+3319. The seven model parameters are inner and out bound-
+aries [푟0, 푟1] in the units of 푅퐺 (Schwarzschild radius), inclination
+angle 푖 of disk-like BLRs, eccentricity 푒, orientation angle 휙0 of
+elliptical rings, local broadening velocity 휎퐿 in units of km/s, line
+emissivity slope 푞 ( 푓푟
+∝ 푟−푞). Meanwhile, we have also applied
+the very familiar elliptical accretion disk model in our more recent
+studies on double-peaked lines in Zhang (2021c, 2022a), and there
+are no further discussions on the elliptical accretion disk model in
+the manuscript. Then, in order to obtain more reliable uncertainties
+of model parameters in the complicated model functions, rather than
+the Levenberg-Marquardt least-squares Minimization technique, the
+Maximum Likelihood method combining with the MCMC (Markov
+Chain Monte Carlo) technique (Foreman-Mackey et al. 2013) is
+applied. The evenly prior distributions of the seven model pa-
+rameters in the elliptical accretion disk model are accepted with
+the following limitations, log(푟0) ∈ [2, 4],
+log(푟1) ∈ [2, 6]
+(푟1
+> 푟0), log(sin(푖)) ∈ [−3, 0], log(푞) ∈ [−1, 1], log(휎퐿) ∈
+[2, 4],
+log(푒) ∈ [−5, 0], log(휙0) ∈ [−5,
+log(2 × 휋)]. The
+MNRAS 000, 1–6 (2022)
+
+4
+Zhang
+Table 1. parameters of the emission line components
+model parameters of elliptical accretion disk model for broad H훼
+푟0 = 2035 ± 240, 푟1 = 3766 ± 500, sin(푖) = 0.71 ± 0.04
+푞 = 3.35 ± 0.19, 푒 = 0.81 ± 0.08, 휎퐿 = 796 ± 70km/s, 휙0 = 190 ± 6◦
+model parameters of Gaussian emission components
+line
+휆0
+휎
+flux
+broad H훼
+6505.6±1.1
+41.4±1.2
+897±25
+6643.9±1.1
+34.9±1.2
+699±24
+Narrow H훼
+6564.2±0.5
+5.6±0.6
+311±54
+Narrow H훽
+4862.4±0.3
+4.2±0.4
+45±8
+[O iii]휆5007Å
+5008.8±0.3
+3.9±0.3
+172±10
+[O i]휆6300Å
+6301.9±1.1
+7.6±1.2
+113±14
+[N ii]휆6583Å
+6585.5±0.2
+6.5±0.3
+642±55
+[S ii]휆6716Å
+6719.2±0.9
+6.6±0.9
+260±33
+[S ii]휆6731Å
+6734.5±0.8
+4.6±0.7
+157±29
+Notice: For the Gaussian emission components, the first column shows which
+line is measured, the Second, third, fourth columns show the measured line
+parameters: the center wavelength 휆0 in unit of Å, the line width (second
+moment) 휎 in unit of Å and the line flux in unit of 10−17 erg/s/cm2.
+determined best fitting results and corresponding residuals to the
+emission line around H훼 are shown in right panels of Fig. 2 with
+휒2/푑표 푓 ∼ 0.48. The MCMC technique determined posterior dis-
+tributions of the model parameters in the elliptical accretion disk
+model are shown in Fig. 3. And the half width at half maximum of
+each parameter distribution is accepted as uncertainty of the param-
+eter. The determined parameters and corresponding uncertainties of
+each model parameter are listed in Table 1. Moreover, as discussed
+in Zhang (2022a), clean double-peaked broad line emission features
+can lead to solely determined model parameters in the elliptical ac-
+cretion disk model. Therefore, there are no further discussions on
+whether is there solely determined model parameter of sin(푖).
+4 MAIN DISCUSSIONS
+In the section, two points are mainly considered. First, it is neces-
+sary to determine that the accretion disk origination is favoured to
+explain the double-peaked broad H훼 in SDSS J1607+3319, rather
+than a BBH system. Second, it is necessary to determine that the
+large broad Balmer decrement (flux ratio of broad H훼 to broad
+H훽) is due to serious obscurations, rather than due to local phys-
+ical conditions, because that BLRs modeled with relatively low opti-
+cal depths and low ionization parameters can reproduce large broad
+Balmer decrements, as well discussed in Kwan & Krolik (1981);
+Canfield & Puetter (1981); Goodrich (1990) without considering se-
+rious obscurations and see the unobscured central regions in a Type-
+1.9 AGN in Barcons et al. (2003).
+For the first point on BBH system, the following discussions are
+given. The double-peaked broad H훼 can also be well described by
+two broad Gaussian functions shown as dashed purple lines in top
+right panel of Fig. 2 with model parameters listed in Table 1. Under
+the assumption of BBH system in SDSS J1607+3319, considering
+the strong linear correlation between broad H훼 luminosity and con-
+tinuum luminosity as discussed in Greene & Ho (2005), there are to-
+tally equal (ratio about 897:699 from emission fluxes of the two broad
+Figure 4. CSS V-band light curve of SDSS J1607+3319. Horizontal solid
+and dashed red lines show the mean value and corresponding 2RMS scatters
+of the light curve.
+Gaussian components) continuum luminosities related to central two
+BH accreting systems, indicating there should be strong variabilities
+with QPOs due to orbital rotating effects. However, there are none
+variabilities in the collected 8.4years-long CSS (Catalina Sky Survey,
+Drake et al. (2009)) V-band light curve shown in Fig. 4 with almost
+all data points lying within 2RMS scatter ranges. Therefor, rather
+than the BBH system, the elliptical accretion disk model is preferred
+to explain the double-peaked broad H훼 in SDSS J1607+3319.
+For the second point, properties of virial BH mass are mainly
+discussed. Based on accepted virialization assumptions to prop-
+erties of observed broad H훼 as discussed in Vestergaard (2002);
+Peterson et al. (2004); Greene & Ho (2005); Shen et al. (2011);
+Mejia-Restrepo et al. (2022), virial BH mass can be estimated by
+푀퐵퐻 = 15.6 × 106(
+퐿퐻 훼
+1042erg/s)0.55(
+휎퐻 훼
+1000km/s )2.06M⊙
+= (5.5 ± 0.6) × 107M⊙
+(1)
+with 퐿퐻 훼 = (1.39 ± 0.05) × 1041erg/s as line luminosity of ob-
+served broad H훼 and 휎퐻 훼 = (3100 ± 110)km/s as second mo-
+ment of observed broad H훼, after considering more recent em-
+pirical R-L relation to estimate BLRs sizes in Bentz et al. (2013).
+Uncertainty of virial BH mass is determined by uncertainties of
+the 퐿퐻 훼 and 휎퐻 훼. If large broad Balmer decrement was due to
+local physical conditions, the estimated virial BH mass should be
+simply consistent with the 푀BH − 휎 relation (Ferrarese & Merritt
+2000; Gebhardt et al. 2000; Kormendy & Ho 2013; Batiste et al.
+2017; Bennert et al. 2021) expected value, otherwise, there should
+be smaller virial BH mass. Then, Fig. 5 shows virial BH mass prop-
+erties of SDSS J1607+3319 in the 푀BH − 휎 space. In order to show
+clearer results, the 89 quiescent galaxies from Savorgnan & Graham
+(2015) and the 29 reverberation mapped (RM) AGN from Woo et al.
+(2015) and the 12 tidal disruption events (TDEs) from Zhou et al.
+(2021) are considered to draw the linear correlation between stellar
+velocity dispersion and BH mass
+log( 푀퐵퐻
+M⊙
+) = (−2.89 ± 0.49) + (4.83 ± 0.22) × log( 휎★
+km/s)
+(2)
+through
+the
+Least
+Trimmed
+Squares
+robust
+technique
+(Cappellari et al. 2013). And then the 3휎, 4휎 and 5휎 confi-
+dence bands to the linear correlation are determined and shown
+in Fig. 5. Therefore, the estimated viral BH mass of SDSS
+J1607+3319 is lower than 푀BH − 휎 expected value with confidence
+level higher than 4휎. Therefore, locate physical conditions are
+MNRAS 000, 1–6 (2022)
+
+Opening angle of Dust Torus
+5
+Figure 5. On the correlation between stellar velocity dispersion measured
+through absorption features and virial BH mass of SDSS J1607+3319.
+Solid five-point-star in dark green shows the virial BH mass of SDSS
+J1607+3319 determined by properties of observed broad H훼. Dot-dashed
+lines in magenta and in black represent the 푀BH − 휎 relations through the
+quiescent galaxies in Kormendy & Ho (2013) and through the RM AGNs in
+Woo et al. (2015), respectively. Solid circles in red, in blue and in pink show
+the values for the 89 quiescent galaxies in Savorgnan & Graham (2015), the
+29 RM AGNs in Woo et al. (2015) and the 12 TDEs in Zhou et al. (2021), re-
+spectively. Thick solid red line shows the best fitting results to all the objects,
+and thick dashed, dotted and dot-dashed red lines show corresponding 3휎,
+4휎 and 5휎 confidence bands to the best fitting results.
+disfavored to explain the large broad Balmer decrement in SDSS
+J1607+3319.
+Based on the double-peaked broad H훼 in the Type-1.9 DPAGN
+SDSS J1607+3319, half opening angle of central dust torus is well
+estimated as (46±4)◦ (sin(푖) ∼ 0.71 ± 0.04), roughly consistent with
+statistical mean value in Zhuang et al. (2018). Therefore, it is inter-
+esting to study properties of opening angles of dust torus through
+Type-1.9 DPAGN in the near future, after many efforts to disfavour
+BBH systems to explain their double-peaked broad H훼 and to dis-
+favour local physical conditions to explain disappearance of broad
+H훽.
+Before ending of the manuscript, an additional point is noted.
+Before giving clear physical information of materials in the central
+dust torus, it is hard to confirm that the accretion disk origination
+determined inclination angle is completely consistent with the half
+opening angle of the central dust torus in Type-1.9 DPAGN. If ma-
+terial densities in regions around upper boundary of the central dust
+torus were too low to lead the broad H훽 being totally obscured, the
+determined inclination angle should be lower than the intrinsic half
+opening angle of the central dust torus. Moreover, it is not clear
+whether are there different radial dependent material densities in the
+direction perpendicular to the equatorial plane related to central AGN
+activities, which should also have effects on the consistency between
+the accretion disk origination determined inclination angle and the
+half opening angle of the central dust torus in AGN with different
+central AGN activities. In the near future, through studying a sample
+of Type-1.9 DPAGN as one of our ongoing projects, clearer clues
+and detailed discussions will be given on the consistency between
+the inclination angle and the half opening angle of the central dust
+torus.
+5 CONCLUSIONS
+An independent method is proposed to estimate the opening an-
+gle of the central dust torus in Type-1.9 DPAGN through unique
+double-peaked features of broad H훼, accepted the assumptions of
+obscurations of the central dust torus on BLRs leading to disappear-
+ance of broad H훽 and of the double-peaked broad H훼 with accre-
+tion disk originations. Then, among the reported DPAGN, the SDSS
+J1607+3319 is collected due to its apparent broad double-peaked
+broad H훼 but no broad H훽. Moreover, long-term optical variabilities
+can be applied to disfavour the BBH system in SDSS J1607+3319 to
+explain the double-peaked broad H훼. And properties of virial BH
+mass can be applied to determine that local physical conditions are
+not favoured to explain the large broad Balmer decrement in SDSS
+J1607+3319. Then, based on the well applied elliptical accretion
+disk model applied to describe the double-peaked broad H훼 in SDSS
+J1607+3319, the half opening angle of the central dust torus can be
+well estimated as (46±4)◦ in SDSS J1607+3319. The results in the
+manuscript strongly indicate that the proposed independent method
+is practicable, and can be applied to study detailed properties of the
+opening angles of the central dust torus through a sample of Type-1.9
+DPAGN, which will be studied in the near future.
+ACKNOWLEDGEMENTS
+Zhang
+gratefully
+acknowledges
+the
+anonymous
+referee
+for
+giving us constructive comments and suggestions to greatly
+improve our paper. Zhang gratefully acknowledges the kind
+funding
+support
+NSFC-12173020.
+This
+research
+has
+made
+use of the data from the SDSS (https://www.sdss.org/)
+funded by the Alfred P. Sloan Foundation, the Participating
+Institutions, the National Science Foundation and the U.S. De-
+partment of Energy Office of Science, and use of the data from
+CSS
+http://nesssi.cacr.caltech.edu/DataRelease/.
+The
+research
+has
+made
+use
+of
+the
+MPFIT
+package
+https://pages.physics.wisc.edu/~craigm/idl/cmpfit.html,
+and
+of
+the
+LTS_LINEFIT
+package
+https://www-astro.physics.ox.ac.uk/~cappellari/software/,
+and of the emcee package https://pypi.org/project/emcee/.
+DATA AVAILABILITY
+The data underlying this article will be shared on request to the
+corresponding author ([email protected]).
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+

ENE0T4oBgHgl3EQfQgDK/content/tmp_files/2301.02195v1.pdf.txt

@@ -0,0 +1,954 @@
+Towards Autoformalization of Mathematics and Code Correctness:
+Experiments with Elementary Proofs
+Garett Cunningham
+School of EECS
+Ohio University
+Athens, OH 45701
+[email protected]
+Razvan C. Bunescu
+Department of Computer Science
+UNC Charlotte
+Charlotte, NC 28223
+[email protected]
+David Juedes
+School of EECS
+Ohio University
+Athens, OH 45701
+[email protected]
+Abstract
+The ever-growing complexity of mathemati-
+cal proofs makes their manual verification by
+mathematicians very cognitively demanding.
+Autoformalization seeks to address this by
+translating proofs written in natural language
+into a formal representation that is computer-
+verifiable via interactive theorem provers. In
+this paper, we introduce a semantic parsing ap-
+proach, based on the Universal Transformer
+architecture, that translates elementary math-
+ematical proofs into an equivalent formaliza-
+tion in the language of the Coq interactive the-
+orem prover.
+The same architecture is also
+trained to translate simple imperative code dec-
+orated with Hoare triples into formally veri-
+fiable proofs of correctness in Coq.
+Exper-
+iments on a limited domain of artificial and
+human-written proofs show that the models
+generalize well to intermediate lengths not
+seen during training and variations in natural
+language.
+1
+Introduction
+To the uninitiated, the notion of mathematical proof
+represents simply an argument written by people
+to convince others of mathematical truth. How-
+ever, in a real sense, mathematical proof must have
+formal underpinnings that go beyond the written
+argument. Arguments that lack such underpinnings
+might have fatal errors or even logical inconsisten-
+cies (see, for example, Russell’s Paradox (Irvine
+and Deutsch, 2021)). Nevertheless, mathematical
+arguments written in natural language are the norm
+and they have great value.
+In Tymoczko (1979)’s well-known paper that dis-
+cusses a somewhat controversial (at the time) proof
+of the Four Color Theorem (Appel and Haken,
+1977; Appel et al., 1977), he explores “what is a
+mathematical proof?” He posits that all mathemati-
+cal proofs must be (i) convincing, (ii) surveyable,
+and (iii) formalizable. The first two points are for
+the reader—proofs must be convincing to and com-
+prehensible by mathematicians. For the third point,
+he notes that, “Most mathematicians and philoso-
+phers believe that any acceptable proof can be for-
+malized. We can always find an appropriate formal
+language and theory in which the informal proof
+can be embedded and ‘filled out’ into a rigorous
+formal proof.” For most mathematicians, this third
+part is crucial for ensuring that subtle, but fatal,
+errors in logic do not exist in mathematical proof.
+Great progress has been made since the 1970’s
+in fully formalizing significant mathematical re-
+sults. For instance, the Feit-Thompson Theorem
+(Gonthier et al., 2013; Gonthier, 2013) and the Four
+Color Theorem (Gonthier, 2008) have been for-
+mally verified using the proof assistant Coq (Bertot
+and Castéran, 2013), and the Kepler Conjecture
+(Hales, 2005; Hales et al., 2017) has been formally
+verified using the proof assistants Isabelle and HOL
+Light (Nipkow et al., 2002). Moreover, proof assis-
+tants have demonstrated immense utility for soft-
+ware verification, such as the full certification of a
+C compiler (Leroy, 2009). Proofs demonstrating
+the correct behavior of code share a similar struc-
+ture to proofs in pure mathematics, where systems
+like Hoare logic replace standard first-order logic.
+Thus, Tymoczko’s criteria for mathematical proof
+can be extended to the verification of programs.
+For many experts, LaTeX provides an excellent
+tool for satisfying the first two criteria. In addition,
+carefully written LaTeX (Higham, 2020) provides
+a rich structure for establishing the third criterion.
+The vast majority of modern mathematics is ex-
+pressed using natural language (NL), with the over-
+whelming majority typeset in LaTeX. Fully for-
+malizing mathematics using proof assistants is still
+a difficult and time consuming task. This paper
+takes some preliminary steps toward bridging this
+gap by exploring how modern machine learning
+techniques can be used to convert carefully writ-
+ten LaTeX into equivalent, and formally verified
+arXiv:2301.02195v1  [cs.CL]  5 Jan 2023
+
+mathematics in Coq, a process referred to as auto-
+formalization in the literature (Szegedy, 2020).
+Wang et al. (2018, 2020) explored the similar
+task of translating mathematical statements from
+LaTeX into Mizar, using LSTM-based models with
+attention. To generate aligned LaTeX-Mizar pairs,
+they use a tool (Bancerek, 2006) that translates
+top-level Mizar statements into artificial LaTeX
+sentences, a task that is facilitated by the fact that
+Mizar is human readable and similar in length with
+the corresponding LaTeX version. Carman (2021)
+evaluated the competency of LSTMs toward for-
+malizing a restricted set of artificially generated the-
+orems about simple arithmetic expressions, report-
+ing reasonable success over expression lengths seen
+during training. More recently, Wu et al. (2022)
+evaluated Codex and PaLM on a significantly more
+limited, but human-written set of theorems in alge-
+bra and number theory.
+In contrast to prior work, we address the auto-
+formalization of both theorems and their proofs,
+and extend the scope to proofs of code correctness.
+We use a number of manually written mathemati-
+cal statements to abstract a complex grammar that
+is then used to generate a dataset of substantially
+longer and more diverse mathematical theorems
+and proofs.
+We develop an architecture based
+on the Universal Transformer (Dehghani et al.,
+2018) and adapt a copying mechanism (Gu et al.,
+2016) to handle arbitrary numbers and variable
+names at test time. The models are evaluated exten-
+sively on their ability to systematically generalize
+to statement lengths not seen during training, for
+which we report sequence-level accuracy as well
+as a semantic-level accuracy calculated by combin-
+ing sequence-level accuracy for the theorem and
+running Coq to determine if the generated proof
+is correct. Code and data are made available at
+https://github.com/gc974517/autoformalization.
+2
+Dataset of Theorems and Proofs
+We create two independent datasets of mathemat-
+ical statements that overall correspond to four
+classes of theorems and proofs: the first dataset con-
+tains three classes of arithmetic statements (EVEN-
+ODD, COMPOSITES, and POWERS), described in
+detail in Section 2.1, and the second dataset contain-
+ing statements about code correctness via Hoare
+logic (POLY), described in detail in Section 2.2.
+In each example, the input theorem-proof pair is
+given in LaTeX, whereas the formalized output is
+represented in Coq. This work focuses on the proof
+assistant Coq (Bertot and Castéran, 2013) because
+(a) there is a rich set of mathematical libraries that
+have been developed for it, (b) it has been used
+successfully to reason about significant computa-
+tion artifacts, such as the ComperCert C compiler
+(Leroy, 2009)), and (c) it benefits from a rich set of
+training material for the proof assistant related to
+software verification (Pierce et al., 2010).
+Each class of examples demonstrates features
+necessary for the successful autoformalization of
+mathematical theorems and proofs. For example,
+POWERS and COMPOSITES examples may define
+useful terminology to make the theorems shorter,
+e.g. proving that 4 is a square, or conversely they
+may state theorems directly without any prelim-
+inary definitions, e.g. proving ∃n. n2 = 4. As
+shown in Figures 3 and 4, this corresponds in Coq
+to aliasing propositions using the Definition key-
+word. Additionally, the examples in the dataset
+provide a stress test of the copying mechanism de-
+scribed in Section 3.1, testing its ability to learn
+the correct order and number of terms to include
+in mathematical expressions, as well as their place-
+ment in theorems and proofs, in a way that general-
+izes to arbitrary tokens in mathematical language.
+For each of the four classes of theorems and
+proofs, we manually created a few examples our-
+selves in order to guide the construction of a com-
+plex grammar that is then used to generate a dataset
+of substantially longer and more diverse mathemat-
+ical theorems and proofs. Each dataset is generated
+using its corresponding grammar in an identical
+way. First, a random seed is sampled that controls
+the overall structure of the theorem, proof, and
+definition, if any. Then, the skeleton structure of
+the proof is completed with phrases that are sam-
+pled from a separate context-free grammar. The
+coarse control of the skeleton structure allows the
+construction of examples with interesting features
+like sublemmas, forward or backward proof direc-
+tion, coreference, or additional conditions for the
+theorem, among others.
+Many of the difficulties in formalizing mathe-
+matical statements from NL into Coq stem from
+the wide variability in the level of detail of mathe-
+matical proofs, and the frequent mismatch between
+what is considered an acceptable inference step in
+NL proofs vs. an inference step in Coq. Further-
+more, there may be multiple Coq proofs for any
+given theorem, at different levels of granularity. We
+
+LaTeX Input Sequence
+Theorem. 28M + 308 is even.
+Proof. We know the summation between even numbers
+in N will be an even number. Observe that 308 is known
+to be even. Additionally, note that the pair M × 28 is
+trivially even. This is true because the coefficient 28 is
+even.
+Coq Output Sequence
+Require Import Arith.
+Theorem M28_308:
+forall M : nat, Nat.even(28 * M + 308) = true.
+Proof.
+intros.
+repeat rewrite Nat.even_add.
+assert (H1: Nat.even 308 = true).
+{ auto. }
+assert (H2: Nat.even 28 = true).
+{ auto. }
+assert (H3: Nat.even (28 * M) = true).
+{ rewrite Nat.even_mul.
+rewrite H2.
+auto. }
+rewrite H1.
+rewrite H3.
+auto.
+Qed.
+Figure 1: Generated example from the EVEN-ODD set.
+address this ambiguity by requiring the structure of
+the Coq proof to match the overall structure of the
+NL proof. This is achieved by quasi-synchronously
+generating the LaTeX and Coq versions of mathe-
+matical statements, while still allowing for some
+simple re-orderings in order to improve general-
+ization performance, e.g. swapping arguments of
+commutative operations.
+In total, the grammar-based method for gen-
+erating examples can theoretically produce over
+283 million unique arithmetic examples and over
+491,000 unique code examples, before consider-
+ing variations in phrasing by sampling from the
+context-free grammar.
+2.1
+Arithmetic Statements
+We generated three classes of mathematical state-
+ments, i.e. theorem-proof pairs:
+• EVEN-ODD: an expression is even or odd.
+• COMPOSITES: a number is composite.
+• POWERS: a number is an integer power of n.
+EVEN-ODD examples contain arithmetic expres-
+sions of n variables with even coefficients that are
+summed with a constant term, meaning that the
+parity of this constant determines the parity of the
+LaTeX Input Sequence
+Theorem. 450 + a · 192 + j · 462 is guaranteed to be
+even for any natural terms j, and a.
+Proof. It can be justified that 192 · a + j · 462 is trivially
+even. Note that 192a is an even number in N because
+multiplying between an even integer with an arbitrary
+number in N is guaranteed to be even. Likewise, 462j
+is trivially an even number in N. The claim is proven as
+a consequence of the fact that the sum of even numbers
+with an even number will be in itself an even number.
+Therefore, our theorem holds.
+Coq Output Sequence
+Require Import Arith.
+Theorem a450_192j_450_even:
+forall j a : nat,
+Nat.even (192 * a + 462 * j + 450) = true.
+Proof.
+intros.
+rewrite Nat.even_add.
+assert (H1: Nat.even (192 * a) = true).
+{ rewrite Nat.even_mul.
+auto. }
+assert (H2: Nat.even (462 * j) = true).
+{ rewrite Nat.even_mul.
+auto. }
+assert (H3: Nat.even
+(192 * a + 462 * j) = true).
+{ repeat rewrite Nat.even_add.
+rewrite H1.
+rewrite H2.
+auto. }
+rewrite H3.
+auto.
+Qed.
+Figure 2: Instance of sublemma use in the EVEN-ODD
+dataset. The proof that the sum of non-constant terms is
+even (assertion H3) is given before proving the theorem.
+whole expression. Proofs make use of this fact
+with varying rigor based on our manually designed
+grammar, an example of which is shown by Figure
+1. The Coq program is generated concurrently with
+the paired LaTeX example. The example shown in
+Figure 2 illustrates the generation and use of prior
+facts to prove an implicit sublemma, in both the
+natural language and matching Coq version.
+Examples of theorems and proofs for POWERS
+and COMPOSITES share a similar structure in both
+their LaTeX and Coq forms, as shown in Figures 3
+and 4, respectively. The theorems assert the ex-
+istence of a natural number such that a defining
+property holds and their proofs are constructive,
+with the distinction that examples for composites
+prove factorization into n factors.
+For both training and testing, we generate 5,000
+even-odd, 5,000 composites, and 2,000 powers ex-
+
+LaTeX Input Sequence
+Definition. We define that w ∈ N is a composite natural
+number if taking some R, Q ∈ N we have Q, R ≥ 2 and
+Q × R = w.
+Theorem. 35 is a composite whole number.
+Proof. Remember that a composite natural number is the
+multiplication between Q and R such that Q and R ≥ 2.
+Allow R = 7, Q = 5. We justify the result is valid as
+35 = Q · R.
+Coq Output Sequence
+Require Import Lia.
+Definition composite
+(w : nat) :=
+exists R Q : nat,
+(Q >= 2) /\ (R >= 2)
+/\ (Q * R = w).
+Theorem w_composite:
+composite 35.
+Proof.
+unfold composite.
+exists 7.
+exists 5.
+lia.
+Qed.
+Figure 3: Generated COMPOSITES example.
+LaTeX Input Sequence
+Definition. We say o is a square whenever there exists
+some whole number Z such that Z ≥ 2 and o = Z2.
+Theorem. o = 64 is a square.
+Proof. Let Z = 8. Observe that 64 = 82. Also notice
+Z = 8 is more than or equal to 2. This yields 64 is a
+square whole number.
+Coq Output Sequence
+Require Import Lia.
+Definition square
+(o : nat) :=
+exists Z : nat,
+(Z >= 2) /\ (o = Z^2).
+Theorem square_64:
+square 64.
+Proof.
+unfold square.
+exists 8.
+assert (H1: 8 >= 2).
+{ lia. }
+repeat split.
+apply H1.
+Qed.
+Figure 4: Generated example from the POWERS set.
+amples. We train on values of n ∈ {2, 3, 5, 7, 9}
+and test on values n ∈ {2, 3, . . . , 12}, where n rep-
+resents the number of variables in the arithmetic
+expression, the number of factors, or the power,
+respectively. This is done in order to evaluate the
+model’s ability to generalize to unseen arithmetic
+expression lengths and numbers of factors.
+2.1.1
+Handwritten Examples
+We also created a small collection of 45 human-
+written LaTeX theorem-proof pairs to evaluate per-
+formance on examples outside of our manually
+generated grammar. These are distinct from the
+original manually written examples that were used
+to guide the development of the generative gram-
+mar. There are 15 examples for each type of proof
+from the arithmetic set, using the same vocabulary
+with a number of unseen grammatical structures.
+2.2
+Code Correctness Statements
+We create a dataset of correctness proofs about
+short programs written in the imperative program-
+ming language Imp (Pierce et al., 2018), which we
+call POLY. The programs represent various algo-
+rithms for evaluating a polynomial, and their proofs
+of correctness verify that the programs correctly
+model the polynomial as a mathematical function.
+Proofs are conducted as either fully decorated pro-
+grams or as sequences of Hoare triples with natural
+language justifying steps in between. An example
+is shown in Figure 5.
+For both training and testing data, we generate
+5,000 examples. We train on programs containing
+2, 3, 5, 7, 9, and 11 lines, then test on programs con-
+taining from 2 up to 14 lines to evaluate the model’s
+ability to generalize to novel program lengths.
+3
+Semantic Parsing Architecture
+To formalize LaTeX statements into Coq, we de-
+veloped an encoder-decoder architecture based on
+the Universal Transformer (Dehghani et al., 2018).
+Similar to Csordás et al. (2021), we do so by adding
+recursive passes into the encoder and decoder of
+a base Transformer (Vaswani et al., 2017), thus
+making the model analogous to a Universal Trans-
+former without adaptive computation time (ACT).
+Further, we introduce a copying mechanism and
+support for out-of-vocabulary mathematical terms.
+3.1
+Copying Mechanism
+Mathematical language contains features uncom-
+mon or non-existent in natural language, such as
+numbers, variables, and carefully defined terminol-
+ogy. In order to address the use of general math-
+ematical jargon, these tokens are replaced in the
+
+LaTeX Input Sequence
+Coq Output Sequence
+Theorem. Consider the following series of
+commands such that
+S := 3;
+S := 3 + S * Z;
+S := 1 + S * Z
+Allow Z = y, for any natural number y, ahead
+of running this code then S = 3×y2+3×y+1
+after the set of instructions has executed.
+Proof. By application of usual Hoare logic:
+{Z = y}
+S := 3;
+{Z = y ∧ S = 3}
+S := 3 + S * Z;
+{Z = y ∧ S = 3 × y + 3}
+S := 1 + S * Z
+{Z = y ∧ S = 3 × y2 + 3 × y + 1}
+Hence, this program is shown to be correct.
+Require Import String.
+From PLF Require Import Imp.
+From PLF Require Import Hoare.
+Theorem poly_code_correct:
+forall y : nat,
+{{ Z = y }}
+S := 3;
+S := 3 + S * Z;
+S := 1 + S * Z
+{{ S = 3 * y ^ 2 + 3 * y + 1 }}.
+Proof.
+intros.
+apply hoare_seq with
+(Q := (
+(Z = y /\ S = 3)
+)%assertion).
+apply hoare_seq with
+(Q := (
+(Z = y /\ S = 3 * y + 3)
+)%assertion).
+apply hoare_seq with
+(Q := (
+(Z = y /\ S = 3 * y^2 + 3 * y + 1)
+)%assertion).
+all: eapply hoare_consequence_pre;
+try (apply hoare_asgn || assn_auto'').
+Qed.
+Figure 5: Generated POLY example: [Left] the Hoare logic proof; [Right] the code correctness proof in Coq.
+LaTeX input with generic forms denoting their us-
+age, such as <var1> up to <varN> for variables,
+which effectively ensures generalization to vari-
+able renaming (Ferreira et al., 2022), <nat1> up to
+<natN> for numbers, or <def> for definitions, cou-
+pled with the use of a copying mechanism adapted
+from Gu et al. (2016). Note that a different generic
+token is introduced for each unique numerical con-
+stant or variable literal in the theorem and its proof,
+and the corresponding generic token is used in
+the Coq version. For example, considering the
+⟨LaTeX, Coq⟩ pair in Figure 3, <nat1>, <nat2>,
+<nat3>, and <nat4> would be used to replace the
+constants 2, 35, 7, and 5 respectively, everywhere in
+the LaTeX and Coq statements. Similarly, <var1>,
+<var2>, and <var3> were used to replace variable
+literals w, R, and Q. This is in contrast to using
+just two generic tokens <nat> and <var> every-
+where, which would make all numbers coreferent
+and all variables coreferent. Preliminary experi-
+ments validated the utility of encoding these dis-
+tinctions while maintaining the correct coreference
+in both LaTeX and Coq statements.
+Overall, by using generic tokens for numbers,
+variables, and definitions, only a limited set of em-
+beddings need to be trained and the model is forced
+to utilize contextual information in order to appro-
+priately copy tokens into the Coq output. In this
+way, the model has the ability to generalize to un-
+seen numbers or variable and definition names.
+The original CopyNet (Gu et al., 2016) used an
+encoder-decoder architecture with a copying mech-
+anism to calculate the probabilities of generating
+in-vocabulary tokens vs. copying tokens from the
+input sequence to the output. Our autoformaliza-
+tion task guarantees mutual exclusivity between
+generating (g) and copying (c) tokens, which al-
+lows using a simplified formula for calculating the
+probability of producing a token yt at time step t.
+Letting Vc denote the Coq vocabulary, X denote
+the input sequence of LaTeX tokens, and X denote
+the collection of unique tokens in X, we calculate
+the probability of producing yt as:
+p(yt) =
+�
+�
+�
+�
+�
+�
+�
+p(yt, g) = 1
+Zt
+eψg(yt),
+yt ∈ Vc
+p(yt, c) = 1
+Zt
+�
+xj∈X:xj=yt
+eψc(xj), yt ∈ X
+where Zt =
+�
+yt∈Vc
+eψg(yt) +
+�
+xj∈X
+eψc(xj). The scor-
+
+ing functions are given by ψg(yt) = v⊤
+ytWost and
+ψc(xj) = tanh
+�
+h⊤
+j Wc
+�
+st, where vyt is a one-
+hot encoding of yt, hj is the hidden encoder state
+for the input token xj, st is the decoder state at step
+t, and Wo and Wc are learnable parameters.
+3.2
+Encoder-Decoder Architecture
+We diverge from the standard Transformer archi-
+tecture in a few crucial ways:
+• Probabilities are calculated via p(yt) above.
+• Absolute positional encodings are removed.
+• Self-attention uses relative positional repre-
+sentations as in Shaw et al. (2018).
+• Stacks of N encoder/decoder blocks have T
+recurrent passes.
+All other aspects of the model remain unchanged
+from the original Transformer. We emphasize rel-
+ative positional information over absolute in our
+model architecture. Preliminary evaluations on the
+EVEN-ODD dataset showed that Transformer mod-
+els that use absolute positional encodings obtain
+0% sequence-level accuracy on expression lengths
+that are not seen at training time. Removing re-
+liance on absolute position resolves this type of
+systematic generalization. The use of relative posi-
+tional encodings for the Transformer-based models
+was thus essential for achieving stronger systematic
+generalization, which also agrees with the findings
+of Csordás et al. (2021) on other NLP tasks.
+4
+Experimental Evaluations
+To evaluate the performance of trained models, we
+ran two primary experiments: first on the collection
+of arithmetic examples, then on the collection of
+code correctness examples. All models are eval-
+uated in terms of sequence-level accuracy, where
+an example is considered correctly processed only
+if the generated Coq sequence for both the theo-
+rem and its proof perfectly matches token by to-
+ken the ground truth sequence. We also report
+semantic-level accuracy, for which the generated
+Coq theorem needs to attains a perfect sequence-
+level accuracy and the Coq engine verifies that the
+generated Coq proof truly proves the generated
+Coq theorm, regardless of whether it matches the
+ground truth version of the proof. This empha-
+sizes that the model was able to capture the general
+meaning of the natural language proof by correctly
+translating the theorem and successfully proving it
+EVEN-ODD
+COMPOSITES
+POLY
+n
+Seq
+Sem
+Seq
+Sem
+Both
+2
+99.6
+99.8
+76.7
+97.6
+100.0
+3
+99.4
+99.6
+64.6
+94.2
+100.0
+4
+99.4
+99.4
+56.1
+93.9
+82.1
+5
+99.2
+99.6
+54.9
+94.4
+99.2
+6
+98.8
+98.8
+57.1
+94.3
+45.1
+7
+99.1
+99.5
+58.5
+93.4
+96.5
+8
+93.8
+94.0
+53.5
+88.3
+15.7
+9
+98.6
+98.6
+53.7
+93.7
+98.2
+10
+7.0
+7.0
+1.2
+1.6
+35.6
+11
+0.0
+0.0
+0.0
+0.0
+93.5
+12+
+0.0
+0.0
+0.0
+0.0
+0.0
+POWERS
+Seq = 100.0
+Sem = 100
+Table 1:
+Sequence-level (Seq) and semantic-level
+(Sem) accuracy (%) on test examples, split by expres-
+sion length, with the exception of POWERS.
+using the natural language version as a guide.
+All experiments were performed on one NVIDIA
+RTX-A6000 GPU with 48GB of memory.
+4.1
+Arithmetic Statements
+We evaluate a Transformer model on the full data
+combining EVEN-ODD + COMPOSITES + POWERS
+and using both the theorem and its proof in each
+sequence. We tune a model with embedding and
+state sizes of 32, a feed forward width of 256, 4
+encoder and decoder blocks with 4 recurrent passes,
+4 attention heads, and a clipping value of 2 for self-
+attention. We trained this model over minibatches
+of size 20, optimized with Adam using β1 = 0.9,
+β2 = 0.98, ε = 1e − 9, and an initial learning rate
+of 0.001, annealed by a factor of 1/
+√
+10 based on
+training loss plateaus with a patience of 5 epochs.
+The results in Table 1 show that the model gener-
+alizes well to the intermediate lengths of {4, 6, 8},
+with a small number of correctly translated exam-
+ples longer than the maximum of 9 used in training.
+Otherwise, the model fails to generalize to longer
+unseen lengths, which is not surprising, given that
+Transformer models are known to fail dramatically
+at systematic generalization on longer inputs for
+various NLP tasks (Csordás et al., 2021), or to in-
+cur substantial decrease in accuracy for longer sym-
+bolic integration problems (Welleck et al., 2022).
+Switching to semantic-level evaluation leads to a
+significant increase in accuracy for COMPOSITES,
+with a more modest increase for EVEN-ODD.
+
+4.2
+Code Correctness Statements
+We extend our scope to include data representing
+proofs of program correctness using the language
+of Hoare logic. We train a separate model with
+the same embedding and state sizes, feed forward
+width, and learning rates as in Section 4.1. Depth
+is increased to 8 encoder and decoder blocks with 8
+recurrent passes, 8 attention heads, and a clipping
+value of 8. The model is trained over minibatches
+of size 1 with Adam, with a patience of 3 epochs.
+The POLY results shown in Table 1 demonstrate
+that the model is able to generalize to program line
+counts of {4, 6, 8, 10} unseen during training with
+diminishing returns as the program length grows,
+eventually failing to generalize for lengths longer
+than the maximum seen in training. We observe
+that increasing the depth of the model significantly
+improved generalization.
+A model with identi-
+cal hyperparameters to the arithmetic experiment
+yielded less then half the sequence-level accuracy
+for intermediate program lengths. Therefore, fur-
+ther increasing the depth of the model could push
+performance closer to optimal generalization to in-
+termediate lengths at the cost of significantly more
+computing resources. Additionally, POLY exam-
+ples are far less prone to non-fatal token swapping
+errors. We observe that semantic-level accuracy is
+identical to sequence-level, as all copying errors
+compromised the validity of the proof. Therefore,
+accuracies are shown as one column (Both).
+4.3
+Handwritten Examples
+We also evaluate the semantic-level accuracy of
+the trained models on the collection of 45 human-
+written LaTeX theorem-proof pairs. This is done
+by manually verifying that the generated Coq the-
+orem corresponds to the LaTeX version and that
+the subsequent proof is correct according to the
+Coq interpreter. The fully trained model achieved
+53.3% for both EVEN-ODD and COMPOSITES, and
+73.3% for POWERS.
+Mistakes in almost all cases are confined to the
+mishandling of out-of-vocabulary tokens, such as
+mis-copying a variable within a definition or the
+omission of an assertion in the proof tied to a term.
+The model otherwise generated syntactically sound
+Coq code. Mistakes strongly correlate with exam-
+ples that deviate significantly from the grammatical
+structure of the artificial data. Thus, pre-trained lan-
+guage models as evaluated by Wu et al. (2022) or
+pre-training new models on mathematical corpora
+like MATH (Hendrycks et al., 2021) may serve to
+alleviate the problems caused by the scarcity of
+aligned natural and formal mathematics data.
+5
+Concluding Remarks
+As we have seen, it is feasible to train machine
+learning models to perform autoformalization over
+very restricted domains of math and code correct-
+ness proofs. These models show capability to sys-
+tematically generalize to new expression lengths
+and program sizes. Moreover, these models were
+able to translate previously unseen hand written
+natural language examples, albeit with lower ac-
+curacy. We are hopeful that this approach can be
+applied to autoformalization of a larger segment of
+mathematics and code verification.
+As mentioned by Szegedy (2020), "Autoformal-
+ization is not just a challenge: successful autofor-
+malization would represent a breakthrough for gen-
+eral AI with significant implications in various do-
+mains." We see an especially significant impact in
+education, where integration of autoformalization
+into proof assistants for introductory mathematics
+and software verification courses would enable the
+detection of missing steps or misconceptions in
+students’ proofs.
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+

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+Locally adaptive aggregation of organisms under death risk in rock-paper-scissors models
+J. Menezesa,b, E. Rangelb
+aInstitute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
+bSchool of Science and Technology, Federal University of Rio Grande do Norte
+Caixa Postal 1524, 59072-970, Natal, RN, Brazil
+cDepartment of Computer Engineering and Automation, Federal University of Rio Grande do Norte, Av. Senador Salgado Filho 300, Natal, 59078-970, Brazil
+Abstract
+We run stochastic simulations of the spatial version of the rock-paper-scissors game, considering that individuals use sensory
+abilities to scan the environment to detect the presence of enemies. If the local dangerousness level is above a tolerable threshold,
+individuals aggregate instead of moving randomly on the lattice. We study the impact of the locally adaptive aggregation on the
+organisms’ spatial organisation by measuring the characteristic length scale of the spatial domains occupied by organisms of a
+single species. Our results reveal that aggregation is beneficial if triggered when the local density of opponents does not exceed
+30%; otherwise, the behavioural strategy may harm individuals by increasing the average death risk. We show that if organisms
+can perceive further distances, they can accurately scan and interpret the signals from the neighbourhood, maximising the effects
+of the locally adaptive aggregation on the death risk. Finally, we show that the locally adaptive aggregation behaviour promotes
+biodiversity independently of the organism’s mobility. The coexistence probability rises if organisms join conspecifics, even in
+the presence of a small number of enemies. We verify that our conclusions hold for more complex systems by simulating the
+generalised rock-paper-scissors models with five and seven species. Our discoveries may be helpful to ecologists in understanding
+systems where organisms’ self-defence behaviour adapts to local environmental cues.
+Keywords: population dynamics, cyclic models, stochastic simulations, behavioural strategies
+1. Introduction
+Behavioural biology has revealed the mechanisms that or-
+ganisms use to improve their fitness, being fundamental for the
+stability of the rich biodiversity in nature[1–4]. There is plenty
+of evidence that self-preservation strategies are properly exe-
+cuted because of the organism’s evolutionary ability to scan the
+environment cues, perceiving the presence of a nearby enemy
+and the energy expended in the action[5–9]. In this scenario,
+living in groups facilitates the defence action since individual
+protection against enemies is maximised by collective effort in
+surveillance and resistance, demanding less individual energy
+expenditure on defense against enemies [10–19].
+Cyclic models of biodiversity have been studied using the
+rock-paper-scissors game rules, which successfully describe the
+nonhierarchical competition interactions found in many biolog-
+ical systems [20–29]. However, experiments with bacteria Es-
+cherichia coli revealed that the cyclic dominance among three
+bacteria strains is insufficient to stabilise the system. It has been
+discovered that coexistence is ensured only if individuals inter-
+act locally [30]. This shows the central role of space in the sta-
+bility of biological systems, as it has been also observed in com-
+munities of lizards and systems of competing coral reefs [31–
+33]. Furthermore, cyclic dominance has been shown to play a
+fundamental role in the spatial interactions in social systems,
+public good with punishment, and human bargaining [34, 35].
+There is plenty of evidence that organisms’ mobility plays a
+central role in promoting or jeopardising biodiversity in struc-
+tured populations [36–46].
+Evidence shows that organisms’
+foraging behaviour may affect biodiversity in the spatial rock-
+paper-scissors game [22, 28]. Organisms’ moving to escape
+their enemies and find natural resources to the species perpet-
+uation may unbalance the cyclic game or decelerate the pop-
+ulation dynamics, thus jeopardising or promoting biodiversity
+[28, 29, 47–49].
+Recently, it has been shown that aggregation behaviour is an
+efficient antipredator strategy in tritrophic predator-prey cyclic
+models [49]. Numerical simulations of the Lotka-Volterra ver-
+sion of the rock-paper-scissors game revealed that individu-
+als’ predation risk decreases if organisms execute a gregarious
+movement, instead of exploring the territory to found prey and
+reproduce. In contrast with the standard model where organ-
+isms move in a random direction, the grouping strategy pro-
+duces spiral-type patterns with organisms of the same species
+living in spatial domains whose characteristic length depends
+on the the distance the individuals can scan their neighbour-
+hood, and their cognitive ability to perform the directional self-
+preservation movement tactic [49].
+Although the revealing details of the complexity of the spa-
+tial interactions, the model in Ref. [49] considers exclusively a
+non-adaptive aggregation tactic, i.e., individuals cannot smartly
+adapt their movement to trigger the grouping strategy only
+when pressured by an imminent enemy’ attack, as happens, for
+example, in spider mites communities [9]. In this case, the
+unnecessary expenditure is avoided since organisms can con-
+Preprint submitted to Journal of LATEX Templates
+January 5, 2023
+arXiv:2301.01729v1  [q-bio.PE]  4 Jan 2023
+
+1
+3
+2
+1
+2
+3
+Figure 1: The rock-paper-scissors model rules. The black arrows illustrate the
+dominance in the spatial game: individuals of species i eliminate organisms of
+species i+1, with i = 1, 2, 3 and i±3 = i. Organisms of the same species aggre-
+gate when attacked and move randomly when not in danger. Dark blue, pink,
+and green represent individuals of species 1, 2, and 3 moving gregariously;
+light blue, pink, and green indicate organisms of species 1, 2, and 3 moving
+randomly.
+tinue freely advancing on the lattice to conquer territory, allow-
+ing the population growth [39]. In this work, we sophisticate
+the stochastic model to simulate a locally adaptive aggregation
+where organisms move gregariously only under death risk [49].
+We also consider that the decision to aggregate is the individual
+competence, meaning that each organism acts autonomously
+according to its own local reality. Therefore, each individual
+can decide if moving gregariously or randomly, with the con-
+gregation being triggered only if the local density of enemies
+is higher than a tolerable threshold. In addition, we implement
+the behavioural survival strategy using the May-Leonard imple-
+mentation of the spatial rock-paper-scissors game. This allows
+the generalisation of our results to systems where competition
+for natural resources is the goal of the cyclic game [50].
+We aim to answer the following questions: i) how does the
+locally adaptive aggregation modify the spiral patterns, char-
+acteristic of the standard May-Leonard implementation of the
+rock-paper-scissors model?; ii) how does the aggregation trig-
+ger influence the organisms’ spatial organisation altering the
+size of the typical single-species domains?; iii) how does adap-
+tive grouping benefit individuals by reducing the average death
+risk?; iv) how does the locally adaptive congregation behaviour
+impact species coexistence probability?
+The outline of this paper is as follows. In Sec. 2, we in-
+troduce our stochastic model and present the methods used to
+implement the locally adaptive grouping in our simulation algo-
+rithm. In Sec. 3, the changes in the spatial patterns are studied
+for various values of aggregation trigger; the autocorrelation
+function and characteristic length scales are addressed in Sec.
+4. The reduction in the organisms’ average death risk is com-
+puted in Sec. 5 for a range of aggregation triggers and mobil-
+ity probabilities. Finally, the coexistence probability in terms
+of the individual’s mobility is investigated in Sec. 6, while our
+comments and conclusions appear in Sec. 7.
+2. The Model
+We study a cyclic model of three species that outcompete
+each other according to the rock-paper-scissors game rules, il-
+lustrated in Fig. 1. This means that individuals of species i elim-
+inate organisms of species i + 1, with i = 1, 2, 3, with the cyclic
+identification i = i + 3 β, where β is an integer. Our model con-
+siders that organisms of the same species aggregate to minimize
+the probability of being killed in the spatial game. The gre-
+garious movement is locally adaptive, triggered whenever the
+density of enemies in the organisms’ neighbourhood is higher
+than a tolerable threshold. This means that each individual of
+species i can scan the environment to perceive the presence of
+organisms of species i + 1, thus, accurately deciding if the best
+strategy is to search for refuge joining their conspecifics. or
+continue moving randomly to explore the territory. The dark
+colours in Fg.1 stand for individuals executing the gregarious
+movement, whereas the light colours represent organisms mov-
+ing randomly.
+2.1. Numerical simulations
+To perform the numerical simulations, we use square lattices
+with periodic boundary conditions; the number of grid sites is
+N. We use the May-Leonard implementation, where the total
+number of individuals is not conserved. Therefore, as each grid
+point is occupied by at most one individual (or it is empty),
+the maximum number of organisms in the system is the total
+number of grid points N.
+Initially, the organisms are randomly distributed in the lat-
+tice: each individual is allocated at a random grid site. The ini-
+tial conditions are prepared so that the number of individuals is
+the same for every species is the same. We define the number of
+individuals of each species at the initial state as one-third of the
+total number of organisms: Ii(t = 0) ≈ N/3, with i = 1, 2, 3;
+the rest of grid sites are left empty in the initial conditions.
+Once the random initial conditions are ready, the algorithm
+stochastically implements the interactions following the von
+Neumann neighbourhood, where each organism can interact
+with one of its four immediate neighbours. The spatial inter-
+actions are:
+• Selection: i j → i ⊗ , with j = i + 1, where ⊗ means
+an empty space: an individual of species i eliminates a
+neighbour of species i + 1 following the rules illustrated in
+Fig.1 - the grid site occupied by the eliminated individual
+is left empty.
+• Reproduction: i ⊗ → i i : an empty space is filled by a new
+organism of any species.
+• Mobility: i ⊙ → ⊙ i , where ⊙ means either an individual
+of any species or an empty site. An organism moves by
+switching positions with another individual of any species
+or an empty space.
+The interactions are implemented following a fixed set of
+probabilities which is the same for every species: s (selec-
+tion probability), r (reproduction probability), and m (mobility
+probability). During the interaction implementation, the code
+follows the steps:
+1. an active individual of any species is drawn among all or-
+ganisms in the lattice;
+2. one interaction is randomly chosen following the set of
+probabilities rates (s, r, and m);
+2
+
+(a)
+(b)
+(c)
+(d)
+Figure 2: Snapshots captured from simulations of the rock-paper-scissors game with individuals’ locally adaptive aggregation. The realisations ran in lattice with
+2002 grid points for a timespan of 2000 generations, with R = 3, r = s = 0.25 and m = 0.5. Figures 2a, 2b, 2c, and 2d show the organisms’ spatial organisation at
+the end of Simulation A (ϕ = 1.0), B (ϕ = 0.1), (ϕ = 0.025), and D (ϕ = 0.0), respectively. The colours follow the scheme in Fig. 1, with blue, pink, and green
+depicting individuals of species 1, 2, and 3, respectively. Dark and light colours distinguish organisms performing the congregation strategy and moving randomly.
+Yellow dots depict empty sites.
+3. one of the four immediate neighbours is drawn to suffer
+the action (selection, reproduction, and random mobility)
+- the only exception is the adaptive gregarious movement,
+where the organism move towards the direction with more
+conspecifics.
+Every time an interaction is implemented, one timestep is
+counted. After N timesteps, one generation is completed - our
+time unit is one generation.
+To understand the population dynamics during the simula-
+tions, we calculate the density of organisms of species i, ρi(t),
+with i = 1, 2, 3. This is defined as the fraction of the lattice oc-
+cupied by individuals of the species i at time t, ρi(t) = Ii(t)/N.
+Also, the temporal dependence of the density of empty spaces
+is computed as ρ0 = 1 − ρ1 − ρ2 − ρ3.
+2.2. Implementing the locally adaptive aggregation strategy
+To implement the locally adaptive grouping tactic, we define
+the perception radius, R, to represent the maximum distance an
+organism of species i can scan the environment to be aware of
+the presence of enemies. Thus, the local density of organisms of
+each species is computed within a circular area of radius R, cen-
+tred in the organism of species i [29, 49]. In addition, we intro-
+duce the aggregation trigger, ϕ, to represent the minimum local
+density of individuals of species i − 1 (enemies) that stimulates
+the organism of species i to move gregariously. This means that
+if the local density of organisms of species i − 1 is lower than
+ϕ, the individual moves randomly.
+The numerical implementation of the gregarious movement
+is performed by dividing the observing disc into four circular
+sectors, each section in the directions of the one nearest neigh-
+bour of the von Neumann neighbourhood [22, 25, 26, 28, 49,
+51]. Next, it is computed how many individuals of species i
+exist within each circular sector, with organisms on the circu-
+lar sector borders assumed to be part of both circular sectors.
+Finally, the organism switches positions with the immediate
+neighbour in the direction with more conspecifics; a draw in
+the event of a tie.
+3. Spatial Patterns
+Our first goal is to understand the effects of the locally adap-
+tive congregation strategy in spatial patterns. Therefore, we ran
+a single simulation for four values of the aggregation trigger:
+• Simulation A: ϕ = 1.0 - the absence of organisms’ group-
+ing behaviour, i.e., individuals do not aggregate even under
+death risk;
+• Simulation B: ϕ = 0.1 - organisms’ agglomeration occurs
+if, at least, 10% neighbours are enemies;
+• Simulation C: ϕ = 0.025 - an individual move gregariously
+if at least, 2.5% neighbours are enemies;
+• Simulation D: ϕ = 0.0 - the gregarious movement is not
+locally adaptive, with individuals always grouping inde-
+pendently of the presence of enemies surrounding them.
+The realisations were performed in lattices with 2002 grid sites,
+running for a timespan of 2000 generations. We set the param-
+eters to s = r = 0.25, m = 0.5, and R = 3.
+Figures 2a, 2b, 2c, and 2d show the individuals’ spatial or-
+ganisation at the end of Simulations A, B. C, and D, respec-
+tively. To depict each organism, we use the same colours of
+the scheme in Fig. 1: blue, pink, and green dots show the in-
+dividuals of species 1, 2, and 3, respectively. The organisms
+performing the aggregation strategy are highlighted using dark
+colours, while the individuals moving randomly appear in light
+shades. We also quantified the dynamics of the species densities
+for Simulation A, B, C, and D, which are depicted in Figs. 3a,
+3b, 3c, and 3d, respectively. As in Fig.1, blue, pink, and green
+lines shows the temporal dependence of densities of individuals
+of species 1, 2, and 3, respectively;
+Let us first focus on Simulation A, where individuals do not
+aggregate to protect themselves against enemies (Fig. 2a). Be-
+cause of the random initial conditions, selection interactions are
+frequent at the beginning of the simulation. After that, spatial
+patterns are formed with organisms of the same species occu-
+pying departed patches. Since organisms are unaware of the
+3
+
+0.2
+0.25
+0.3
+0.35
+0.4
+0
+500
+1000
+1500
+2000
+ϕ = 1.0
+ρi
+t (generations)
+1
+2
+3
+(a)
+0.2
+0.25
+0.3
+0.35
+0.4
+0
+500
+1000
+1500
+2000
+ϕ = 0.1
+ρi
+t (generations)
+1
+2
+3
+(b)
+0.2
+0.25
+0.3
+0.35
+0.4
+0
+500
+1000
+1500
+2000
+ϕ = 0.025
+ρi
+t (generations)
+1
+2
+3
+(c)
+0.2
+0.25
+0.3
+0.35
+0.4
+0
+500
+1000
+1500
+2000
+ϕ = 0.0
+ρi
+t (generations)
+1
+2
+3
+(d)
+Figure 3: Dynamics of species densities during the simulations in Fig. 2. The blue, pink, and green lines in Figs. 3a, 3b, 3c, and 3d depict the temporal dependence
+of the density of individuals of species 1, 2, and 3, in Simulations A, B, C, and D, respectively.
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+0
+500
+1000
+1500
+2000
+ρ0
+t (generations)
+ϕ = 0.000
+ϕ = 0.025
+ϕ = 0.100
+ϕ = 1.000
+Figure 4: Temporal dependence of the density of empty spaces in simulations
+of Fig. 2. The grey, orange, yellow, and brown lines show the dynamics of
+empty sites in Simulations A, B, C, and D, respectively.
+neighbourhood, they move randomly, independently of the risk
+of being caught. This results in faster dynamics of species den-
+sities, with organisms being destroyed and newborns appearing
+at a high rate. Consequently, the species densities’ frequency
+and amplitude are high, as shown in Fig. 3a.
+In addition to the usual pattern formation process driven by
+the cyclic game rules, the gregarious movement performed by
+individuals under death risk promotes the formation of self-
+protection clusters on the border that is attacked by enemies,
+as shown in Figs. 2b and 2c. For example, the organisms of
+species 2 aggregating (dark pink dots) are concentrated on the
+border with spatial domains of species 1 (blue areas). The self-
+preservation movement tactic produces a deformation of the
+spiral patterns, with individuals concentrating in patches with
+smaller sizes since they abdicate to explore extensive areas of
+the territory to form clumps. Because of this, the population dy-
+namics are decelerated, with reduced frequency and amplitude,
+as depicted in Figs.3b and 3c.
+Finally, the snapshot in Fig. 3d reveals what occurs in the
+case of the non-adaptive aggregation strategy (ϕ = 0.0) - indi-
+viduals move gregariously even if no enemy surrounds them.
+In this scenario, the population dynamics are altered since the
+individuals neglect the conquest of new territories to focus ex-
+clusively on the survival movement strategy. This induces a
+contraction of the spatial domains occupied by organisms of
+a single species, since individuals do not advance in the terri-
+tory even if they are not under death risk. Finally, Fig. 4 shows
+the temporal dependence of the density of empty spaces, ρ0, in
+Simulations A (grey line), B (orange line), C (green line), and
+D (brown line). The results show that the density of empty
+spaces decreases after an initial period of pattern formation.
+Furthermore, the locally congregation reduces the organisms’
+death risk. Because of this, the lower the aggregation trigger,
+the more the density of empty spaces is reduced.
+4. Autocorrelation Function
+Let us now quantify the scale of spatial domains in the pres-
+ence of locally adaptive aggregation. For this, we compute the
+spatial autocorrelation function. The autocorrelation function
+is computed from the inverse Fourier transform of the spectral
+density as
+C(⃗r′) = F −1{S (⃗k)}
+C(0)
+,
+(1)
+where S (⃗k) is given by
+S (⃗k) =
+�
+kx,ky
+Φ(⃗κ),
+(2)
+4
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+25
+30
+35
+C
+r
+ϕ = 0.0
+ϕ = 0.1
+ϕ = 1.0
+Figure 5: Autocorrelation functions in terms of the radial coordinate. The grey,
+orange, and brown lines depict the results for the standard model (ϕ = 1.0),
+aggregation triggered when at least 10% of neighbours are enemies ϕ = 0.1, and
+the non-adaptive aggregation (ϕ = 0.0), respectively. The error bars indicate
+the standard deviation; the dashed black line shows the threshold assumed to
+calculate the characteristic length scale. The interaction probabilities are r =
+s = 0.25 and m = 0.5; the perception radius is R = 3.
+and Φ(⃗κ) is Fourier transform
+Φ(⃗κ) = F {φ(⃗r) − ⟨φ⟩}.
+(3)
+The function φ(⃗r) represents the spatial distribution of individ-
+uals of species 1, with φ(⃗r) = 0 and φ(⃗r) = 1 indicating the
+absence and the presence of an individual of species 1 in at the
+position ⃗r in the lattice, respectively). The spatial autocorrela-
+tion function is given by
+C(r′) =
+�
+|⃗r′|=x+y
+C(⃗r′)
+min �2N − (x + y + 1), (x + y + 1)�.
+(4)
+Moreover, we compute the spatial domains’ scale for C(l) =
+0.15, where l is the characteristic length.
+We calculated the spatial autocorrelation function in terms
+of the radial coordinate r for three cases: absence of group-
+ing behaviour (ϕ = 1.0), aggregation triggered when the neigh-
+bourhood is, at least, 10% hostile (ϕ = 0.1), and non-adaptive
+aggregation (ϕ = 0.0). The outcomes were obtained by run-
+ning sets of 100 simulations with different random initial con-
+ditions in lattices with 5002 grid sites for a time span of 5000
+generations. To calculate the autocorrelation function, we used
+the spatial configuration at the end of the simulation (t = 5000
+generations). Because organisms of every species can perform
+the locally adaptive congregation, the autocorrelation function
+is the same irrespective of the species; thus, we used the data
+from species 1. In all simulations, we considered the interac-
+tions probabilities s = r = 0.25 and m = 0.5; the perception
+radius was set to R = 3.
+The brown, orange, and grey lines in Figure 5 show C as a
+function of the radial coordinate r for ϕ = 0.0, ϕ = 0.1, and ϕ =
+0.0, respectively; the error bars indicate the standard deviation.
+The horizontal dashed black line indicates the threshold used to
+calculate the length scale: C(l) = 0.15. The results confirm
+that once organisms move gregariously, the average size of the
+spatial domains inhabited by a single species decreases.
+Figure 6 shows the relative variation of the characteristic
+length scale ˜l, defined as ˜l = (l − l0)/l0, where l0 is the value in
+the absence of the adaptive aggregation (ϕ = 1.0). We repeated
+the set of 100 simulations - starting from different initial condi-
+tions - for 0 ≤ ϕ ≤ 0.4, with intervals of δϕ = 0.05. The error
+−40
+−35
+−30
+−25
+−20
+−15
+−10
+−5
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+˜l(%)
+ϕ
+Figure 6: The relative change in the characteristic length scale of the typical
+single-species spatial domain as a function of the threshold used to trigger the
+gregarious movement compared with the standard model. The simulations ran
+in lattices with 5002 grid sites, running until 5000 generations for r = s = 0.25
+and m = 0.5; the perception radius is R = 3. The outcomes were averaged from
+sets of 100 simulations starting from different initial conditions; the error bars
+show the standard deviation. We assumed the probabilities r = s = 0.25 and
+m = 0.5.
+bars show the standard deviation; the parameters are the same
+used in the simulations in Fig. 5. The outcomes show that the
+average group size decreases compared to the standard model,
+with the reduction becoming significant for ϕ = 0.0. This hap-
+pens because all individuals group themselves, independently
+of what is happening in their surroundings, as we observed in
+Fig. 2d.
+5. The role of the locally adaptive aggregation in the organ-
+isms’ death risk
+We now investigate the effects of locally adaptive grouping
+to reduce the organisms’ death risk. For this purpose, we in-
+troduce the death risk, which is calculated as follows: i) it is
+counted as the total number of individuals of species i at the
+beginning of each generation; ii) the number of organisms of
+species i killed by individuals of species i − 1 during the gen-
+eration is computed; iii) the death risk, ζ is defined as the ratio
+between the number of eliminated organisms and the amount
+at the beginning of each generation. Due to the symmetry of
+the rock-paper-scissors game rules, the average death risk is the
+same for individuals of every species; thus, we choose the re-
+sults for species 1 to represent the individuals’ death risk.
+5.1. The influence of the aggregation trigger
+First, we study the influence of the aggregation trigger ϕ in
+the relative decrease of the individuals’ death risk by running
+sets of 100 simulations starting from different initial conditions
+for 0 ≤ ϕ ≤ 1.0 in intervals of δϕ = 0.1. This experiment was
+conducted for two values of perception radius: R = 3 and R = 5;
+the interaction probabilities are s = r = 0.25 and m = 0.5. To
+guarantee the quality of the results, we remove the data from
+the initial pattern formation stage, thus calculating the average
+organisms’ death risk in the second half of each realisation.
+The purple and red lines in Figure 7 show the organisms’
+death risk in terms of the aggregation trigger for R = 3 and
+R = 5, respectively; the standard deviation is shown by error
+bars. The outcomes reveal that for ϕ ≥ 0.6, the locally adap-
+tive strategy is ineffective in reducing the organisms’ death risk
+5
+
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+ζi
+ϕ
+R = 5
+R = 3
+Figure 7: Organisms’ death risk in terms of the aggregation trigger. The simu-
+lations were performed in lattices with 5002 grid sites, running for a timespan
+of 5000 generations. The red and purple lines show the outcomes for organisms
+with perception radius R = 3 and R = 5, respectively. The results were aver-
+aged from sets of 100 simulations starting from different initial conditions; the
+standard deviation is depicted by error bars. The interaction probabilities are
+s = r = 0.25 and m = 0.5.
+−50
+−45
+−40
+−35
+−30
+−25
+−20
+−15
+−10
+0.05
+0.15
+0.25
+0.35
+0.45
+0.55
+0.65
+0.75
+0.85
+0.95
+˜ζ(%)
+m
+Figure 8: Relative change in the individuals’ death risk in terms of the mobility
+probability in simulations running in lattices with 5002 grid sites, running for a
+timespan of 5000 generations. We averaged the outcomes sets of 100 simula-
+tions starting from different initial conditions; the standard deviation is shown
+by error bars. The perception radius is R = 3; the interaction probabilities are
+to s = r = (1 − m)/2).
+compared with the standard model (ϕ = 1.0). This happens be-
+cause most of organism of species i whose neighbourhood con-
+tains 60% or more of organisms of species i − 1 is far from the
+spatial domain dominated by their conspecifics; thus, grouping
+may not be possible to be executed before the individual being
+eliminated by enemies.
+Our findings show that the locally adaptive aggregation jeop-
+ardises the organisms’ safety for intermediate values of ϕ. As
+shown in Fig. 7, for R = 3, the organisms’ death risk in-
+creases for 0.4 ≤ ϕ < 0.6, while for R = 5, ζ increases for
+0.4 ≤ ϕ < 0.3. Therefore, the adaptive is beneficial only if
+the threshold assumed to move gregariously is in the interval
+0 ≤ ϕ < 0.4 for R = 3 and 0 ≤ ϕ < 0.3 for R = 5, with the
+relative reduction of ζ increasing as the ϕ is lowered.
+The results in Fig. 7 show how the complexity of the spatial
+interactions is influenced by the organism’s ability to make an
+accurate decision, triggering the adaptive tactic correctly. Our
+findings show that if organisms can perceive further distances,
+they can more easily: i) identify the presence of invading en-
+emies beyond the border of their territory; ii) distinguish the
+direction with more conspecifics in case of need to move gre-
+gariously. Because of this, the relative variation in the organ-
+isms’ death risk is more accentuated for R = 5 than for R = 3
+in Fig. 7.
+5.2. The interference of organisms’ mobility
+The locally adaptive grouping is profitable for the organ-
+isms because of the death risk reduction, as shown in Fig. 7
+for m = 0.5.
+Now, we repeated the simulations to explore
+how the benefits of the locally adaptive aggregation depend
+on the organism’s mobility. For this purpose, we ran sets of
+100 realisations starting from different initial conditions for
+0.05 ≤ m ≤ 0.95, in intervals of δm = 0.05. The selection and
+reproduction probabilities are set to s = r = (1 − m)/2; the per-
+ception radius is R = 3, and the aggregation trigger is ϕ = 0.05.
+We implemented the simulations in lattices with 5002 grid sites,
+running until 5000 generations.
+Figure 8 shows the relative change of the organisms’ death
+risk: ˜ζ = (ζ − ζ0)/ζ0, where ζ0 is the death risk in the absence
+of grouping behaviour (ϕ = 1.0). For 0.05 ≤ m ≤ 0.085, the
+relative reduction in the organisms’ death risk is more signifi-
+cant for individuals that explore greater fractions of the lattice
+per time unit [39]. This happens because high-mobile individ-
+uals are more vulnerable to being eliminated by enemies in the
+cyclic game, thus, benefitting more from the self-preservation
+movement strategy. However, if m > 0.085, the relative vari-
+ation in ζ decreases because the selection probability becomes
+very low, becoming the effect less significant.
+6. Coexistence Probability
+Now, we focus on the impact of locally adaptive flocking on
+biodiversity in cyclic games. In this study, we ran sets of 1000
+simulations in lattices with 1002 grid points for 0.05 < m <
+0.95 in intervals of δ m = 0.05; selection and reproduction
+probabilities were set to s = r = (1 − m)/2. For each set
+of simulations, each realisation began from different random
+initial conditions, running until 10000 generations. If at least
+one species is extinguished before the simulation ends, biodi-
+versity is lost. Thus, the coexistence probability is the frac-
+tion of the simulations where all species are present at the end.
+We extended the investigation to quantify the impact of locally
+adaptive aggregation in more complex systems by simulating
+the generalised rock-paper-scissors models with five and seven
+species. Figures 9a, 9b and 9c depict the coexistence probabil-
+ity for ϕ = 0.0 (brown line), ϕ = 0.05 (green line), ϕ = 0.1
+(orange line), ϕ = 0.2 (blue line), and ϕ = 1.0 (grey line) for
+the models with N = 3, N = 5, and N = 7 species, respectively.
+Overall, species biodiversity is more threatened for systems
+with highly mobile individuals, independent of the number of
+species in the cyclic game. The outcomes also show the benefits
+of the locally adaptive aggregation for biodiversity: the lower
+the aggregation trigger, the higher is the coexistence proba-
+bility. This conclusion holds independently of the number of
+species in the cyclic game Furthermore, the outcomes show
+that the more complex the system is, the more favourable it
+is for biodiversity loss. By comparing the same color lines in
+Fig. 9a, 9b and 9c, one observes that the coexistence probabil-
+ity is lower for the system with N = 9 species, independently
+of the organisms’ mobility. Finally, we observe that all simula-
+tions resulted in coexistence when individuals agglomerate with
+6
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Coexistence Probability
+m
+ϕ = 0.00
+ϕ = 0.05
+ϕ = 0.10
+ϕ = 0.20
+ϕ = 1.00
+(a)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Coexistence Probability
+m
+ϕ = 0.00
+ϕ = 0.05
+ϕ = 0.10
+ϕ = 0.20
+ϕ = 1.00
+(b)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Coexistence Probability
+m
+ϕ = 0.00
+ϕ = 0.05
+ϕ = 0.10
+ϕ = 0.20
+ϕ = 1.00
+(c)
+Figure 9: Coexistence probability as a function of the mobility probability
+for the generalised rock-paper-scissors game with organisms’ locally adaptive
+aggregation. Figures 9a, 9b, and 9c show the outcomes for the cyclic model
+with three, five, and seven species, respectively. The results were obtained by
+running 1000 simulations in lattices with 1002 grid points running until 10000
+generations for R = 3 and s = r = (1 − m)/2. The brown, green, orange, blue,
+and grey lines depict the results for ϕ = 0.0, ϕ = 0.05, ϕ = 0.1, ϕ = 0.2, and
+ϕ = 1.0, respectively.
+their conspecifics irrespective of the local densities of enemies.
+According to the brown lines in Figs. 9a, 9b and 9c.
+7. Comments and Conclusions
+Aggregation behaviour is found in many systems where or-
+ganisms adapt their movement, grouping with their conspecifics
+when in death risk. We investigate cyclic models described by
+the rock-paper-scissors game rules, where individuals can scan
+their environment and adapt their movement to environmental
+cues. In our stochastic simulation, each organism freely ex-
+plores the territory without precaution if there is no nearby en-
+emy but prevents damage from enemy attack moving gregarious
+to join the biggest group of conspecific in the neighbourhood.
+To execute the locally adaptive grouping, each individual scans
+their vicinity, thus triggering the gregarious movement if the lo-
+cal density of enemies reaches a prefixed threshold. Running a
+series of simulations, we investigate the role of adaptive aggre-
+gation in transforming the organisms’ spatial organisation. The
+results show that the characteristic length scale of the spatial
+domains occupied by organisms of a single species is not ac-
+centuated if the threshold is not inferior to 10%. Otherwise, the
+typical group size decreases significantly, being minimal in the
+case of organisms flock even when not under death risk pres-
+sure.
+We discover that the gregarious movement does not interfere
+with organisms’ safety if the grouping is only triggered when
+more than 70% neighbourhood is occupied by enemies. Coun-
+terintuitively, if the self-preservation movement tactic is cali-
+brated to be triggered if between 30% and 60% neighbours are
+enemies, the individuals’ death risk increases instead of bene-
+fiting the organisms. Our outcomes show that the behavioural
+strategy is profitable only if each organism aggregates with con-
+specifics when detecting the fraction of opponents in the vicin-
+ity using a threshold inferior to 30%. In addition, we find that
+if organisms can perceive further distances, they can accurately
+scan and interpret the signals from the neighbourhood, increas-
+ing the effects of the adaptive aggregation on the death risk.
+Moreover, we study the impact of mobility on the benefits of
+adaptive congregation considering low, intermediate and high-
+mobile individuals. Our simulations provided evidence that lo-
+cally adapting their movement to aggregate when under death
+risk is more advantageous as the more mobile the organisms,
+provided that the individuals’ mobility is not superior to 85%;
+otherwise, the relative death risk reduction diminishes as the
+mobility grows.
+Finally, we study the influence of locally adaptive aggre-
+gation on biodiversity maintenance.
+Our findings show that
+the coexistence probability increases independently of the or-
+ganism’s mobility, being maximal in the case of non-adaptive
+grouping, where the gregarious movement is executed even
+when there is no local death risk for the individual. This re-
+sult holds for more complex systems where an arbitrary odd
+number of species participate in the cyclic game. Extending
+our algorithm to implement the generalised rock-paper-scissors
+model with five and seven species, we confirm that the gregari-
+ous movement promotes biodiversity, being more beneficial for
+low adaptive aggregation triggers. Our discoveries may be help-
+ful to ecologists in understanding systems where organisms’
+self-defence behaviour adapts to local environmental cues. Our
+results may also clarify the role of the local phenomena in com-
+plex systems in other areas of nonlinear science.
+Acknowledgments
+We thank CNPq, ECT, Fapern, and IBED for financial and
+technical support.
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+Adapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Bin Han
+[email protected]
+University of Washington
+Seattle, USA
+Bill Howe
+[email protected]
+University of Washington
+Seattle, USA
+ABSTRACT
+We adapt image inpainting techniques to impute large, irregular
+missing regions in urban settings characterized by sparsity, variance
+in both space and time, and anomalous events. Missing regions
+in urban data can be caused by sensor or software failures, data
+quality issues, interference from weather events, incomplete data
+collection, or varying data use regulations; any missing data can
+render the entire dataset unusable for downstream applications. To
+ensure coverage and utility, we adapt computer vision techniques
+for image inpainting to operate on 3D histograms (2D space + 1D
+time) commonly used for data exchange in urban settings.
+Adapting these techniques to the spatiotemporal setting requires
+handling skew: urban data tend to follow population density pat-
+terns (small dense regions surrounded by large sparse areas); these
+patterns can dominate the learning process and fool the model into
+ignoring local or transient effects. To combat skew, we 1) train
+simultaneously in space and time, and 2) focus attention on dense
+regions by biasing the masks used for training to the skew in the
+data. We evaluate the core model and these two extensions using
+the NYC taxi data and the NYC bikeshare data, simulating differ-
+ent conditions for missing data. We show that the core model is
+effective qualitatively and quantitatively, and that biased masking
+during training reduces error in a variety of scenarios. We also ar-
+ticulate a tradeoff in varying the number of timesteps per training
+sample: too few timesteps and the model ignores transient events;
+too many timesteps and the model is slow to train with limited
+performance gain.
+CCS CONCEPTS
+• General and reference → Empirical studies; • Computing
+methodologies → Computer vision; • Applied computing;
+KEYWORDS
+image inpainting, urban computing, spatial-temporal, missing data
+ACM Reference Format:
+Bin Han and Bill Howe. 2023. Adapting to Skew: Imputing Spatiotemporal
+Urban Data with 3D Partial Convolutions and Biased Masking. In Proceedings
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
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+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+Conference’17, July 2017, Washington, DC, USA
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+of ACM Conference (Conference’17). ACM, New York, NY, USA, 12 pages.
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+High-quality, longitudinal, and freely available urban data, coupled
+with advances in machine learning, improve our understanding
+and management of urban environments. Although conventional
+machine learning techniques are common in urban applications [35,
+50, 55], neural architectures are opening new opportunities by
+adapting convolutional, recurrent, and transformer architectures to
+spatiotemporal settings [17, 27, 33, 43, 54, 60, 63, 64]; see Grekousis
+2020 for a recent survey [14]. For example, spatio-temporal neural
+architectures have been used in predictions of rideshare demand
+[44, 53], traffic conditions [34, 56], and air quality [23, 32]. But these
+models depend on access to complete, longitudinal datasets. Such
+datasets are inconsistent in availability and quality, limiting the
+opportunity for understanding cities as the complex systems they
+are [2, 15, 22, 51].
+Figure 1: A histogram of taxi pickups in Manhattan. We
+adapt imagine inpainting techniques to reconstruct missing
+and corrupted data in urban settings: The improved model
+(upper left) uses biased masking and temporal context to
+capture local effects (red circle). The basic model (lower left)
+uses ordinary masking and is insensitive to local effects.
+Baseline methods that ignore space (lower middle) or time
+(lower right) are not competitive. Classical linear methods
+such as kriging and inverse-distance weighting (not shown)
+cannot impute large irregular regions in dynamic settings.
+arXiv:2301.04233v1  [cs.CV]  10 Jan 2023
+
+L1 Error: 5.642
+L1 Error: 7.308
+L1 Error: 23.446
+L1 Error: 16.994Conference’17, July 2017, Washington, DC, USA
+Bin Han and Bill Howe
+This inconsistency persists despite significant investments in
+open data. Over the last two decades, cities have increasingly re-
+leased datasets publicly on the web, proactively, in response to
+transparency regulation. For example, in the US, all 50 states and
+the District of Columbia have passed some version of the federal
+Freedom of Information (FOI) Act. While this first wave of open
+data was driven by FOI laws and made national government data
+available primarily to journalists, lawyers, and activists, a second
+wave of open data, enabled by the advent of open source and web
+2.0 technologies, was characterized by an attempt to make data
+“open by default" to civic technologists, government agencies, and
+corporations [49]. While open data has indeed made significant
+data assets available online, their uptake and use has been weaker
+than anticipated [49], an effect attributable to convenience sam-
+pling effects [24]: We release what we can, even if portions are
+missing, corrupt, or anomalous.
+In this paper, we consider a neural data cleaning strategy based
+on masking out corrupted regions and using a trained model to
+reconstruct the masked region. These masks are necessarily large,
+irregular, and extend in both time and space; they may represent po-
+litical boundaries (municipal zoning, zip codes, city blocks), sensor
+or software failures [26, 62, 65], varying legal restrictions [1, 39],
+or unusual events (adverse weather). These missing patches can
+destroy the utility of the entire dataset for applications that assume
+coverage. By modeling missing or corrupted data by an arbitrary
+mask, we afford user control: any areas can be masked and recon-
+structed, regardless of the reason. We envision tools to improve the
+coverage and quality of data for use in downstream urban learning
+tasks [23, 32, 34, 44, 53, 56].
+Following the literature, we represent spatiotemporal event data
+in a 2D or 3D raster form (e.g., a histogram). Our basic model uses
+the partial convolution approach from Liu et al [29] to handle the
+irregular boundaries of missing data (e.g., districts), which focuses
+model attention on the valid regions while shrinking the masked
+region, layer-by-layer, to obtain a complete prediction. More recent
+approaches to image inpainting on the web emphasize eliminating
+perceptual artifacts rather than numerical accuracy and are there-
+fore less relevant to our setting. Our contribution is to extend the
+basic model to the 3D spatiotemporal setting and propose a training
+regime that adapts to the skewed distribution found in practice.
+Spatiotemporal interpolation of missing data has been widely
+studied in the earth sciences [38, 45], especially in remote sensing
+where weather effects can obscure measurement [46, 65]. Con-
+ventional statistical approaches to impute missing values, such as
+global/local mean imputation, interpolation, and kriging, are essen-
+tially linear, and therefore limited in their ability to capture the non-
+linear dynamics needed to impute large irregular missing regions.
+Neural image inpainting techniques [29, 57] can recover missing
+patches via training on large datasets of independent images, such
+that the reconstructed images appear realistic. These approaches
+have shown promising results with global climate data [48], but
+have not been adapted to the urban setting in which data are not
+smooth functions of space and time, but are rather histograms of
+events constrained by the built environment.
+The goal of inpainting for natural images is to produce a subjec-
+tively recognizable image free from perceptible artifacts. But the
+goal in our setting is quantitative accuracy: we intend for our recon-
+structed results to be used numerically in downstream applications.
+The distribution is relatively stable, but exhibits skew and sparsity
+that can obscure local, dynamic features (Figure 2).
+The challenge for imputation in the urban setting is skew: urban
+data tend to follow population density patterns — small dense
+regions surrounded by large sparse areas. These population patterns
+can dominate the learning process and fool the model into ignoring
+numerical accuracy in dense regions, even while aggregate error
+may remains low. To combat skew, we 1) bias the training process
+to focus on populated regions by seeding the mask in non-zero
+areas; (2) use 3D convolutions and vary the number of timesteps in
+each 3D training sample to capture transient events. Together, these
+two techniques complement each other: biased masking focuses
+attention on dense regions, and 3D convolutions with a large chunk
+size focus attention on sparse regions.
+We evaluate these techniques on the NYC taxi data (a popular
+dataset for its coverage and quality) and a NYC bikeshare dataset
+(less dominated by the built environment). We find that the basic
+model is effective for urban data imputation, while biased masking
+reliably reduces error over random masking, both globally and
+locally. Additionally, we find that the number of timesteps per
+training sample exhibits a tradeoff: too few timesteps and the model
+ignores transient patterns, while too many timesteps significantly
+increases training time without enhancing the inpainting results.
+We evaluate specific local scenarios (high-traffic locations, low-
+traffic locations, high-variability locations, anomalous events) to
+reflect the use cases distinct from image inpainting on the web
+(where subjective quality is all that matters).
+Figure 2: Urban data (bottom row) exhibits skewed, sparse,
+yet stable distributions that can dominate learning, in con-
+trast with the diversity of natural images (top row).
+In summary, we make the following contributions:
+• We evaluate a basic model adapting image inpainting techniques
+to urban histograms characterized by skew and sparsity effects
+due to constraints by the built environment, demonstrating qual-
+itative and quantitative accuracy relative to classical methods.
+• We improve on this basic model by extending to the 3D spatiotem-
+poral setting to better recognize transient events; we analyze the
+training time and performance tradeoffs of varying the number
+of timesteps per training sample.
+• We propose a self-supervised training process called biased mask-
+ing to encourage the model to attend to dense population regions
+
+Adapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Conference’17, July 2017, Washington, DC, USA
+and thereby improve accuracy on the highly dynamic regions
+typical in urban environments; we show that biased masking
+reliably improves convergence.
+• We evaluate these techniques on two real mobility datasets (NYC
+taxi trips and NYC bikeshare trips), both globally and locally
+in varying traffic conditions, weather events, and disruptions.
+Finally, we show that the model can be used to remove or syn-
+thesize anomalous events through targeted masking.
+2
+RELATED WORK
+Our work is informed by techniques in image inpainting and geospa-
+tial interpolation.
+Image Inpainting Image inpainting, or image completion, is a
+task of synthesizing missing pixels in images, such that the recon-
+structed images are visually credible and semantically realistic. In
+computer vision, there are two broad categories of inpainting tech-
+niques. The first category contains diffusion-based or patch-based
+methods, which utilize low-level image features to recover the miss-
+ing pixels. The second category contains learning-based methods
+that generally involve the training of deep neural networks.
+Diffusion-based methods [4, 6, 25] propagate information from
+neighboring valid pixels to missing pixels, typically from border to
+the center of the missing regions. Those techniques are convenient
+to apply, but are limited to small missing regions. Recently, Saharia
+et al. [42] developed an image-to-image translation framework
+based on conditional diffusion models. The evaluation on inpainting
+task outperformed several learning-based methods. Patch-based
+inpainting techniques [7, 10, 12, 16] function by searching similar
+patches from the valid regions of the same image or from other
+images, and then paste the patches to the target missing region.
+However, this process could induce high computational costs. A
+milestone of patch-based approach, PatchMatch [5], speeds up the
+search process with a new nearest neighbor algorithm.
+Learning-based methods are trained to learn image patterns with
+large volume of image data, thus being capable of recovering miss-
+ing regions, as well preserving the semantics of the imagery. Pathak
+et al.[36] proposed context encoder, which was the first work to
+combine CNN with generative adversarial network. It applied the
+encoder-decoder architecture and used both ℓ2 reconstruction loss
+and generative adversarial loss in the objective function. Lizuka
+et al. [18] improved on their work by incorporating global and
+local discriminator, which improved content consistency between
+the valid and missing region. Additionally, they replaced general
+convolutional layers with dialated convolutional layers to better
+capture information from distant pixels. Yu et al. [58] proposed
+proposed a two-stage coarse-to-fine model architecture and incor-
+porated contextual attention layer to attend to related features from
+spatially distant regions. They also replaced general generative ad-
+versarial loss with WGANS loss. Liu et al. [29] proposed partial
+convolution, allowing inpainting models to be used on irregular
+holes rather than just rectangular missing regions. On top the work
+of partial convolution, Yu et al. [57] proposed gated convolutional
+layers to automatically learn and update the masks as opposed to
+rule-based update. To further address the problems of blurry tex-
+tures and distorted structures in the inpainted images, Liu et al. [30]
+proposed coherent semantic attention layer, which can both pre-
+serve contextual structure and capture semantic relevance between
+hole features. Zhou et al.[66] incorporated dual spatial attention
+modules into the U-Net architecture, which can capture the corre-
+lations between facial textures at different scales. Seven different
+discriminators are utilized to ensure realistic local details as well
+as global consistency. Yu et al. [59] designed spatial region-wise
+normalization (RN) to overcome the problem of mean and variance
+shifts. RN computes the mean and variances separately for the
+missing and valid regions. Xu et al. [52] combined the paradigms
+of both patch-based and learning-based methods, and inpainted
+missing regions using textures of patch samples from unmasked
+regions. Additionally, they proposed patch distribution loss to en-
+sure the quality of synthesized missing regions. Zeng et al. [61]
+introduced aggregated contextual transformation GAN, aiming to
+improve content reasoning from distant pixels and enhance details
+of synthesized textures. For more image inpainting works, we refer
+reader to the following surveys [19, 31, 37].
+The recent trajectory in image inpainting involves reducing or
+eliminating perceptual artifacts such as discontinuous edges and
+blurred patches using new loss terms, image preprocessing, or train-
+ing regimes that favor subjective quality over numerical accuracy.
+For example, the work of Liu et al.[30], Yu et al. [59], and Xu et al.
+[52] all propose extensions to partial convolutions to repair blurred
+boundaries between missing and valid regions. Since our focus is
+on numerical accuracy and downstream utility of the synthesized
+data, we base our approach on partial convolutions from Liu et al.
+[29]. Additionally, we aim to design and study architecture-agnostic
+training regimes that can be used with newer models when appli-
+cable.
+Geospatial Missing Data Imputation Classical spatio-temporal
+interpolation methods, generally variants of inverse-distance or
+nearest-neighbor weighting [9, 41], kriging [3, 28], or matrix fac-
+torization [13] are variations of linear methods that do not attempt
+to (and cannot) interpolate within large, arbitrary, irregular re-
+gions, and typically do not seamlessly consider both space and time.
+Physics-based models based on computational fluid dynamics [8] or
+agent-based models that directly encode human behavior [11, 47]
+have been used to infer mobility dynamics, but must be designed
+separately for each application rather than learned automatically
+from data. Gong et al. [13] solve multi-variable non-negative ma-
+trix factorization to impute urban data, but assume the availability
+of multiple variables and do not consider arbitrary irregularities.
+Zhang et al. [65] were concerned about the malfunction of satellites
+and poor atmospheric conditions (e.g. thick cloud), which could
+produce missing regions in remote sensing data. They proposed
+unified spatial-temporal-spectral deep CNN architecture to recover
+the missing information in satellite images. Kang et al. [21] mod-
+ified the architecture from [58] to restore the missing patterns of
+sea surface temperature (SST) from satellite images. Tasnim and
+Mondal [48] also adopted the coarse-to-fine inpainting architecture
+from [58] to restore satellite images. The innovation of their work is
+the abandonment of coarse-inpainting pipeline. Instead, they used
+another highly correlated temporal image as an auxiliary input to
+go through the refinement pipeline. Additionally, Kadow, Hall and
+Ulbrich [20] borrowed the architecture from [29] to reconstruct
+missing climate information. In the geo-spatial domain, most of the
+
+Conference’17, July 2017, Washington, DC, USA
+Bin Han and Bill Howe
+literature that we found applied image inpainting techniques on
+remote sensing data. As far as we acknowledge, there is no prior
+work that has taken advantage of image inpainting methods to
+reconstruct missing values in urban data.
+3
+REPRESENTATIVE DATASETS
+We worked with two mobility datasets: NYC taxi data and NYC
+bikeshare data. Although potential applications of the proposed
+model are widely available, datasets on which to evaluate the model
+are rare: we need longitudinal coverage to provide ground truth,
+sufficient complexity to study both global and local fidelity, and
+accessibility to a general audience for expository purposes. Mobility
+data achieves all three goals.
+• NYC Taxi Data. NYC taxi trip data were collected from NYC
+Open Data portal from 2011 to 20161. The year 2011 — 2015 cover
+the trips throughout the entire year, while 2016 only covers the
+first half of the year until June 30. The raw data are presented
+in tabular format. Each record from the data summarizes the
+information for one single taxi trip, which contains the longitude
+and latitude of the location where the taxi took off. Each record
+can be viewed as one taxi demand count.
+• NYC Bikeshare Data: NYC bikeshare data were collected from
+NYC DOT from 2019 to 2021 portal.2 All three datasets cover
+the bike trips throughout the entire year. Similar to the taxi data,
+the raw data are presented in tabular format. Each data point
+summarizes the information for one single bike trip, including
+the longitude and latitude of the location where the bike was
+unlocked. Each record can be viewed as one bike demand count.
+Figure 3: Left: Taxi pickups in 2011 overlaid on a regional
+map. The distribution of taxi demand count is skewed —
+high demand in Manhattan, and low demand in the sur-
+rounding areas. Right: Taxi pickups aggregated into a 64×64
+histogram.
+We aggregate both datasets into a 3D histogram by defining a
+rectangular region, then binning mobility events into a regular grid
+to create a 2D histogram amenable to image techniques. These
+2D images are stacked to create 3D blocks. The temporal depth of
+the block is a parameter we study in this paper. We defined the
+NYC region as shown in Figure 3. At this resolution, the region
+has a relatively balanced coverage of areas with different levels of
+1https://opendata.cityofnewyork.us/data/
+2https://ride.citibikenyc.com/system-data
+demand counts: high demand in Manhattan and near airports, and
+low demand elsewhere. This skew is common in urban applications
+and represents both an opportunity and a challenge for neural
+prediction: the patterns are relatively stable, but the sparse regions
+can dilute the learning process.
+We then define a 64×64 grid over the region of interest. Then,
+for each hour of each day, we count the number of taxi/bike trips
+that began within each pixel. The value is commonly interpreted
+as an estimate of demand. We do not consider multiple resolutions
+in this paper. After processing, we have 30,648 training images and
+3,620 test images in NYC taxi data, and we have 25,560 training
+images and and 720 test images in NYC bikeshare data. Figure 3
+shows the defined region and an example of the corresponding taxi
+demand histogram.
+4
+INPAINTING MODEL
+In this section, we describe the basic model for using partial con-
+volutions for inpainting spatiotemporal urban histograms. Each
+sample consists of a masked region with unknown, corrupted, or in-
+accurate values to be reconstructed and a valid region with known
+values. The task is to predict values in the masked region to match
+the original image. Training is self-supervised by creating random
+masks for any input image; we consider the manner in which the
+masks are created in this paper.
+4.1
+Model Architecture
+We adapt the architecture from Liu et al.[29], which proposed par-
+tial convolutional layers to accommodate irregular masks. Partial
+convolutions ignore the masked region, but the mask is updated af-
+ter each partial convolution layer: after several partial convolution
+layers, all the values in the mask will be set to one such that the
+entire output is considered valid. We use the U-Net architecture
+with skip connections [40], with all the convolution layers replaced
+with partial convolution layers. Web images only contain 2D in-
+Figure 4: The model architecture is a U-Net extended to 3D,
+partial convolutional layers [29] to ignore masked regions
+during training. In the decoding branch, multiple 3D up-
+convolutional layers are utilized and skip-connections are
+applied. In total, there are six encoding layers and six decod-
+ing layers.
+formation (ignoring RGB channels), but urban histograms vary in
+both space and time. As a result, our training data is essentially one
+massive 3D block rather than a large number of independent train-
+ing images. We therefore have a design choice of how to “shred”
+
+QUEENSUrban
+3D
+Data
+Kernel
+Block
+3D ConV,
+3D Up-
+ReLU
+Conv
+Copy &
+More 3D
+Concaten
+Conv
+ate
+LayersAdapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Conference’17, July 2017, Washington, DC, USA
+this block into training samples. In this paper, we consider only
+the temporal extent in 3D; varying spatial resolution, bounds, or
+overlap during rasterization of the source data is left for future
+work.
+If we slice the input into individual timesteps, the model cannot
+exploit temporal consistency. We therefore extend all convolutional
+layers, inputs, and masks, to 3D, and consider the effect of varying
+the number of timesteps per training sample. The inputs are 3D
+image blocks of dimension 𝑇 × 𝑊 × 𝐻, where 𝑇 represents the
+temporal dimension. The masks are also in 3D blocks with the same
+shape as the image block. The model architecture is illustrated in
+Figure 4. The parameters of each convolutional layer appear in
+Table 1.
+Layers
+Channel
+Kernel Size
+Stride
+Padding
+encoder 1
+64
+(1,3,3)
+(1,2,2)
+(0,1,1)
+encoder 2
+128
+(1,3,3)
+(1,2,2)
+(0,1,1)
+encoder 3
+256
+(1,3,3)
+(1,2,2)
+(0,1,1)
+encoder 4
+512
+(1,3,3)
+(1,2,2)
+(0,1,1)
+encoder 5
+512
+(T,3,3)
+(2,2,2)
+(2*((T-1)//4),1,1)
+encoder 6
+512
+(T,3,3)
+(2,2,2)
+(2*((T-1)//4),1,1)
+decoder 1
+512
+(1,3,3)
+(1,1,1)
+(0,1,1)
+decoder 2
+512
+(1,3,3)
+(1,1,1)
+(0,1,1)
+decoder 3
+256
+(1,3,3)
+(1,1,1)
+(0,1,1)
+decoder 4
+128
+(1,3,3)
+(1,1,1)
+(0,1,1)
+decoder 5
+64
+(1,3,3)
+(1,1,1)
+(0,1,1)
+decoder 6
+1
+(1,3,3)
+(1,1,1)
+(0,1,1)
+Table 1: Parameters of 3D convolutional layers. T represents
+the temporal dimension of the image block.
+4.2
+Loss function
+We used ℓ1 loss as the objective function for pixel-wise reconstruc-
+tion accuracy. The ℓ1 loss term bridges the absolute gap between the
+reconstructed value and the ground truth. We adopt the following
+notation
+I𝑔𝑡 ∈ R𝑇×𝑊 ×𝐻: the block of ground truth images. 𝑇 represents
+the temporal dimension of the block.
+I𝑜𝑢𝑡 ∈ R𝑇×𝑊 ×𝐻 : the block of reconstructed images.
+M ∈ R𝑇×𝑊 ×𝐻 : the block of binary masks.
+𝑁I = 𝑇 ∗𝑊 ∗ 𝐻: the total number of pixels in the image block.
+𝑁valid: the total number of valid pixels in the image block.
+𝑁hole: the total number of missing pixels in the image block.
+Following Liu, we separate the valid and hole regions in the
+ℓ1 loss. Even though the valid region has available data and we
+therefore typically would not use the predicted values in practice,
+we want to include this loss during training to improve continuity
+across mask boundaries (and therefore improve overall error). The
+ℓ1 loss is calculated as
+L𝑡𝑜𝑡𝑎𝑙 = L𝑣𝑎𝑙𝑖𝑑 + 𝜆Lℎ𝑜𝑙𝑒
+where
+Lℎ𝑜𝑙𝑒 =
+1
+𝑁hole
+||(1 − M) ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1
+L𝑣𝑎𝑙𝑖𝑑 =
+1
+𝑁valid
+||M ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1
+4.3
+Biased Masking
+By default, masks can be generated by randomly select a starting
+point in the image and then conducting a random walk for a fixed
+number of step. We call this process random masking. However,
+since urban data is constrained by the built environment and is
+therefore highly skewed toward populated areas, random masks
+tend to include a large number of zero-valued cells, squandering
+opportunities to learn from the steep gradients in dense, high-traffic
+regions; Figure 5a illustrates an example. To focus attention on pop-
+ulated areas, we use a biased masking approach: 1) Given an input
+image, apply Gaussian blur to blend the pixel values and increase
+the region of potential starting points. 2) Select a threshold (e.g.,
+90% percentile of the image values) to identify populous regions.
+3) Randomly select a starting location from one of the detected
+areas and generate masks via random walk. The probability of se-
+lecting one of the detected areas is proportional to the size of the
+area. These steps are illustrated in Figure 5b. The biased masking
+approach makes the learning problem more challenging by increas-
+ing “contrast”: ensuring that masks tend to include dense, dynamic
+regions, but also include sparse, stable regions. To compare the
+performance of the two masking approaches, we generated two
+masks (one random and one biased) for each training sample.
+(a) Random masking. For each image (left), randomly select a
+starting point (orange dot, middle), then grow a mask via random
+walk to generate a masked region (right).
+(b) Biased masking. For each image (left), we first apply Gaussian
+blur and then threshold the image (middle images), then select
+a starting point at random in the thresholded region and grow a
+mask via random walk (right).
+Figure 5: Comparison of the random and biased masking
+regimes.
+5
+EXPERIMENTAL EVALUATION
+We consider the following questions:
+(Q1) Is the core 3D model qualitatively & quantitatively effective
+at inpainting missing data? (Section 5.1, Figure 6, Table 2)
+(Q2) Does increasing the number of timesteps per training sample
+generally improve performance? (Section 5.2, Figure 7)
+(Q3) Does biased masking improve performance overall, and in
+specific regions? (Section 5.3, Figure 8)
+
+Conference’17, July 2017, Washington, DC, USA
+Bin Han and Bill Howe
+(Q4) Does varying the number of timesteps per training sample
+influence the spatial distribution of error between sparse and
+dense regions? (Section 5.2, Figure 9)
+(Q5) Does the model faithfully reconstruct local, dynamic condi-
+tions in specific areas of interest? (Section 5.5, Figure 11)
+With NYC taxi data, we trained the models on both mask types
+— random and biased, and with different temporal dimension T =
+{1,2,3,5,7,10,15}. Based on initial experiments on both mask types
+and at lower temporal chunk sizes, we found that 𝜆 = 12 offered
+effective performance; we fix 𝜆 to be 12 for all experiments on the
+taxi data. The batch size and initial learning rate are set to 16 and
+0.01 respectively. Learning rate decays every 500 training iterations
+at rate of 0.9. Unless otherwise stated, we evaluate the model on the
+test set using ℓ1,ℎ𝑜𝑙𝑒, which is the sum of the absolute value of the
+difference between the ground truth and predictions at the masked
+positions only.
+We compare our models with baseline statistical methods:
+• Temporal Global Mean: On the training data, we calculate the
+average taxi demand at each pixel, for each hour of the day. On
+the test data, we assign each masked pixel the corresponding
+global mean computed from the training data.
+• Nearest Neighbor (NN) Interpolation: We assign each masked
+pixel the value of the nearest unmasked pixel. We experimented
+with both 2D and 3D implementations using scipy.3
+• RBF Interpolation We interpolate using radial basis functions
+(RBF) on observations at points sampled outside the masked
+region. We experimented with both 2D and 3D RBF interpolation
+with RBF Python implementation.4
+We considered 3D kriging, but found the poor scalability to be
+prohibitive: the estimated time to complete the computation for an
+experiment with T=2 was about two weeks on a typical platform.
+Moreover, kriging is a linear method, and we have no reason to
+believe that it can reconstruct data across large, irregular regions.
+Another approach, which we did not study, is to use physics-
+based models based on computational fluid dynamics [8] or agent-
+based models that directly encode human behavior [11, 47] to cap-
+ture macro traffic dynamics. These approaches can potentially "fill"
+large missing regions, but must be designed separately for each
+application rather than learned automatically from data.
+5.1
+Model Effectiveness (Q1)
+We find that for both taxi and bikeshare datasets the proposed model
+faithfully captures qualitative visual patterns and also significantly
+outperforms baseline methods on multiple metrics.
+5.1.1
+Qualitative Analysis. We first present some visual examples
+of inpainting results on NYC taxi data in Figure 6. The left figure
+shows taxi demand at four different hours of the day (8AM, 2PM,
+8PM, and 2AM). From left to right, we show the ground truth,
+the (biased) mask, the mask applied to the ground truth, and the
+reconstructed image. The inpainting model was trained with 5
+timesteps per training sample and with biased masking.
+3https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.griddata.
+html#scipy.interpolate.griddata
+4https://github.com/treverhines/RBF
+For all hours and all masks, the model is effective at reconstruct-
+ing missing data, even when the majority of the signal is obscured.
+The reason is clear: the patterns are sufficiently stable from timestep
+to timestep as to allow the model to infer missing values from tempo-
+ral patterns as well as spatial patterns. The model is also responsive
+to the time of day: We see fewer rides at 2AM than at 2PM, as
+expected, suggesting that the model has learned temporally local
+patterns as opposed to relying on global spatial patterns. The transi-
+tion across the mask boundary is also smooth, suggesting the model
+was able to consider local spatial patterns appropriately. Overall,
+we find that the model is perceptually effective at reconstructing
+missing values, even in challenging cases.
+The right plot in Figure 6 visually shows corresponding results
+for bikeshare data. The model was trained with bikeshare data using
+T=3, biased masking and 𝜆 = 4. We observe similar observations as
+the results from taxi data — at all times of day and for all masks,
+the reconstructed images are visually similar to the ground truth
+images, indicating the consistent effectiveness of our model.
+5.1.2
+Quantitative Analysis. Table 2 contains quantitative results
+of baseline models and our neural models in different evaluation
+metrics. We observe that: 1) Our neural models, trained with either
+masking type or with any temporal dimension, always outperform
+the baseline models. The 2D baseline models that ignore the tempo-
+ral dimension are especially ineffective. Global mean ignores spatial
+effects and just models a function 𝑝𝑖𝑥𝑒𝑙,ℎ𝑜𝑢𝑟 → 𝑣𝑎𝑙𝑢𝑒. 2D- and
+3D- nearest neighbor methods perform poorly when the nearest
+neighbors may be far away; 2D- and 3D-RBF methods assume rela-
+tively uniform sampling across the region, which is not possible in
+our setting of wide-area missing data. 2) At T=5 and 7, our method
+performs similarly and achieves the best performances — almost
+50% lower ℓ1 error and 66% lower ℓ2 error than the best baseline.
+3) SSIM does not significantly distinguish different models; while
+popular in image inpainting, this metric is designed to capture per-
+ceptual similarity of natural images, which are not relevant for the
+spatiotemporal aggregations we study. 4) The model training time
+increases by about 9 minutes for every additional hour included in
+a chunk. At T=5, the model takes 55 minutes to train. The baseline
+heuristic-based methods — global mean and 2D- and 3D-NN — are
+very fast (completing in a few minutes) but very inaccurate given
+that they do not attempt to model global dynamics. The 3D-RBF
+method is inefficient: T=2 required over 24 hours to train.
+5.2
+Temporal Dimension Tradeoff (Q2)
+Figure 7 shows the prediction errors for NYC taxi data, evaluated
+on random masks (top plot) and biased masks (bottom plot). The
+y-axis is the ℓ1 loss considered for the masked region only ("Hole").
+The x-axis varies the number of timesteps included per training
+sample (Temporal dimension), ranging from 1 to 15. (a) When tested
+with random masks, the average mask covers the entire region,
+concentrated at the center. Models trained with biased masking
+reduces error at all sizes. The ℓ1 error decreases as the number
+of timesteps increases up until T=7, then starts to increase again
+(T=5 and T=7 have similar performances when trained with biased
+masking.) At T=2, the model begins to make use of the temporal
+dependency between the data by applying 3D convolutions. With
+both biased and random masking, the ℓ1 loss decreases sharply
+
+Adapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Conference’17, July 2017, Washington, DC, USA
+Figure 6: Reconstructed results of taxi demand images (Left) and bike demand images (Right) at different hours time trained
+with biased masking and 3D partial convolutions (T=5 for taxi data and T=3 for bikeshare). From left to right, each column
+displays the ground truth image, mask, masked ground truth, and reconstructed data. From top to bottom, each row presents
+the taxi demand at 8AM, 2PM, 8PM, and 2AM, respectively.
+when T changes from 1 to 5. (b) When tested with biased masks,
+the average masked cells are concentrated at the upper left due to
+the bias toward populated regions. The plot has a similar U-shape
+as that of random masking.
+5.3
+Biased Masking is Effective (Q3)
+Figure 7, as discussed, compares the effects of biased masking to
+random masking at various value of T; we see that at all tested tem-
+poral dimensions, models trained with biased masking outperform
+those trained with random masking, indicated by smaller ℓ1 errors.
+Model
+Mask Type
+ℓ1,ℎ𝑜𝑙𝑒
+ℓ2,ℎ𝑜𝑙𝑒
+SSIM
+PSNR
+Train (m)
+Global Mean
+-
+1.2644
+55.3298
+0.9973
+61.4880
+<5
+2D-RBF
+-
+3.1442
+284.8807
+0.9890
+54.8346
+70
+2D-NN
+-
+3.1179
+318.6575
+0.9884
+54.0717
+<5
+3D-RBF
+-
+1.6653
+94.7708
+0.9956
+57.9921
+>24h
+3D-NN
+-
+1.3632
+84.0529
+0.9964
+59.1652
+<5
+Ours, 𝑇 = 1
+biased
+0.9081
+37.8468
+0.9984
+62.4268
+18
+random
+0.9406
+40.3730
+0.9983
+62.4679
+18
+Ours, 𝑇 = 2
+biased
+0.8551
+32.6429
+0.9986
+63.1815
+27
+random
+0.8979
+35.2923
+0.9985
+63.1056
+27
+Ours, 𝑇 = 3
+biased
+0.7847
+25.8374
+0.9987
+63.6445
+35
+random
+0.7950
+26.4765
+0.9989
+63.7221
+35
+Ours, 𝑇 = 5
+biased
+0.7196
+18.7080
+0.9991
+64.4028
+55
+random
+0.7606
+20.6116
+0.9990
+64.1000
+55
+Ours, 𝑇 = 7
+biased
+0.7185
+18.6746
+0.9990
+64.3407
+75
+random
+0.7489
+20.0100
+0.9990
+64.2656
+75
+Ours, 𝑇 = 10
+biased
+0.7537
+24.8383
+0.9986
+63.3329
+75
+random
+0.7820
+26.1138
+0.9985
+63.1288
+75
+Ours, 𝑇 = 15
+biased
+0.7729
+25.3386
+0.9985
+63.1885
+140
+random
+0.7849
+21.9446
+0.9989
+63.8721
+140
+Table 2: Model training time and performance.
+Figure 7: Evaluation of models trained with biased masking
+against those trained with random masking, at seven tem-
+poral dimensions, with two different masking scenarios —
+random and biased masking.
+In addition to the measurement of overall error, we also inspected
+the convergence rates under both training regimes, as measured
+by the validation set with our selected scenarios (Figure 8). The
+scenario masks are chosen to evaluate local accuracy in high-traffic,
+low-traffic, high-variability, and semantically important locations.
+See 5.5 for masks of the scenarios and detailed evaluations.
+Overall, when we tested with random and biased masks, the
+model trained with biased masks converged faster and had smaller
+errors, indicating that biased masking is beneficial to the imputation
+task under skewed distributions (upper left). Evaluating the 5th
+Avenue and Penn station scenarios, the model trained with biased
+
+Ground Truth, Time: 8AM
+Mask
+Masked Ground Truth
+Prediction, Time: 8AM
+Ground Truth, Time: 2PM
+Mask
+Masked Ground Truth
+Prediction, Time: 2PM
+Ground Truth, Time: 8PM
+Mask
+Masked Ground Truth
+Prediction. Time: 8PM
+Ground Truth. Time: 2AM
+Mask
+Masked Ground Truth
+Prediction. Time: 2AMGround Truth, Time: 8AM
+Mask
+Masked Ground Truth
+Prediction, Time: 8AM
+Ground Truth. Time: 2PM
+Mask
+Masked Ground Truth
+Prediction. Time: 2PM
+Ground Truth. Time: 8PM
+Mask
+Masked Ground Truth
+Prediction. Time: 8PM
+Ground Truth, Time: 2AM
+Mask
+Masked Ground Truth
+Prediction. Time: 2AMTrained With Biased Masking
+Trained With Random Masking
+Trained With Biased Masking
+Trained With Random MaskingConference’17, July 2017, Washington, DC, USA
+Bin Han and Bill Howe
+masking displayed similar patterns — they converged faster and
+achieved better results than the model trained with random masks.
+Those two scenarios are representative of dense and busy areas.
+We conjecture that biased masking avoids rewarding the model for
+trivially predicting zero in sparse regions and ignoring the dynamics
+in dense regions. We consider this result an initial foray: encoding
+domain knowledge and data patterns into the masking strategy
+appears to be a powerful, easy, and architecture-agnostic means of
+improving model performance, aligned with emerging principles of
+data-centric AI. The other three scenarios — airport, lower east side,
+and Astoria, represent sparse regions with relatively light traffic.
+The convergence lines for them are less stable, and no benefit of
+biased masking is realized. We conjecture that variants of biased
+masking to weight both dense and sparse (yet non-zero) areas may
+further improve the model, as would specialized training on regions
+of interest (though that approach could be considered data leakage
+from training to test).
+Figure 8: Convergence plots of the models trained with ei-
+ther biased or random masking, and tested with random
+masks, biased masks and other five additional scenarios
+maskings.
+5.4
+Spatial distribution of errors (Q4)
+We hypothesized that the original 2D partial convolution archi-
+tecture (corresponding to T=1, Figure 7(a)) would be insufficient
+to capture transient events. For example, taxi rides occur in the
+suburbs, but they are infrequent and less predictable; we expected
+the model to be less capable of accurately predicting these events.
+Increasing the temporal dimension is also expected to be helpful
+with the dense region as well.
+We can inspect the spatial distribution of the error for T=1 in
+Figure 9 to check this hypothesis: Each map is the average of 3000
+timesteps, and is colored by the difference between the predicted
+value and the ground truth: a blue cell indicates an underestimate
+and a red cell represents an overestimate. We see that the suburban
+regions are consistently underestimated, while the dense region is
+overestimated. At T=5, we observe similar pattern, but with both
+underestimation and overestimation errors significantly reduced.
+The suburbs are still underestimated, but the dense regions are
+Figure 9: Aggregated spatial errors between predicted and
+ground truth values, from models trained with different
+temporal dimensions. Red areas indicate overestimation,
+while blue areas represent underestimation.
+effectively improved when more temporal dimensions are incorpo-
+rated. At T=15, the spatial error distribution is almost identical to
+T=5, with slightly higher underestimation and lower overestima-
+tion. However, T=15 requires prohibitive training time due to very
+large training samples, so this approach is undesirable with just
+slightly better performance. This tradeoff in temporal scope reflects
+a subtle characteristic of the source data; we hypothesize that T=5
+corresponds to the window size needed to capture dynamic traffic
+periods; e.g., morning and evening commutes.
+5.5
+Scenario Based Evaluation (Q5)
+Spatiotemporal patterns of missing data in practice are unlikely to
+resemble random walks. Instead, outages will correlate with envi-
+ronmental features: sensors may fail in certain weather conditions,
+transient events may prevent data acquisition, or legal restrictions
+on data availability may follow political boundaries. To demonstrate
+the applicability of our inpainting models in real-world situations,
+we evaluate the inpainting methods based on specific locations
+representing varying conditions. We tested five different scenarios
+to cover various spatial locations, temporal variances, and social
+events. The five scenarios include the masking of 5th Avenue, Penn
+Station, airport, lower east side, and Astoria. The masks are visual-
+ized in Figure 10.
+Figure 10: Scenario masks overlaid on NYC map. Annotation:
+The ratio of masked-to-unmasked area.
+
+Trained With Biased Masks
+Trained With Random MasksSpatial Error Distribution - T=l, Mask=biased
+Spatial Error Distribution - T=3, Mask=biased
+Total Overestimation Value: 93.57
+Total Overestimation Value: 101.0
+Total Underestimation Value: -249.85
+Total Underestimation Value: -194.18
+Total Absolute Error Value: 343.42
+Total Absolute Error Value: 295.18
+Spatial Error Distribution - T=5, Mask=biased
+Spatial Error Distribution - T=15, Mask=biased
+0
+-1
+-2
+Total Overestimation Value: 74.6
+Total Overestimation Value: 77.38
+Total Underestimation Value: -179.77
+Total Underestimation Value: -179.04
+Total Absolute Error Value: 254.37
+Total Absolute Error Value: 256.42Sth Avenue
+Airport
+Penn station
+Lower East Side
+Astoria
+MaskingRatio:0.49%
+MaskingRatio:0.8%
+Masking Ratio:0.1%
+MaskingRatio:0.42%MaskingRatio:2.17%Adapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Conference’17, July 2017, Washington, DC, USA
+As mentioned in Section 5.3, 5th Avenue and Penn station are rep-
+resentative of busy and dense areas with heavy traffic. 5th Avenue
+can also show the impacts of certain social events on traffic patterns:
+The Pride Parade showed an anomalous intervention where traffic
+was zero on the parade route. Lower East Side is away from central
+Manhattan, with relatively lighter traffic than the first two cases.
+The scenario of airport and Astoria represent the sparse regions
+where traffic is light.
+We chose two periods for those scenarios to cover temporal
+variance – Feb. 1st to Feb. 15th, 2016, and June, 18th to June 29th,
+2016. A snowstorm from Feb 5th to 8th in New York City is evident
+in the data (Figure 11). On June 26th, 2016, the Pride Parade in
+New York City started at 5th Avenue, and moved downtown to 8th
+Street. The event blocked all traffic along the route and affected the
+surrounding traffic as well. Therefore, testing in the selected June
+period can help evaluate the model’s response to anomalies.
+We test three inpainting models — our model trained with biased
+masking at T=5, the same model but trained with random masking
+at T=5, and the global mean approach. We plotted the ground truth
+and predicted values at the average pixel level in the missing region,
+for each hour during the selected periods. The visualizations are
+provided in Figure 11. The average absolute errors between the
+ground truth and predicted values, over the missing region and
+during the evaluation periods, are reported in Table 3. We have the
+following observations:
+Scenarios
+G.T.- Biased
+G.T. - Random
+G.T. - Mean
+02/01/2016 — 02/15/2016
+5th Avenue
+4.2
+6.2
+17.0
+Penn Station
+19.3
+33.5
+30.0
+Lower East Side
+2.5
+2.8
+8.2
+Airport
+2.3
+1.6
+1.8
+Astoria
+0.8
+0.7
+0.4
+06/18/2016 — 06/30/2016
+5th Avenue
+3.6
+4.8
+22.53
+Penn Station
+21.6
+37.5
+30.0
+Lower East Side
+1.7
+2.1
+7.4
+Airport
+2.4
+1.9
+2.0
+Astoria
+0.8
+0.7
+0.4
+Table 3: Average absolute error between the predicted values
+and ground truth, over the missing regions, and during the
+selected evaluation periods.
+• For three scenarios — 5th Avenue, Penn Station, and Lower East
+Side, our models — whether trained with biased or random mask-
+ing — have much smaller gaps between the predicted values and
+the ground truth, compared with the temporal mean approach.
+This benefit holds for both evaluated periods, as shown in both
+Table 3 and Figure 11. For the airport and Astoria scenarios, the
+temporal mean is slightly better, with much smaller magnitude
+in comparison with other three cases.
+• rom Table 3, we see that for both evaluation periods, the model
+trained with biased masking has smaller average errors than the
+model trained with random masking, other than the scenario of
+airport during June.
+• During the snow days (02/05-02/08/2016), it is expected that
+the traffic in the dense regions would be significantly impacted,
+which can be supported by the trough seen from the ground
+truth line in the scenario of Penn Station (other scenarios are not
+Figure 11: Temporal line plots of evaluations for five sce-
+narios. In each plot, we visualize the ground truth, predic-
+tion from model trained with biased masking and random
+masking, and predictions from temporal mean method. Two
+evaluation periods, Feb. and June are selected. The irregular
+events, extreme snow days and pride parade, are annotated
+with grey regions.
+heavily impacted by the snow.) The model trained with biased
+masking is responsive to the irregular traffic caused by extreme
+weather, unlike the temporal mean baseline.
+• During the event pride parade, the traffic on 5th Avenue was
+all diverted to other routes, creating an anomaly in the traffic
+patterns. Therefore, we saw a dip in the traffic counts. Similar
+observation as the snow day, the temporal mean baseline does not
+recover the missing values . However, even though the inpainting
+results from our model are close to the ground truth values, they
+slightly overestimate the results.
+Overall, the reconstruction accuracy is compelling at specific
+locations, but not perfect. For 5th Avenue scenario, the parade can
+be seen as an anomaly, which is rare in the training stage and hard
+to be detected. But this scenario represents another application
+
+Pride Parade
+Ground Truth
+Biased Prediction
+Random Prediction
+Temporal Mean
+Snow Days
+Ground Truth
+Biased Prediction
+Random Prediction
+Temporal Mean
+Snow Days
+Ground Truth
+Biased Prediction
+Random Prediction
+Temporal Mean
+Snow Days
+Ground Truth
+Biased Prediction
+Random Prediction
+Mear
+Snow Days
+Ground Truth
+Biased Prediction
+Random Prediction
+Temporal MeanConference’17, July 2017, Washington, DC, USA
+Bin Han and Bill Howe
+usage of our model: rather than assuming that ground truth data
+is “correct". We use the masking to intentionally repair known bad
+data, and reconstruct global patterns in a semantically reasonable
+way. This “airbrushing” of flaws in the data can be used to improve
+the quality of training sets for downstream applications, such as
+biofouled or errant sensors and faulty telemetry. For example, from
+the top visualization in Figure 12, we visualize the 5th Avenue
+scenario: The first column shows the taxi counts along 5th Avenue
+during parade day, zoomed in on the Manhattan region. Several
+locations of missing data (white dots) can be seen on the avenue.
+We masked out the 5th Avenue altogether and used our inpainting
+model to reconstruct the missing values. The use case is to enable
+policymakers and researchers to conduct counterfactual studies:
+what would have taxi demand been like were it not for the parade?
+The results, as shown in the forth column, recover the missing
+regions in a realistic way.
+Alternatively, the model might be used to synthesize parade-day
+traffic rather than removing its effects. By masking the surrounding
+area and retaining the parade disruption, the model can attempt to
+represent the influence of the disruption elsewhere in the city. As
+shown from the bottom visualization in Figure 12, the generated
+results are smaller in magnitude, but overall the pattern is matched
+faithfully, suggesting this use case is viable for synthesizing scenar-
+ios that may not be present in the data record (natural disasters,
+proposed construction, accidents, etc.). Penn Station is a train sta-
+Figure 12: Top: “Airbrushing” the parade event (white pix-
+els) to remove its effect on the data. Bottom: Inferring traf-
+fic effects of the parade by reconstructing data everywhere
+except 5th Avenue to produce qualitatively realistic results.
+tion and represents a high-demand area for taxis. Our model tends
+to underestimate the high demand at this location, though biased
+masking improves the prediction. For Lower East Side, there are a
+few anomalous spikes, to which the proposed models are respon-
+sive. For airport and Astoria, our models are no better than the
+temporal mean approach. We conjecture that for airport, the highly
+variable rides in and out of the airport confound the model. For
+Astoria, the much lower demand is harder to predict; note the lower
+scale of the y-axis.
+6
+DISCUSSION
+Our study is motivated by the inconsistent availability of urban data
+caused by missing, corrupt, or inaccurate data, which hinders their
+use in downstream tasks, especially learning tasks, that require
+coverage and accuracy. We designed and implemented a model
+based on partial convolutions that can tolerate irregular missing
+regions — zip codes, geographical boundaries, congrssional districts,
+or other regions that may correlated with data absence or quality.
+To capture the temporal dependency in urban data, we replaced 2D
+convolutional layers in the model with 3D convolutional layers and
+experimented with varying the number of timesteps per training
+sample, finding non-trivial tradeoffs and a local optimum around
+T=5 for taxis and T=3 for bikeshare, potentially interpretable as the
+autocorrelation period of traffic (i.e., about 5 hours of rush hour).
+To address the spatial skew in human activity, we proposed a
+masking approach that can reflect the skew in the distribution. By
+encouraging the model to attend to dense, dynamic regions (via a
+percentile threshold), the model learns faster and is not rewarded
+for accurate predictions in trivially inactive areas. Biased mask-
+ing showed improved performance across all values of 𝑇, multiple
+global evaluation strategies, and most local evaluation scenarios.
+This approach suggests a broader family of related masking strate-
+gies to help users encode domain knowledge about the data and
+setting. For example, encoding correlations between high-traffic
+areas (e.g., subway stops and train stations during lunch time) as
+masks may help the model learn these correlations with less data.
+Qualitatively, we confirmed from the visual examples that im-
+age inpainting techniques can be used to reconstruct data in large,
+irregular regions in space and time. Quantitatively, we confirmed
+that extending the model architecture to 3D benefits improves per-
+formance, as supported by the sharp decrease in ℓ1 when T changes
+from 1 to 2. Second, we observe that increasing the temporal di-
+mension to a certain threshold improves performance in general,
+regardless of masking strategy; ignoring the temporal dimension
+in this setting is untenable.
+Additionally, we evaluated performance in local settings, demon-
+strating that the model is not just learning an average value, but is
+responsive to subtle spatial variation. The model captures irregular
+traffic patterns caused by transient events, such as extreme weather
+and the Pride Parade, and showed that biased masking can improve
+performance in local settings. Additionally, the scenario evaluations
+also showcased the better results introduced by the biased masking
+than the random masking.
+7
+LIMITATIONS & FUTURE WORK
+There are several limitations of our study that represent directions
+for future work. First, our results on mobility data may extend
+to other urban activity (e.g., 311 calls, crowd movement, business
+permits, public safety events, housing events, and more). We do not
+consider the generalizability of these methods to multiple variables,
+or variables that do not follow the same spatial patterns; there are
+opportunities to exploit correlations between variables to improve
+performance. Additionally, the taxi dataset is exceptionally large
+and complete; understanding how these techniques behave in low-
+data regimes is important for practical applications. Integration
+of masked multi-variate data may be an opportunity: given the
+shared built environment, models trained on one variable may
+transfer to predictions of other variables. Second, rasterizing event
+data to a form amenable to computer vision techniques involves a
+number of design choices we did not study: resolution, overlap, and
+
+Adapting to Skew: Imputing Spatiotemporal Urban Data
+with 3D Partial Convolutions and Biased Masking
+Conference’17, July 2017, Washington, DC, USA
+irregular boundaries may present opportunities or challenges. In
+particular, data associated with census blocks, tracts, or individual
+trajectories lose information when regridded as histograms. In
+these cases, graph neural networks may be more appropriate to
+represent the spatial adjacency relationships. Third, even with the
+best model configuration, we consistently overestimate in the city
+region and underestimate in the sparse suburban region. Some
+model architectures (attention mechanism, multi-view learning) or
+loss functions may improve performance, as may more specialized
+masking and training regimes.
+8
+CODE AVAILABLITY
+Our code is available at [anonymized for review].
+REFERENCES
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+Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping
+Albrecht Kurze
+Chemnitz University of Technology, [email protected]
+Have you ever typed particularly powerful on your keyboard, maybe even harsh, to write and send a message with some emphasis of 
+your emotional state or message? Did it work? Probably not. It didn't affect how you typed or interacted with your mouse. But what 
+if you had other, connected devices, with other modalities for inputs and outputs? Which would you have chosen, and how would  
+you characterize your interactions with them? We researched with our multisensory and multimodal tool, the Loaded Dice, in co-
+design workshops the design space of IoT usage scenarios: what interaction qualities users want, characterized using an interaction 
+vocabulary, and how they might map them to a selection of sensors and actuators. We discuss based on our experience some  
+thoughts of such a mapping.
+CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI)
+Additional Keywords and Phrases: interaction, vocabulary, emotions, qualities, design, ideation, tools, methods, IoT
+ACM Reference Format:
+Albrecht Kurze. 2022. Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping. In Workshop 
+The Future of Emotion in Human-Computer Interaction (CHI’22). April 13-14, 2022. 4 pages.
+1
+INTRODUCTION
+Some years ago we designed and developed the Loaded Dice [9,10], a multisensory and multimodal hybrid toolkit to 
+ideate IoT devices and scenarios, e.g. for the ‘smart’ home, and with different groups of co-designers  [3,8,9]. The 
+Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.g. KnowCards [1], and functional but 
+tinkering based tools, e.g. littleBits [2], for multisensory and multimodal exploration, ideation and prototyping.
+We will introduce a) our adapted and extended interaction vocabulary, b) how we use it in a method to explore and 
+the describe the WHY and HOW of interactions with and through connected devices; and c) we introduce the Loaded 
+Dice, what sensing and actuating functions they have, and how we use the to let participants map interaction qualities  
+to modalities to align the WHY and the HOW of interactions.
+This brings us to our core question: Is it possible to derive a (somehow universal) mapping between certain interaction 
+qualities, i.e. emotional ones, and specific modalities and actions?
+
+1st
+2nd
+Goals
+Actors
+Spaces
+3rd
+precise
+frendly
+..
+4publie
+private
+covered
+apparent
+deterministic
+chaotic
+close
+distant
+fluent
+stepwise
+prosaic
+poetic
+papeas
+binary
+precise
+approximate
+metaphorical
+factual
+spotial
+spatial
+proximity
+separation
+direct
+mediated
+familiar
+strange
+targeted
+incidental
+constant
+inconstant
+srabbing
+powerful
+gentle
+instant
+delayed
+harsh
+tender
+diverging
+uniform
+fast
+slow
+angn
+friendlySENSORDIE
+Temperature Sensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanINPUTFigure 1: left: the types of cards of our co-design method, 1st setting a goal (why), 2nd context of interaction (actors and space), 
+3rd defining desired interaction qualities; right: cards with terms of the extended interaction vocabulary
+2
+ADAPTED AND EXTENDED INTERACTION VOCABULARY
+Diefenbach  [4] introduced in 2013 a first interaction vocabulary to describe interaction qualities in a user 
+perspective. The original vocabulary consisted of 11 pairs of adjectives and antonyms, e.g.  fast and  slow. An 
+example of how the vocabulary was intended to describe interaction qualities: When we switch the light in a 
+room, this happens in a binary way at the switch (on/off) with an instant effect in the same way at the lamp 
+distant to the switch. With a dimmer the input and output are graded in a fluent or stepwise manner (vocabulary 
+terms in italics). The original vocabulary was intended for use as a semantic differential on a graded scale, e.g. in a 
+questionnaire.
+Our intention was to not only characterize interactions with a single object but also in complex connected 
+interaction scenarios. In scenarios as the IoT allows them for smart connected things, across multiple devices and 
+shared between multiple involved actors (typically human users but not limited to them).
+While Diefenbach’s intention for the original interaction vocabulary was first to describe “the HOW of 
+interaction” [4] they also had drawn a first conclusion between HOW and WHY of interaction. We put this first. It 
+became clear to us that it is often not meaningful to isolate the HOW from the WHY of interaction. Therefore we 
+embedded the interaction vocabulary methodically in a goal-actors-properties driven scenario creation to match 
+the IoT design space [3]. We adapted and extended the original vocabulary. We did this in the same way as the 
+original vocabulary was constructed – as pairs of adjectives and antonyms. Additionally, we introduced to the 
+vocabulary an extension with some more emotional terms to grasp interaction qualities often beyond a non-
+judgmental dimension [4]. We discuss their role following on with mappings to specific modalities. We have 
+already iterated the actual terms based on what we have learned in co-design workshops using the vocabulary. 
+We see the vocabulary still as work in progress.
+We created based on the vocabulary a set of cards for the use in co-design workshops. On the front face an  
+adjective  and on the  back the  antonym  (fig. 1b). We introduced a subtle differentiation  between  the  two 
+categories. The non-judgmental terms (including the original terms) are in black letters on colored background 
+while the slightly more emotional terms  are in white  letters. In contrast to Diefenbach’s graded semantic 
+differential we decided with the cards only for the extremes - an ‘either or’. However, this stimulates in the co-
+design workshops a verbalization how something is meant – often not in the extremes but then user defined 
+graded.
+2
+
+1st
+2nd
+Goals
+Actors
+Spaces
+3rd
+precise
+frendly
+..
+4publie
+private
+covered
+apparent
+deterministic
+chaotic
+close
+distant
+fluent
+stepwise
+prosaic
+poetic
+papeas
+binary
+precise
+approximate
+metaphorical
+factual
+spotial
+spatial
+proximity
+separation
+direct
+mediated
+familiar
+strange
+targeted
+incidental
+constant
+inconstant
+srabbing
+powerful
+gentle
+instant
+delayed
+harsh
+tender
+diverging
+uniform
+fast
+slow
+angn
+friendlySENSORDIE
+Temperature Sensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanINPUTFigure 2: left: faces and functions of the Loaded Dice – sensors and actuators [9]; right: an example of using the cards to 
+characterize and map input and output qualities of an ideated connected product with help of the Loaded Dice [6]
+3
+INTERACTION MODALITIES
+The Loaded Dice are a set of two cubical devices wirelessly connected (fig. 2a). Each cube has six sides, offering 
+in one cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and 
+multimodal environmental and user interactions. The sensor cube normalizes a raw sensor value meaningfully, 
+transmits it, and then the other cube actuates it mapped on an output. The cubical shape communicates the 
+intuitive reading that the top side is active, like a die, offering an easy and spontaneous way to re-combine sensors 
+and actuators. Every sensor-in and actuator-out combination is possible resulting in 36 combinations in total. [6] 
+New  multisensory  interaction  modalities,  not  yet  implemented,  e.g.  smell,  have  the  potential  to  broaden 
+interaction qualities even further and especially in an emotional way [7]. 
+4
+MAPPING INTERACTION QUALITIES
+Last step in our co-design method is a mapping of desired interaction qualities by participants to interaction 
+modalities represented by the Loaded Dice (fig. 2b). The extended interaction vocabulary allows characterizing the 
+intended interactions very well while the Loaded Dice allow participants to try them out up to a certain degree. 
+This way our workshops often brought up a number of unconventional ideas of multisensory interactions with 
+devices, often far beyond ordinary inputs and especially outputs. In our experience sensory sensations and 
+modalities do not need to be perfect, at least for ideation of interactions and scenarios as well as for mapping 
+interaction qualities. It is about bringing the idea and the core concept behind it to the co-design activities. A 
+demonstration of a technical possibility for sensing and actuating as a stimulus is often enough to trigger further 
+thinking and verbalization of how something might be used.
+We found some repeating themes when it comes to a mapping between certain interaction characteristics and 
+suitable sensing as well as actuating possibilities. For example, the thermo-element was not only associated with 
+slow and warmth literally but also with ‘love’ and tender in a poetic way, while loud sound and bright light were 
+selected for powerful, attention-grabbing and sometimes even harsh interactions etc. Participants often chose non-
+visible and non-audible modalities for  private interactions,  covered and not easily perceivable by others, only 
+noticeable  to  a mentioned  one, e.g. using heat  or vibration  in  ideated  wearable  devices. In  another  case 
+participants mapped the vibration motors and the associated sound caused when having the Loaded Dice placed 
+on a wooden table to  attention-grabbing and  harsh, associated to feelings of being alarmed and named it 
+“electronic rattlesnake”.
+We also found similar patterns for inputs. The distance sensor can detect a hand in proximity in different ways. 
+In a graded kind, if done slowly, allowing for gentle gestures, e.g. swiping with the hand through the air above the 
+sensor,  without  touching something,  without  any force. As these  gestures  can be  very  similar to  petting 
+3
+
+1st
+2nd
+Goals
+Actors
+Spaces
+3rd
+precise
+frendly
+..
+4publie
+private
+covered
+apparent
+deterministic
+chaotic
+close
+distant
+fluent
+stepwise
+prosaic
+poetic
+papeas
+binary
+precise
+approximate
+metaphorical
+factual
+spotial
+spatial
+proximity
+separation
+direct
+mediated
+familiar
+strange
+targeted
+incidental
+constant
+inconstant
+srabbing
+powerful
+gentle
+instant
+delayed
+harsh
+tender
+diverging
+uniform
+fast
+slow
+angn
+friendlySENSORDIE
+Temperature Sensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanINPUTsomething they were associated with this action in a poetic and very tender way. On the other hand, a fast and 
+sudden movement is also detectable, like a punch, being very powerful, targeted and harsh. While the distance 
+sensor allows for such a differentiation based on the speed of hand movement the PIR movement detection sensor 
+allows not – what also might be wanted, e.g. for an only binary type of input.
+Both ways of using the distance sensor for proximity-based hand gestures are possible and meaningful. 
+However, the HOW of the interaction is then depending on the emotional state of the user and the WHY of 
+interaction. Therefore, it makes a difference what a user tries to express and in an end-to-end view of interactions 
+‘through’ devices [6], from one device to another device, as communication to another actor. Has the user the 
+intention to send a message with a positive emotion, a non-verbal equivalent of “I love you tender”, or to send a 
+message associated with a negative emotion, e.g. with an equivalent in “Turn the damn music down”? In terms of 
+Hassenzahl’s model of interactions for experience design  [5], as a hierarchy of WHY, WHAT, and HOW of 
+interactions, it is clear that this WHY, the motive, defines the WHAT and the HOW. Therefore, it will not be a 
+simple  one-dimensional  mapping  between  an  emotional  interaction  quality  and  specific  sensor  /  actuator 
+modalities. The motive of use (as higher-level use goal) and the expressed or implied associate actions (e.g. petting 
+or punching) must also be considered.
+5
+CONCLUSION
+While we see especially big potential in the use of an interaction vocabulary and different modalities for 
+intending or expressing emotional interaction qualities, it still needs further exploration to identify certain 
+patterns for a mapping. 
+ACKNOWLEDGMENTS
+This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.
+References
+[1]
+Tina Aspiala and Alexandra Deschamps-Sonsino. 2016. Know Cards: Learn. Play. Collect. Know Cards. Retrieved December 6, 2016 from 
+http://know-cards.myshopify.com/
+[2]
+Ayah Bdeir. 2009. Electronics As Material: LittleBits. In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction 
+(TEI ’09), 397–400. https://doi.org/10.1145/1517664.1517743
+[3]
+Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker. 2019. The Inflatable Cat: Idiosyncratic 
+Ideation  Of  Smart  Objects  For  The  Home.  In  CHI  Conference  on  Human  Factors  in  Computing  Systems  Proceedings. 
+https://doi.org/10.1145/3290605.3300631
+[4]
+Sarah Diefenbach, Eva Lenz, and Marc Hassenzahl. 2013. An Interaction Vocabulary. Describing the How of Interaction. In CHI ’13 Extended 
+Abstracts on Human Factors in Computing Systems (CHI EA ’13), 607–612. https://doi.org/10.1145/2468356.2468463
+[5]
+Marc Hassenzahl. 2010. Experience Design: Technology for All the Right Reasons. Synthesis Lectures on Human-Centered Informatics 3, 1: 1–
+95. https://doi.org/10.2200/S00261ED1V01Y201003HCI008
+[6]
+Albrecht  Kurze.  2021.  Interaction  Qualities  For  Interactions  With,  Between,  And  Through  IoT  Devices. 
+https://doi.org/10.1145/3494322.3494348
+[7]
+Albrecht Kurze. 2021. Scented Dice: New interaction qualities for ideating connected devices. In Workshop Smell, Taste, and Temperature 
+Interfaces at Conference on Human Factors in Computing Systems (CHI ’21). Retrieved from https://arxiv.org/abs/2201.10484
+[8]
+Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger. 2016. Explorative Co-Design-Werkzeuge 
+zum  Entwerfen  von  Smart  Connected  Things  am  Beispiel  eines  Workshops  mit  Blinden  und  Sehbehinderten.  In  Technische 
+Unterstützungssysteme, die die Menschen wirklich wollen, 395–400. Retrieved January 19, 2017 from http://tinyurl.com/janya26
+[9]
+Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger.  2016. Loaded Dice: 
+Exploring the Design Space of Connected Devices with Blind and Visually Impaired People. In Proceedings of the 9th Nordic Conference on 
+Human-Computer Interaction (NordiCHI ’16), 31:1-31:10. https://doi.org/10.1145/2971485.2971524
+[10]
+Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger. 2017. Loaded Dice: How to cheat your 
+way to creativity. In Proceedings of the 3rd Biennial Research Through Design Conference. https://doi.org/10.6084/m9.figshare.4746976.v1
+4
+
+1st
+2nd
+Goals
+Actors
+Spaces
+3rd
+precise
+frendly
+..
+4publie
+private
+covered
+apparent
+deterministic
+chaotic
+close
+distant
+fluent
+stepwise
+prosaic
+poetic
+papeas
+binary
+precise
+approximate
+metaphorical
+factual
+spotial
+spatial
+proximity
+separation
+direct
+mediated
+familiar
+strange
+targeted
+incidental
+constant
+inconstant
+srabbing
+powerful
+gentle
+instant
+delayed
+harsh
+tender
+diverging
+uniform
+fast
+slow
+angn
+friendlySENSORDIE
+Temperature Sensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanINPUT

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+page_content='de  Have you ever typed particularly powerful on your keyboard, maybe even harsh, to write and send a message with some emphasis of  your emotional state or message?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Did it work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Probably not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=" It didn't affect how you typed or interacted with your mouse." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' But what  if you had other, connected devices, with other modalities for inputs and outputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Which would you have chosen, and how would  you characterize your interactions with them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We researched with our multisensory and multimodal tool, the Loaded Dice, in co-  design workshops the design space of IoT usage scenarios: what interaction qualities users want, characterized using an interaction  vocabulary, and how they might map them to a selection of sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We discuss based on our experience some  thoughts of such a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI)  Additional Keywords and Phrases: interaction, vocabulary, emotions, qualities, design, ideation, tools, methods, IoT  ACM Reference Format:  Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In Workshop  The Future of Emotion in Human-Computer Interaction (CHI’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' April 13-14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  1  INTRODUCTION  Some years ago we designed and developed the Loaded Dice [9,10], a multisensory and multimodal hybrid toolkit to  ideate IoT devices and scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' for the ‘smart’ home, and with different groups of co-designers  [3,8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The  Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' KnowCards [1], and functional but  tinkering based tools, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' littleBits [2], for multisensory and multimodal exploration, ideation and prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  We will introduce a) our adapted and extended interaction vocabulary, b) how we use it in a method to explore and  the describe the WHY and HOW of interactions with and through connected devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content='Potentiometer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Distance Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='ACTUATORDIE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Vibration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Heating Surface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='LED-Bargraph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Loudspeaker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Power-LEDs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='FanINPUTFigure 1: left: the types of cards of our co-design method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 1st setting a goal (why),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2nd context of interaction (actors and space),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  3rd defining desired interaction qualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' right: cards with terms of the extended interaction vocabulary  2  ADAPTED AND EXTENDED INTERACTION VOCABULARY  Diefenbach  [4] introduced in 2013 a first interaction vocabulary to describe interaction qualities in a user  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The original vocabulary consisted of 11 pairs of adjectives and antonyms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' fast and  slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' An  example of how the vocabulary was intended to describe interaction qualities: When we switch the light in a  room, this happens in a binary way at the switch (on/off) with an instant effect in the same way at the lamp  distant to the switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' With a dimmer the input and output are graded in a fluent or stepwise manner (vocabulary  terms in italics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The original vocabulary was intended for use as a semantic differential on a graded scale, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' in a  questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  Our intention was to not only characterize interactions with a single object but also in complex connected  interaction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In scenarios as the IoT allows them for smart connected things, across multiple devices and  shared between multiple involved actors (typically human users but not limited to them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  While Diefenbach’s intention for the original interaction vocabulary was first to describe “the HOW of  interaction” [4] they also had drawn a first conclusion between HOW and WHY of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We put this first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' It  became clear to us that it is often not meaningful to isolate the HOW from the WHY of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Therefore we  embedded the interaction vocabulary methodically in a goal-actors-properties driven scenario creation to match  the IoT design space [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We adapted and extended the original vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We did this in the same way as the  original vocabulary was constructed – as pairs of adjectives and antonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Additionally, we introduced to the  vocabulary an extension with some more emotional terms to grasp interaction qualities often beyond a non-  judgmental dimension [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We discuss their role following on with mappings to specific modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We have  already iterated the actual terms based on what we have learned in co-design workshops using the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  We see the vocabulary still as work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  We created based on the vocabulary a set of cards for the use in co-design workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' On the front face an  adjective  and on the  back the  antonym  (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' We introduced a subtle differentiation  between  the  two  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The non-judgmental terms (including the original terms) are in black letters on colored background  while the slightly more emotional terms  are in white  letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In contrast to Diefenbach’s graded semantic  differential we decided with the cards only for the extremes - an ‘either or’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' However, this stimulates in the co-  design workshops a verbalization how something is meant – often not in the extremes but then user defined  graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  2  1st  2nd  Goals  Actors  Spaces  3rd  precise  frendly  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content='Temperature Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Light Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Microphone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='MovementSensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Potentiometer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Distance Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='ACTUATORDIE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Vibration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Heating Surface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='LED-Bargraph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Loudspeaker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Power-LEDs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='FanINPUTFigure 2: left: faces and functions of the Loaded Dice – sensors and actuators [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' right: an example of using the cards to  characterize and map input and output qualities of an ideated connected product with help of the Loaded Dice [6]  3  INTERACTION MODALITIES  The Loaded Dice are a set of two cubical devices wirelessly connected (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Each cube has six sides, offering  in one cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and  multimodal environmental and user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The sensor cube normalizes a raw sensor value meaningfully,  transmits it, and then the other cube actuates it mapped on an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The cubical shape communicates the  intuitive reading that the top side is active, like a die, offering an easy and spontaneous way to re-combine sensors  and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Every sensor-in and actuator-out combination is possible resulting in 36 combinations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' [6]  New  multisensory  interaction  modalities,  not  yet  implemented,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' smell,  have  the  potential  to  broaden  interaction qualities even further and especially in an emotional way [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  4  MAPPING INTERACTION QUALITIES  Last step in our co-design method is a mapping of desired interaction qualities by participants to interaction  modalities represented by the Loaded Dice (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The extended interaction vocabulary allows characterizing the  intended interactions very well while the Loaded Dice allow participants to try them out up to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  This way our workshops often brought up a number of unconventional ideas of multisensory interactions with  devices, often far beyond ordinary inputs and especially outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In our experience sensory sensations and  modalities do not need to be perfect, at least for ideation of interactions and scenarios as well as for mapping  interaction qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' It is about bringing the idea and the core concept behind it to the co-design activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' A  demonstration of a technical possibility for sensing and actuating as a stimulus is often enough to trigger further  thinking and verbalization of how something might be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  We found some repeating themes when it comes to a mapping between certain interaction characteristics and  suitable sensing as well as actuating possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' For example, the thermo-element was not only associated with  slow and warmth literally but also with ‘love’ and tender in a poetic way, while loud sound and bright light were  selected for powerful, attention-grabbing and sometimes even harsh interactions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Participants often chose non-  visible and non-audible modalities for  private interactions,  covered and not easily perceivable by others, only  noticeable  to  a mentioned  one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' using heat  or vibration  in  ideated  wearable  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In  another  case  participants mapped the vibration motors and the associated sound caused when having the Loaded Dice placed  on a wooden table to  attention-grabbing and  harsh, associated to feelings of being alarmed and named it  “electronic rattlesnake”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  We also found similar patterns for inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The distance sensor can detect a hand in proximity in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  In a graded kind, if done slowly, allowing for gentle gestures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' swiping with the hand through the air above the  sensor,  without  touching something,  without  any force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' As these  gestures  can be  very  similar to  petting  3  1st  2nd  Goals  Actors  Spaces  3rd  precise  frendly  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content='covered  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='apparent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='deterministic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='chaotic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='close  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='distant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='fluent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='stepwise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='prosaic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='poetic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='papeas  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='binary  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='precise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='approximate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='metaphorical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='factual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='spotial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='spatial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='proximity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='separation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='direct  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='mediated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='familiar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='strange  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='targeted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='incidental  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='constant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='inconstant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='srabbing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='powerful  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='gentle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='instant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='delayed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='harsh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='tender  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='diverging  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='uniform  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='fast  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='slow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='angn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='friendlySENSORDIE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Temperature Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Light Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Microphone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='MovementSensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Potentiometer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Distance Sensor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='ACTUATORDIE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Vibration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Heating Surface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='LED-Bargraph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Loudspeaker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='Power-LEDs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='FanINPUTsomething they were associated with this action in a poetic and very tender way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' On the other hand, a fast and  sudden movement is also detectable, like a punch, being very powerful, targeted and harsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' While the distance  sensor allows for such a differentiation based on the speed of hand movement the PIR movement detection sensor  allows not – what also might be wanted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' for an only binary type of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  Both ways of using the distance sensor for proximity-based hand gestures are possible and meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  However, the HOW of the interaction is then depending on the emotional state of the user and the WHY of  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Therefore, it makes a difference what a user tries to express and in an end-to-end view of interactions  ‘through’ devices [6], from one device to another device, as communication to another actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Has the user the  intention to send a message with a positive emotion, a non-verbal equivalent of “I love you tender”, or to send a  message associated with a negative emotion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' with an equivalent in “Turn the damn music down”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In terms of  Hassenzahl’s model of interactions for experience design  [5], as a hierarchy of WHY, WHAT, and HOW of  interactions, it is clear that this WHY, the motive, defines the WHAT and the HOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Therefore, it will not be a  simple  one-dimensional  mapping  between  an  emotional  interaction  quality  and  specific  sensor  /  actuator  modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The motive of use (as higher-level use goal) and the expressed or implied associate actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' petting  or punching) must also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  5  CONCLUSION  While we see especially big potential in the use of an interaction vocabulary and different modalities for  intending or expressing emotional interaction qualities, it still needs further exploration to identify certain  patterns for a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='  References  [1]  Tina Aspiala and Alexandra Deschamps-Sonsino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Know Cards: Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Know Cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Retrieved December 6, 2016 from  http://know-cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='myshopify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='com/  [2]  Ayah Bdeir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Electronics As Material: LittleBits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction  (TEI ’09), 397–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='1145/1517664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='1517743  [3]  Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' The Inflatable Cat: Idiosyncratic  Ideation  Of  Smart  Objects  For  The  Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In  CHI  Conference  on  Human  Factors  in  Computing  Systems  Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' An Interaction Vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Describing the How of Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In CHI ’13 Extended  Abstracts on Human Factors in Computing Systems (CHI EA ’13), 607–612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Experience Design: Technology for All the Right Reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Interaction  Qualities  For  Interactions  With,  Between,  And  Through  IoT  Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content=' Scented Dice: New interaction qualities for ideating connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In Workshop Smell, Taste, and Temperature  Interfaces at Conference on Human Factors in Computing Systems (CHI ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content='10484  [8]  Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Explorative Co-Design-Werkzeuge  zum  Entwerfen  von  Smart  Connected  Things  am  Beispiel  eines  Workshops  mit  Blinden  und  Sehbehinderten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' In  Technische  Unterstützungssysteme, die die Menschen wirklich wollen, 395–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content=' Retrieved January 19, 2017 from http://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
+page_content='com/janya26  [9]  Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+page_content=' In Proceedings of the 9th Nordic Conference on  Human-Computer Interaction (NordiCHI ’16), 31:1-31:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'}
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+arXiv:2301.01276v1  [cs.IT]  3 Jan 2023
+Age of Information of a Power Constrained
+Scheduler in the Presence of a Power
+Constrained Adversary
+Subhankar Banerjee
+Sennur Ulukus
+Anthony Ephremides
+Department of Electrical and Computer Engineering
+University of Maryland, College Park, MD 20742
+[email protected]
+[email protected]
+[email protected]
+Abstract—We consider a time slotted communication network
+consisting of a base station (BS), an adversary, N users and
+Ns communication channels. In the first part of the paper, we
+consider the setting where Ns communication channels Ns are
+heterogeneously divided among N users. The BS transmits an
+update to the ith user on a subset of the communication channels
+Ns,i where Ns,i ∩ Ns,j is not necessarily an empty set. At each
+time slot, the BS transmits an update packet to a user through a
+communication channel and the adversary aims to block the
+update packet sent by the BS by blocking a communication
+channel. The BS has n discrete transmission power levels to
+communicate with the users and the adversary has m discrete
+blocking power levels to block the communication channels.
+The probability of successful transmission of an update packet
+depends on these power levels. The BS and the adversary have a
+transmission and blocking average power constraint, respectively.
+We provide a universal lower bound for the average age of
+information for this communication network. We prove that
+the uniform user choosing policy, the uniform communication
+channel choosing policy with any arbitrary feasible transmission
+power choosing policy is 4 optimal; and the max-age user
+choosing policy, the uniform communication channel choosing
+policy with any arbitrary feasible transmission power choosing
+policy is 2 optimal. In the second part of the paper, we consider
+the setting where the BS chooses a transmission policy and the
+adversary chooses a blocking policy from the set of randomized
+stationary policies and Ns,i = Ns for all i, i.e., all users can
+receive updates on all channels. We show that a Nash equilibrium
+may or may not exist for this communication network, and
+identify special cases where a Nash equilibrium always exists.
+I. INTRODUCTION
+We consider a wireless communication system consisting
+of N users, one base station (BS), Ns communication chan-
+nels and an adversary. A communication channel can have
+different channel gains to different users, and thus, all the sub-
+carriers may not be available to all the users for transmission
+of an update packet. We consider the static setting. Thus,
+the communication channels are divided into N potentially
+overlapping sets, where each set corresponds to a user. We
+denote the set of communication channels available to user
+i as Ns,i. A sub-carrier can be an element of multiple sets,
+and thus, the set Ns,i ∩ Ns,j is not necessarily empty. The
+cardinality of Ns,i is Ns,i. The set of all available channels is
+Ns = �
+i Ns,i, and has cardinality Ns. There are n discrete
+power levels available to the BS for transmission of an update
+packet to the users and m discrete power levels available to
+the adversary to block the transmission of an update packet.
+We consider a slotted time model. At each time slot, the BS
+chooses a transmission power to transmit an update packet to
+a user via a communication channel and the adversary chooses
+a communication channel and a blocking power to block any
+update packet that is being sent on the chosen channel.
+A large amount of work has been done on the analysis
+of age of information for various applications and system
+models, such as, scheduling policies for wireless networks,
+gossip networks, caching systems, source coding problem,
+remote estimation, energy harvesting systems and many more,
+see e.g., [1]–[41]. These papers consider systems without
+an adversary. The age of information in the presence of an
+adversary in a wireless communication network has been
+studied in the recent literature [42]–[50]. In particular, [49],
+[50] consider an adversarial gossip network. In this paper, we
+do not consider a gossip network, rather we consider that a
+central node, i.e., the BS transmits the update packets to the
+users. [42], [43] consider an adversary which decreases the
+signal to noise ratio of a communication link through jamming,
+due to which the rate of the communication decreases which
+results in a higher age for the communication system. In
+this paper, we consider that when the adversary blocks a
+communication channel it completely eliminates the update
+packet with a positive probability. [44] considers an adversary
+which blocks the communication channel for a duration in time
+which increases the average age of the system by disabling
+communication in that interval. In this paper, we consider
+that the adversary blocks the communication channel in a
+time slotted manner. [45], [46] consider an adversary which
+completely eliminates the update packet, however, they do not
+consider any power constraint on the adversary. In this paper,
+we consider a power constrained adversary. [47], [48] consider
+a power constrained adversary which completely eliminates
+the update packet. They have considered that on the time hori-
+zon T , the adversary blocks αT time slots where 0 < α < 1.
+On the contrary, in this paper, we consider that at each time
+slot t, the adversary chooses one of the m blocking power
+levels with a pmf d(t) and the expected power to be less than
+or equal to a power constraint. Different than the adversary in
+[47], [48], the adversary in this paper completely eliminates
+
+the update packet with a positive probability (strictly less than
+1), and this probability depends on the blocking power chosen
+by the adversary and the transmission power chosen by the BS.
+In the first part of this paper, we propose algorithms to
+minimize the average age of information for the described
+wireless communication network. We show that the uniform
+user choosing policy together with the uniform communication
+channel choosing policy and any arbitrary feasible transmis-
+sion power choosing policy is 4 optimal, and in a special case,
+it is 2 optimal. We show that the maximum-age user choos-
+ing policy together with the uniform communication channel
+choosing policy and any arbitrary feasible transmission power
+choosing policy is 2 optimal.
+In the second part of this paper, we relax the system model
+and consider that at each time slot the BS can choose any one
+of the Ns sub-carriers for transmission of an update packet
+to any one of the N users, i.e., Ns,i = Ns, for all i. We
+also restrict the action space of the BS and the action space
+of the adversary only to the stationary policies. If the power
+level choosing algorithms are not fixed for the BS and for the
+adversary and if those are included in the action space of the
+BS and the action space of the adversary, then we show that
+in the stationary policy regime a Nash equilibrium may not
+exist. We give a counter example to prove this. We also show
+a special case in which the Nash equilibrium exists. However,
+when the power level choosing algorithms for the BS and for
+the adversary are fixed, i.e., those are not included in the list
+of the actions of the BS and the list of the actions of the
+adversary, then the Nash equilibrium always exists.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+At each time slot, the BS schedules a user i out of N users,
+N > 1, with a user choosing algorithm πu and chooses a
+communication channel out of Ns,i communication channels,
+Ns,i > 1, with a communication channel choosing algorithm
+πs to transmit an update packet to the scheduled user i. In
+this paper, we use sub-carrier and communication channel
+interchangeably. We consider that n discrete transmission
+powers, namely {p1, p2, · · · , pn} are available to the BS, and
+at each time slot the BS chooses one of these n transmission
+powers, following a power choosing algorithm πp. Thus, an
+action of the BS is a triplet (πu, πs, πp) and we call a valid
+triplet as a BS scheduling algorithm π. We call the set of all
+causal scheduling algorithms as Π. Let us consider that πp is
+such that at time slot t the BS chooses the ith transmission
+power with probability ei(t). We consider the following power
+constraint for the BS,
+n
+�
+i=1
+ei(t)pi ≤ ¯p,
+t ∈ {1, · · · , T }
+(1)
+We consider that an adversary is present in the system as
+well. At each time slot, the adversary chooses a sub-carrier
+out of Ns sub-carriers following an algorithm ψs to block any
+update packet that is being transmitted by the BS in that sub-
+carrier. We consider that m discrete blocking powers, namely
+{p′
+1, p′
+2, · · · , p′
+m} are available to the adversary and at each
+time slot the adversary chooses one of these powers, following
+a blocking power choosing algorithm ψp, to block any update
+packet on the sub-carrier chosen by ψs. Thus, an action of
+the adversary is a pair (ψs, ψp) and we call a valid pair as an
+adversarial action ψ. We call the set of all valid adversarial
+actions as Ψ. Let us consider that ψp is such that at time
+slot t, the adversary chooses the ith blocking power with
+probability di(t). We consider the following power constraint
+for the adversary,
+m
+�
+i=1
+di(t)p′
+i ≤ ˜p,
+t ∈ {1, · · · , T }
+(2)
+We create an n × m matrix F , whose (i, j)th element,
+fi,j, represents the probability of successful transmission of
+an update packet corresponding to the BS transmission power
+pi and adversary blocking power p′
+j. Thus, at time slot t if
+the BS schedules the user k, and chooses the sub-carrier l to
+transmit an update packet with power pi and if the adversary
+blocks the sub-carrier l with power p′
+j, then with probability
+fi,j the age of the kth user at time slot (t+ 1) becomes 1 and
+with probability 1 − fi,j the age of the kth user at time slot
+t + 1 increases by one.
+The age of user i at time slot t is defined as t−ti(t), where
+ti(t) is the last time slot when the ith user has successfully
+received an update packet. Note that the minimum value for
+the age of user i is 1. We consider that at each time slot the BS
+has a fresh update packet to transmit for every user present in
+the system. Here by fresh update packet, we mean the update
+packet for the ith user at time slot t is generated at time slot
+t. As we are interested in freshness, we assume that if the ith
+user does not receive the corresponding update packet at time
+slot t, then that update packet gets dropped at the BS without
+any cost. This is a valid assumption used in [45]–[48].
+The adversary has the knowledge of πu, πs and πp. How-
+ever, as the BS uses a randomized algorithm at time slot t, the
+adversary has no knowledge about which user will get sched-
+uled, which sub-carrier will get chosen and which transmission
+power will get used at time slot t′ when t ≤ t′ ≤ T . However,
+at time slot t it has full knowledge about all these for time slot
+t′ when 1 ≤ t′ < t, and the adversary can optimize its future
+actions based on these available information. The adversary
+has full knowledge about the elements of each set Ns,i. The
+age of user i at time slot t corresponding to a BS scheduling
+algorithm π and adversarial action ψ is denoted as v(π,ψ)
+i
+(t),
+thus, v(π,ψ)
+i
+(t) = t − ti(t), and the expected age of user i
+at time slot t, is denoted as ∆(π,ψ)
+i
+(t). Note that, if the BS
+successfully transmits an update packet to user i at time slot t,
+then v(π,ψ)
+i
+(t+1) = 1, otherwise v(π,ψ)
+i
+(t+1) = v(π,ψ)
+i
+(t)+1.
+The average age of the overall system corresponding to the BS
+scheduling algorithm π and adversarial action ψ is,
+∆(π,ψ) = lim sup
+T →∞
+1
+T
+T
+�
+t=1
+1
+N
+N
+�
+i=1
+∆(π,ψ)
+i
+(t)
+(3)
+For the simplicity of presentation, in the rest of the paper
+we ignore the superscript (π, ψ), unless we specify otherwise.
+
+Now, as the BS has no control over the adversary, we consider
+the following constrained optimization problem,
+∆∗ = sup
+ψ∈Ψ
+inf
+π∈Π
+∆(π,ψ)
+s.t.
+(1), (2)
+(4)
+For the second part of the paper, we consider a relaxed
+system model. We consider that at each time slot, all the
+Ns sub-carriers are available to the BS to transmit an update
+packet to any one of the N users, i.e., Ns,i = Ns for all
+i. The BS chooses a scheduling algorithm and the adversary
+chooses an adversarial action from the corresponding sets of
+stationary randomized policies. In other words, πu is such
+that at each time slot the BS chooses a user following a pmf
+u = [u1, u2, · · · , uN], πs is such that at each time slot the BS
+chooses a sub-carrier following a pmf s = [s1, s2, · · · , sNs]
+and πp is such that at each time slot the the BS chooses a
+power following a pmf e = [e1, e2, · · · , en]. Similarly, ψs is
+such that at each time slot the adversary blocks a sub-carrier
+following a pmf a = [a1, a2, · · · , aNs] and ψp is such that at
+each time slot the adversary chooses a blocking power follow-
+ing a pmf d = [d1, d2, · · · , dm]. Thus, the power constraints
+for the adversary and the BS become �m
+i=1 dip′(i) ≤ ˜p and
+�n
+i=1 eip(i) ≤ ¯p, respectively. When we restrict ourselves
+only to the stationary randomized policies, instead of writing
+∆π,ψ as in (3), we write the average age of the overall system
+corresponding to pmfs u, s, e (these three pmfs are chosen
+by the BS) and the pmfs a, d (these two pmfs are chosen by
+the adversary) as ∆u,s,e,a,d. We denote the expected age of
+user i at time slot t as ∆u,s,e,a,d
+i
+(t). Thus, the average age
+for the ith user becomes
+∆u,s,e,a,d
+i
+= lim sup
+T →∞
+1
+T
+T
+�
+t=1
+∆u,s,e,a,d
+i
+(t)
+(5)
+Let us assume that the set of all valid user choosing pmfs,
+the set of all valid sub-carrier choosing pmfs and the set of all
+valid transmission power choosing pmfs are Fu, Fs and Fe,
+respectively. Similarly, the set of all valid sub-carrier blocking
+pmfs and the set for all valid blocking power choosing pmfs
+are Fa and Fd, respectively. For a given adversarial action,
+namely a sub-carrier blocking pmf a, and a blocking power
+level choosing pmf d, the BS aims to minimize the average
+age of the overall system by selecting a scheduling algorithm,
+namely a user choosing pmf u, a sub-carrier choosing pmf
+s and a transmission power choosing pmf e from the set
+B(a, d), where B(a, d) is defined as follows,
+B(a, d) =
+arg min
+(u∈Fu,s∈Fs,e∈Fe,�n
+i=1 eipi≤¯p)
+∆u,s,e,a,d
+(6)
+Similarly, for a given scheduling algorithm, i.e., a triplet of
+pmfs (u, s, e), the adversary aims to maximize the average
+age by choosing a pair of pmfs, namely (a, d) from the set
+B(u, s, e), where B(u, s, e) is defined as
+B(u, s, e) =
+arg max
+(a∈Fa,d∈Fd,�m
+i=1 dip′(i)≤˜p)
+∆u,s,e,a,d
+(7)
+We call a 5-tuple of pmfs, namely (u, s, e, a, d) as a Nash
+equilibrium point if and only if (u, s, e) ∈ B(a, d) and
+(a, d) ∈ B(u, s, e).
+In the previous Nash equilibrium setting we consider that
+the transmission power choosing pmf e and blocking power
+choosing pmf d are components of the action space of the BS
+and the action space of the adversary, respectively. However,
+if e and d are fixed and not included in the action space of
+the BS and the action space of the adversary, respectively, then
+we define,
+B(a) =
+arg min
+(u∈Fu,s∈Fs)
+∆u,s,e,a,d
+(8)
+Similarly, we write,
+B(u, s) = arg max
+(a∈Fa)
+∆u,s,e,a,d
+(9)
+We call a triplet of pmfs, namely (u, s, a) as a Nash equilib-
+rium point if and only if (u, s) ∈ B(a, ) and a ∈ B(u, s).
+III. ALGORITHM AND ANALYSIS OF AGE
+We find a fundamental lower bound for the optimization
+problem in (4). Let us define x = arg maxi∈{1,··· ,m} p′
+i ≤ ˜p.
+Consider the following adversarial action: at each time slot
+the adversary blocks any one of the Ns sub-carriers with a
+uniform pmf and chooses the power level px. We denote this
+adversarial action as ¯ψ = ( ¯ψs, ¯ψp). At each time slot, if the BS
+schedules the user which has the maximum age and breaks the
+tie with scheduling the lower indexed user, we call that user
+choosing policy as the max-age policy. (In this paper, we will
+present our results in a sequence of lemmas and theorems,
+with some explanations. The proofs are skipped here due to
+space limitations, and will be provided in the journal version.)
+Lemma 1. For the adversarial action ¯ψ, an optimal user
+choosing policy is the max-age policy; and if the ith user gets
+chosen by the max-age policy, then an optimal sub-carrier
+choosing policy is to choose a sub-carrier in Ns,i uniformly.
+Let us define ¯y = arg mini∈{1,··· ,n} pi ≥ ¯p.
+Theorem 1. The average age of the communication network
+defined in (3) is lower bounded by
+(N+1)Ns
+2(Ns−1+f¯
+y,x).
+Now, we consider that at each time slot the BS schedules
+a user i with probability
+1
+N and chooses one of the Ns,i sub-
+carriers with probability
+1
+Ns,i , to transmit an update packet to
+the scheduled user with transmission power py with probability
+β and with transmission power p¯y with probability (1 − β),
+where β satisfies the following identity:
+βpy + (1 − β)p¯y = ¯p
+(10)
+Let us denote this BS scheduling policy as ˆ˜π. Let us define
+¯x = arg mini∈{1,··· ,m} p′
+i ≥ ˜p.
+Theorem 2. The average age of the communication system
+when the BS employs the scheduling algorithm ˆ˜π is upper
+bounded by 2N; when Ns,i = Ns for all i, then the average
+age is upper bounded by
+NNs
+Ns−1+βfy,¯x+(1−β)f¯
+y,¯x .
+
+Now, we consider that at each time slot the BS schedules the
+max-age user, i, and chooses one of the Ns,i sub-carriers with
+probability
+1
+Ns,i . We also consider that the BS chooses power
+py with probability β and power p¯y with probability 1 − β,
+where β satisfies (10). Denote this BS scheduling policy as ˜˜π.
+Theorem 3. The average age of the communication sys-
+tem when the BS employs the scheduling algorithm ˜˜π is
+upper bounded by
+(N+1) ¯
+Ns
+2( ¯
+Ns−1+βfy,¯x+(1−β)f¯
+y,¯x), where
+¯Ns =
+min {Ns,1, Ns,2, · · · , Ns,N}.
+Next, we make some concluding remarks about the findings
+of this section. From Theorem 1 and Theorem 2, we see that
+in the general setting, ˆ˜π is 4N(Ns−1+f¯
+y,x)
+(N+1)Ns
+optimal, where
+4N(Ns − 1 + f¯y,x)
+(N + 1)Ns
+≤ 4
+(11)
+For the special case, when Ns,i = Ns, for all i, ˆ˜π is
+2(N+1)(Ns−1+f¯
+y,x)
+N(Ns−1+fy,¯x)
+optimal, where
+2(N + 1)(Ns − 1 + f¯y,x)
+N(Ns − 1 + fy,¯x)
+≤2(Ns − 1 + f¯y,x)
+(Ns − 1 + fy,¯x)
+(12)
+≤ 2Ns
+Ns − 1
+(13)
+≤4
+(14)
+If Ns is large, then the right side of (13) can be approximated
+as 2. Thus, for the aforementioned special case and for large
+Ns, ˆ˜π is 2 optimal.
+From Theorem 1 and Theorem 3, we see that the scheduling
+policy ˜˜π is
+¯
+Ns
+¯
+Ns−1 optimal and as Ns,i > 1, for all i, ˜˜π is 2
+optimal. Note that when ¯p exactly matches with one of the
+powers from the sets {p1, p2, · · · , pn} and Ns,i = Ns, for all
+i, then ˜˜π is the optimal scheduling policy.
+IV. EQUILIBRIUM POINTS OF THE AVERAGE AGE FOR
+RANDOMIZED STATIONARY ACTION SPACE
+Let us assume that at each time slot the BS chooses a user
+following a pmf u, chooses a sub-carrier following a pmf s,
+chooses a transmission power with a pmf e and the adversary
+chooses a sub-carrier with a pmf a and chooses a blocking
+power following a pmf d. Recall that for this section we use
+a relaxed system model, where we consider that Ns,i = Ns,
+for all i. At some time slot t, user i successfully receives an
+update packet transmitted by the BS and then after waiting for
+Γi time slots it again receives another update packet from the
+BS. Note that Γi is a random variable. The evolution of the
+age for the ith user is a renewal process and Γi is a renewal
+interval. Thus, from the renewal reward theorem,
+∆u,s,e,a,d
+i
+= E
+�
+Γ2
+i + Γi
+�
+2E [Γi]
+(15)
+Let the probability of successful transmission of the update
+packet to user i be qi. Then, Γi is geometrically distributed
+with success probability qi. Thus, (15) simplifies as,
+∆u,s,e,a,d
+i
+= 1
+qi
+(16)
+Theorem 4. The optimal sub-carrier choosing pmf s, for
+a given adversarial action, namely, a pair of pmfs (a, d),
+depends only on a and is independent of user choosing pmf u,
+transmission power choosing pmf e and d. Moreover, if the
+adversary blocks any l sub-carriers with lowest probability
+then the optimal choice for the BS is to choose any subset of
+these l sub-carriers with probability 1. Similarly, the optimal
+user scheduling pmf u does not depend on a, s, d, e. The
+optimal user scheduling pmf is the uniform pmf.
+Theorem 5. The optimal sub-carrier blocking pmf, a, for
+a given BS scheduling policy depends only on s and is
+independent of u, e and d. Moreover, if the BS chooses any
+l sub-carriers with the highest probability, then the optimal
+choice for the adversary is to block any subset of these l sub-
+carriers with probability 1.
+Without loss of generality, let p1 ≤ p2 ≤ · · · ≤ pn and
+p′
+1 ≤ p′
+2 ≤ · · · ≤ p′
+m. Thus, we have f1,j ≤ f2,j ≤ · · · ≤ fn,j
+and fi,1 ≥ fi,2 ≥ · · · ≥ fi,m, i = 1, · · · , n, j = 1, · · · , m.
+Algorithm 1 below provides an optimal transmission power
+choosing pmf e for a given blocking power choosing pmf
+d. The algorithm states that, if ¯p < p1, then there does
+not exist a feasible e; if pn < ¯p, then the optimal e is to
+choose the power pn with probability 1; If these two cases
+do not occur, then we define x = arg maxi∈{1,··· ,n},pi<¯p i
+and y = arg mini∈{1,··· ,n},pi>¯p i. Clearly, x < y. We define
+a constant, gi = �m
+j=1 djfi,j, i = 1, · · · , n. We call the
+constant
+�
+gi + gx
+py−pi
+px−py − gy
+px−pi
+px−py
+�
+as the coefficient for
+power pi, i ∈ {1, · · · , n}\{x, y}. Then, we traverse from
+power py+1 to power pn, we call this procedure as the first
+traversing procedure. During this traversing process, if we find
+that
+�
+gj + gx
+py−pj
+px−py − gy
+px−pj
+px−py
+�
+, j > y, is a strictly positive
+number, then we change the coefficient of the power pk as
+�
+gk + gx
+pj−pk
+px−pj − gj
+px−pk
+px−pj
+�
+, k ∈ {1, · · · , n}\{x, j}. We keep
+on doing this procedure till we reach pn. Let us assume that
+during this traversing procedure pi is the last power for which
+we get a positive coefficient, then we define y = i. Then,
+we start performing a second traversing procedure from the
+power px−1 to the power p1. During this traversing process,
+if we find that the coefficient of pl, l < x, is a strictly
+positive number, then we change the coefficient of the power
+pk as
+�
+gk + gl
+py−pk
+pl−py − gy
+pl−pk
+pl−py
+�
+, k ∈ {1, · · · , n}\{l, y}. We
+keep on doing this procedure till we reach p1. Let us assume
+that during this second traversing procedure pr is the last
+power for which we get a positive coefficient, then we define
+x = r. Now, if ¯p exactly matches one of the powers from
+the set {p1, p2, · · · , pn}, without loss of generality assume
+that pi
+=
+¯p, then we compare the two vectors zi and
+�
+¯p−py
+px−py zx + px−¯p
+px−py zy
+�
+and select the one which minimizes
+(15), otherwise we select
+�
+¯p−py
+px−py zx + px−¯p
+px−py zy
+�
+, where zi is
+the ith basis vector of Rn.
+We note that, Algorithm 1 finds an optimal solution in O(n)
+time. Next, we state the optimality of Algorithm 1.
+
+Algorithm 1 For a given d finding an optimal e
+Inputs: d, F , p, ¯p
+Define:
+g
+=
+(g1, g2, · · · , gn),
+where
+gi
+=
+�m
+j=1 djfi,j,
+x
+=
+arg maxi∈{1,2,··· ,n},pi<¯p i
+and
+y
+=
+arg mini∈{1,2,··· ,n},pi>¯p i,
+zi
+is
+the
+ith
+basis
+vector for Rn, x1 = x, y1 = y
+if ¯p < p1 then
+Return: Solution does not exist
+else if pn < ¯p then
+Return: zn
+for i = y + 1 : n do
+if
+�
+gi + gx
+py−pi
+px−py − gy
+px−pi
+px−py
+�
+> 0 then
+y = i
+for i = 1 : x − 1 do
+if
+�
+gi + gx
+py−pi
+px−py − gy
+px−pi
+px−py
+�
+> 0 then
+x = i
+Define: e =
+�
+¯p−py
+px−py zx + px−¯p
+px−py zy
+�
+if x1 + 1 = y1 − 1 then
+if �n
+i=1 ei
+�m
+j=1 djfi,j ≤ �m
+j=1 djfx1+1,j then
+Return: zx+1
+else
+Return: e
+else
+Return: e
+Theorem 6. For a given blocking power pmf d, Algorithm 1
+gives an optimal transmission power pmf e.
+Algorithm 2 provides an optimal blocking power choosing
+pmf d for a given e. In Algorithm 2, we perform a similar
+traversing procedure as Algorithm 1. The only difference is
+while traversing in Algorithm 1, we change the coefficient
+of a power level if the corresponding coefficient is strictly
+positive, in Algorithm 2, we change the coefficient if it is
+strictly negative. Next, we state the optimality of Algorithm 2.
+Theorem 7. For a given transmission power choosing pmf e,
+Algorithm 2 gives an optimal blocking power pmf d.
+Next, we present a counter example which suggests that
+when the transmission power choosing pmf and the blocking
+power choosing pmf are not fixed and are part of the action
+space of the BS and the action space of the adversary,
+respectively, then a Nash equilibrium may not exist. Consider a
+system where the BS has three power levels and the adversary
+has also three power levels, i.e., n = m = 3. Both the power
+constraint for the BS and the adversary is 3.5 watts. The
+feasible powers for the BS and for the adversary are the same,
+which is [1, 3, 5]. The matrix F is chosen as
+F =
+
+
+0.5
+0.35
+0.2
+0.6
+0.55
+0.4
+0.8
+0.7
+0.65
+
+
+(17)
+We can show that for this example, for a given d, e cannot
+be of the form [e1, e2, e3], where ei > 0, i ∈ {1, 2, 3} and
+satisfy �3
+i=1 eipi ≤ ¯p. Now, from Algorithm 1, we know that
+Algorithm 2 For a given e finding an optimal d
+Inputs: e, F , p, ¯p
+Define:
+g
+=
+(g1, g2, · · · , gm),
+where
+gi
+=
+�n
+j=1 ejfj,i,
+x
+=
+arg maxi∈{1,2,··· ,m},p′
+i<˜p i
+and
+y
+=
+arg mini∈{1,2,··· ,m},p′
+i>˜p i, zi
+is
+the ith
+basis
+function for Rn, x1 = x, y1 = y
+if ˜p < p′
+1 then
+Return: Solution does not exist
+else if p′
+n < ˜p then
+Return: zn
+for i = y + 1 : n do
+if
+�
+gi + gx
+p′
+y−p′
+i
+p′
+x−p′
+y − gy
+p′
+x−p′
+i
+p′
+x−p′
+y
+�
+< 0 then
+y = i
+for i = 1 : x − 1 do
+if
+�
+gi + gx
+p′
+y−p′
+i
+p′
+x−p′
+y − gy
+p′
+x−p′
+i
+p′
+x−p′
+y
+�
+< 0 then
+x = i
+Define: d =
+� ˜p−p′
+y
+p′x−p′y zx + p′
+x−˜p
+p′x−p′y zy
+�
+if x1 + 1 = y1 − 1 then
+if �m
+j=1 dj
+�n
+i=1 eifi,j ≤ �n
+i=1 eifi,x1+1 then
+Return: d
+else
+Return: zx+1
+else
+Return: d
+if the adversary chooses powers 3 and 5, then the optimal
+choice for the BS is to choose powers 3 and 5, similarly, if
+the adversary chooses powers 1 and 5, then the optimal choice
+for the BS is to choose powers 1 and 5. From Algorithm 2,
+we know that if the BS chooses powers 1 and 5, then the
+optimal choice for the adversary is to choose powers 3 and 5,
+similarly, if the BS chooses powers 3 and 5, then the optimal
+choice for the adversary is to choose powers 1 and 5. Thus, a
+Nash equilibrium does not exist for this example.
+In the next theorem, we consider the Nash equilibrium when
+the transmission power choosing pmf and the blocking power
+choosing pmf are not included in the action space of the BS
+and in the action space of the adversary, respectively.
+Theorem 8. The triplet of actions (ˆu, ˆs, ˆa) is the Nash
+equilibrium point, where ˆa and ˆs are the uniform pmfs over
+Ns sub-carriers and ˆu is the uniform pmf over N users.
+Next, we present a special case in which the Nash equi-
+librium exists even when the transmission power choosing
+pmf and the blocking power choosing pmf are part of the
+action space of the BS and the action space of the adversary,
+respectively. Consider that the matrix F has the property,
+fi,j − f1,j = li,
+j ∈ {1, · · · , m}, i ∈ {1, · · · , n}
+(18)
+where li are non-negative constants. Consider a fixed blocking
+power choosing pmf d. Then, gi in Algorithm 1 is
+gi =
+m
+�
+j=1
+djfi,j =
+m
+�
+j=1
+djf1,j + li
+(19)
+
+Thus,
+gi + gx
+py − pi
+px − py
+− gy
+px − pi
+px − py
+=
+
+
+m
+�
+j=1
+djf1,j
+
+
+�
+1 + py − pi
+px − py
+− px − pi
+px − py
+�
++ lx
+py − pi
+px − py
+− ly
+px − pi
+px − py
++ li
+(20)
+Thus, the sign of gi +gx
+py−pi
+px−py −gy
+px−pi
+px−py does not depend on
+d, which implies that the optimal transmission power choosing
+pmf is the same for all d. Similarly, the sign of gi+gx
+p′
+y−p′
+i
+p′x−p′y −
+gy
+p′
+x−p′
+i
+p′x−p′y in Algorithm 2 does not depend on e, in other words
+the optimal blocking power choosing pmf is independent of
+e. Now, run Algorithm 1 for any arbitrary d and denote the
+output as ˆe, similarly run Algorithm 2 for any arbitrary e and
+denote the output as ˆd. Then, using Theorem 8, we have that
+the 5-tuple (ˆb, ˆc, ˆe, ˆa, ˆd) is the unique Nash equilibrium.
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+

HNAzT4oBgHgl3EQfjP3H/content/tmp_files/2301.01514v1.pdf.txt

@@ -0,0 +1,814 @@
+1
+PENDANTSS: PEnalized Norm-ratios Disentangling
+Additive Noise, Trend and Sparse Spikes
+Paul Zheng, Student Member, IEEE, Emilie Chouzenoux, Senior Member, IEEE, Laurent Duval, Senior Member,
+IEEE
+Abstract—Denoising, detrending, deconvolution: usual restora-
+tion tasks, traditionally decoupled. Coupled formulations entail
+complex ill-posed inverse problems. We propose PENDANTSS
+for joint trend removal and blind deconvolution of sparse peak-
+like signals. It blends a parsimonious prior with the hypothesis
+that smooth trend and noise can somewhat be separated by
+low-pass filtering. We combine the generalized quasi-norm ratio
+SOOT/SPOQ sparse penalties ℓp/ℓq with the BEADS ternary
+assisted source separation algorithm. This results in a both
+convergent and efficient tool, with a novel Trust-Region block
+alternating variable metric forward-backward approach. It out-
+performs comparable methods, when applied to typically peaked
+analytical chemistry signals. Reproducible code is provided.
+Index Terms—Blind deconvolution, sparse signal, trend es-
+timation, non-convex optimization, forward-backward splitting,
+alternating minimization, source separation
+I. INTRODUCTION AND BACKGROUND
+Restoration recovers information from observations with
+amplitude distortion, level displacement or random distur-
+bance. A discrete additive-convolutive degradation model is:
+y = s ∗ π + t + n .
+(1)
+Among N sample values, series of spikes (or impulses, events,
+“diracs”, spectral lines) prototype the first component sought
+sparse signal s ∈ RN. Its convolution with an unknown
+short-support kernel π ∈ RL — typically peak-shaped —
+yields the peak-signal x = s ∗ π ∈ RN. Second component
+t ∈ RN displaces the reference level, harming quantitative
+estimations. It can be called baseline, background, continuum,
+drift, wander. We opt here for trend, a reference above which
+peaks are detected, evaluated, measured. This “trend” notion
+goes from mere offsets to slowly-varying amplitude shifts
+(seasonality, calibration distortion, sensor decline), making its
+automated removal challenging. Third component n ∈ RN
+(noise) gathers stochastic residuals. Given (1), one goal is
+to perform jointly denoising, detrending and deconvolution.
+Namely, given y, to retrieve an estimation of the spiky signal
+and the trend. Figure 1 is reminiscent of standard spectral
+subtraction [1], and motivated here by peak-signal retrieval in
+separative analytical chemistry (AC): chromatography, spec-
+trometry, spectroscopy [2], where peak localization, amplitude,
+width or area provide useful chemical quantitative information.
+This work was supported by the European Research Council Starting Grant
+MAJORIS ERC-2019-STG-850925.
+P. Zheng is currently with Chair of Information Theory and Data Analyt-
+ics, RWTH Aachen University, Germany ([email protected]);
+work conducted while at Univ. Paris-Saclay, CentraleSup´elec, CVN, Inria,
+Gif-sur-Yvette, France.
+E. Chouzenoux is with Univ. Paris-Saclay, CentraleSup´elec, CVN, Inria,
+Gif-sur-Yvette, France ([email protected]).
+L. Duval is with IFP Energies nouvelles, France ([email protected]).
+Whether acquired in its natural domain [3] or after spar-
+sification [4], noise/trend/spike models (1) cover many mul-
+tidimensional issues: signal (1D), image (2D), video, volume
+(3D+). We focus here on 1D data common to distinct domains:
+Fourier spectral analysis, econometrics, stocks, biomedical
+measurements (ECG, EEG, EMG), environment, astronomy. ..
+On the one hand, joint denoising and detrending is a long-
+standing preprocessing question from time series analysis to
+imaging. Background issues are commonly solved using a host
+of filling, fitting and filtering methods. We refer to overviews
+in [5], [6], and for AC to background corrections backcor [7]
+and BEADS [8].
+On the other hand, joint denoising and deconvolution
+matters from channel estimation in communications [9] to
+image deblurring [10]. We refer to [11], [12], and especially
+emphasize on sparsity-promoting methods like SOOT [13] and
+SPOQ [14], using smoothed ”scale-invariant” norm ratios.
+PENDANTSS contributions are a fully coupled and solvable
+non-convex formulation for (1) (Section II) and an efficient
+joint disentangling algorithm (forward-backward-based [15],
+[16]) with proved convergence (Section III), validated by its
+comparative performance (Section IV).
+II. PROPOSED PROBLEM FORMULATION
+A. BEADS peak/trend/noise separation paradigm
+We seek estimates (�s, �t, �π) of (s, t, π) to penalized problem
+minimize s,t∈RN
+π∈RL
+1
+2∥y − π ∗ s − t∥2 + R(s, t, π).
+(2)
+The squared loss is supplemented with regularization R,
+incorporating prior knowledge. Disentangling trend and signal
+is tedious [17]. As in BEADS [8], we assume that the trend
+can be recovered from a peakless observation through a low-
+pass filter L:
+�t = L(y − �π ∗ �s),
+(3)
+This motivates the rewriting of the data ���delity term as:
+(∀s ∈ RN)(∀π ∈ RL) ρ(s, π) = 1
+2∥y − Ly − H(π ∗ s)∥2
+= 1
+2∥H(y − π ∗ s)∥2,
+(4)
+where H = IdN − L is a high-pass filter, and IdN the
+identity operator of RN. We introduce a regularization term Ψ,
+promoting signal sparsity. We add two extra terms to constrain
+estimates �s and �π. The indicator function ιA of the non-empty
+convex set A: zero when the value evaluated belongs to A,
++∞ otherwise. Sets C1 ⊂ RN and C2 ⊂ RL limiting the
+arXiv:2301.01514v1  [eess.SP]  4 Jan 2023
+
+2
+search space for the signal and the kernel are assumed closed,
+non-empty and convex. Optimization problem (2) becomes:
+minimize
+s∈RN, π∈RL
+1
+2||H(y−π∗s)||2+ιC1(s)+ιC2(π)+λΨ(s). (5)
+The estimated trend can be obtained from (3) with �π and �s
+obtained by (5).
+B. SPOQ/SOOT (quasi-)norm ratio penalties
+Being scale-invariant, ratios of norms are promising proxies
+for sparsity characterization [18]. We promote sparse solutions
+s by the Smoothed p-Over-q (SPOQ) family of penalties,
+introduced in [14], a generalization to the Smoothed One-
+Over-Two norm (SOOT) ratio [13], for sparse spectroscopic
+signals. Let p ∈]0, 2[ and q ∈ [2, +∞[. We first define
+two smoothed approximations to the ℓp quasi-norm and ℓq
+norm, parameterized by constants (α, η) ∈]0, +∞[2 . For
+s = (sn)1≤n≤N ∈ RN:
+ℓp,α(s) =
+� N
+�
+n=1
+�
+(s2
+n + α2)p/2 − αp��1/p
+,
+(6)
+and
+ℓq,η(s) =
+�
+ηq +
+N
+�
+n=1
+|sn|q
+�1/q
+.
+(7)
+The non-convex SPOQ penalty is given, for β ∈]0, +∞[, as:
+(∀s ∈ RN)
+Ψ(s) = log
+�
+(ℓp
+p,α(s) + βp)1/p
+ℓq,η(s)
+�
+.
+(8)
+Ψ is Lipschitz differentiable on RN [14, Prop. 2] and admits
+0N as a local minimizer when [14, Prop. 1]:
+q > 2,
+or
+q = 2
+and
+η2αp−2 > βp.
+(9)
+Condition (9) is assumed throughout this paper.
+III. PROPOSED OPTIMIZATION ALGORITHM
+A. Problem structure
+The objective function in (5) is the sum of a differentiable
+function (least squares + SPOQ) and terms acting separably
+on s or π (i.e., indicator terms). In the differentiable part
+(∀s ∈ RN)(∀π ∈ RL)
+f(s, π) = ρ(s, π) + λΨ(s),. (10)
+function ρ from (4) is quadratic in s and π. In particular,
+for every π ∈ RL (resp. ∀s ∈ RN), the gradient ∇ρ1(·, π)
+(resp. ∇ρ2(s, ·)) of ρ with respect to its first (resp. sec-
+ond) variable is Lipschitz continuous with constant Λ1(π)
+(resp. Λ2(s)). As aforementioned, ∇Ψ is Lipschitz continuous
+too. The second part of the objective function reads as:
+(∀s ∈ RN)(∀π ∈ RL)
+g(s, π) = ιC1(s) + ιC2(π).
+(11)
+In a nutshell, Problem (5) amounts to minimizing:
+(∀s ∈ RN)(∀π ∈ RL)
+Ω(s, π) = f(s, π) + g(s, π). (12)
+B. Proposed Trust-Region PENDANTSS algorithm
+The structure of (12) suggests using a block alternating
+approach where signal s and kernel π are updated sequentially.
+We hereby generalize the BC-VMFB algorithm [16], also used
+in [13] for blind deconvolution.
+Algorithm 1: TR-BC-VMFB for solving (5)
+Settings: Kmax > 0, ε > 0, I > 0, θ ∈]0, 1[,
+(γs,k)k∈N ∈ [γ, 2 − γ] and (γπ,k)k∈N ∈ [γ, 2 − γ] for
+some (γ, γ) ∈]0, +∞[2, (p, q) ∈]0, 2[×[2, +∞[
+satisfying (9), convex sets (C1, C2) ⊂ RN × RL.
+Initialize: s0 ∈ C1, π0 ∈ C2
+for k = 0, 1, . . . do
+Update of the signal
+for i = 1, . . . , I do
+Set TR radius ρk,i using (17) with parameter θ;
+Construct MM metric using (15):
+A1,ρk,i(sk, πk) = Λ1(πk)Id + λAq,ρk,i(sk)
+Find sk,i ∈ C1 such that (18) holds.
+if sk,i ∈ ¯Bq,ρk,i then
+Stop loop
+end
+end
+sk+1 = sk,i;
+Update of the kernel
+Find πk+1 ∈ C2 such that (20) holds.
+Stopping criterion
+if ∥sk − sk+1|| ≤ ε or k ≥ Kmax then
+Stop loop
+end
+end
+(�s, �π) = (sk+1, πk+1) and �t given by (3);
+Result: �s, �π, �t
+1) Signal update: Let k ∈ N and (sk, πk) ∈ C1 ×C2. The
+computation of sk+1 follows one Majoration-Minimization
+(MM) iteration [19]. First, we build a majorization for Ω(·, πk)
+around sk. Second, sk+1 is defined as a minimizer to the
+majorant. In practice, both steps can be approximated for
+speedup and robustness to numerical errors. As emphasized
+in [14], [20], we need the majorization to be valid only within
+a neighborhood of the current iterate. For ρ ∈ [0, +∞[, the ℓq-
+ball complement set is:
+¯Bq,ρ = {s = (sn)1≤n≤N ∈ RN|
+N
+�
+n=1
+|sn|q ≥ ρq}.
+(13)
+From [14, Prop. 2], we can show that
+(∀s ∈ ¯Bq,ρ ∩ C1)
+Ω(s, πk) ≤ f(sk, πk)
++ (s − sk)⊤∇1f(sk, πk) + 1
+2∥s − sk∥2
+A1,ρ(sk,πk),
+(14)
+where we define the so-called MM metric as:
+A1,ρ(sk, πk) = (Λ1(πk) + λχq,ρ)IdN+
+λ
+ℓp
+p,α(sk) + βp Diag((s2
+n,k + α2)p/2−1)1≤n≤N,
+(15)
+with the constant
+χq,ρ =
+q − 1
+(ηq + ρq)2/q .
+(16)
+In (14), ∥.∥A denotes the weighted Euclidean norm related
+to a symmetric definite positive (SDP) matrix A ∈ RN×N,
+
+3
+i.e., ∀z ∈ RN, ∥z∥A = (z⊤Az)1/2. Since inequality (14)
+only holds on a limited region, we introduce a Trust-Region-
+based (TR) loop [21] to make sure that the minimizer of the
+majorant is indeed in the validity domain of (14). Namely, we
+set I > 0, a maximum number of trials of TR approach. For
+i ∈ {1, . . . , I}, we define the TR radius as:
+ρk,i =
+�
+�
+�
+�
+�
+�N
+n=1 |sn,k|q
+if i = 1 ,
+θρk,i−1
+if 2 ≤ i ≤ I − 1 ,
+0
+if i = I .
+(17)
+We compute the associated MM metric A1,ρk,i(sk, πk) and
+define sk,i as a minimizer of the right term in (14). The loop
+stops whenever sk,i belongs to ¯Bq,ρk,i, which is ensured to
+arise in a finite number of steps according to [14]. There re-
+mains to explain how we practically compute sk,i. Depending
+on the choice for C1, the right term in (14) might not have a
+closed-form minimizer. Actually, as we will show, it appears
+sufficient for convergence purpose to search for sk,i ∈ C1
+satisfying the first order optimality conditions:
+�
+(sk,i−sk)⊤∇1f(sk, πk)+γ−1
+s,k||sk,i−sk||2
+A1,ρk,i(sk,πk) ≤0,
+||∇1f(sk, πk)+r(1)
+k,i|| ≤ κ1||sk,i−sk||A1,ρk,i(sk,πk)
+(18)
+for some r(1)
+k,i
+∈ NC1(sk,i) (i.e., the normal cone of C1 at
+sk,i [22]), and some κ1 > 0. The existence of such an sk,i
+can be shown from [23, Rem. 3.3]. In particular, a minimizer
+over C1 of the right term in (14) satisfies (18).
+2) Kernel update: It follows a similar approach. The main
+difference is that we do not use the TR loop in that case,
+as the function to minimize here is simpler. Let k ∈ N,
+and (sk+1, πk) ∈ C1 × C2. Using the descent lemma, it is
+straightforward to show that:
+(∀π ∈ C2)
+Ω(sk+1, π) ≤ f(sk+1, πk)
++ (π − πk)⊤∇2f(sk+1, πk) + Λ2(sk+1)
+2
+∥π − πk∥2.
+(19)
+The new iterate πk+1 is then defined as a minimizer of the
+right term of (19). Hereagain, we can solve this problem in
+an inexact manner, that is to search for some πk+1 ∈ C2
+satisfying
+�
+�
+�
+�
+�
+(πk+1 − πk)⊤∇2f(sk+1, πk)
++γ−1
+π,kΛ2(sk+1)∥πk+1 − πk∥2 ≤ 0,
+∥∇2f(sk+1, πk) + r(2)
+k ∥ ≤ κ2
+�
+Λ2(sk+1)∥πk+1 − πk∥,
+(20)
+for some r(2)
+k
+∈ NC2(πk+1) and κ2 > 0. The existence of
+πk+1 can be shown from [23, Rem. 3.3]. In particular, a
+minimizer over C2 of the right term in (19) satisfies (20).
+C. Convergence Result
+We establish the following convergence theorem for Algo-
+rithm 1. Its proof is provided in the supplementary material.
+Theorem 1. Let (sk)k∈N and (πk)k∈N be sequences gener-
+ated by Algorithm 1. If C1 and C2 are semi-algebraic sets then
+the sequence (sk, πk)k∈N converges to a critical point (�s, �π)
+of Problem (5).
+The above result extends [14, Theo.1] to the block alternat-
+ing case using proof ingredients reminiscent from [16], [24].
+IV. NUMERICAL RESULTS
+A. Datasets
+Two datasets A and B were considered. The original sparse
+signal s and the observed signal y are shown in Fig. 1, both
+of size N = 200. The observed signal y is obtained from (1)
+where π is a normalized Gaussian kernel with standard devi-
+ation 0.15 and size L = 21. The noise n is zero-mean white
+Gaussian with variance σ2 either equals 0.5 % or 1.0 % of
+xmax defined as the maximum amplitude of x = π ∗s. Signal
+and kernel convolution is implemented with zero padding.
+B. Algorithmic settings
+We choose C1 = [0, +∞[N and C2 the simplex unit set,
+i.e. C2 =S ={π =(πℓ)1≤ℓ≤L ∈ [0, +∞[L
+s.t.
+�L
+ℓ=1 πℓ =
+1}. For such choices, and giving the fact that the metric (15) is
+diagonal, the resolution of (18) and (20) is straightforward, by
+[22, Prop. 24.11] and [25, Cor. 9]. Namely, for every k ∈ N,
+and i ∈ {1, . . . , I},
+�
+sk,i =ProjC1
+�
+sk−γs,kA1,ρk,i(sk, πk)−1∇1f(sk, πk)
+�
+,
+πk+1 = ProjC2
+�
+πk − γπ,kΛ2(sk+1)−1∇2f(sk+1, πk)
+�
+.
+Hereabove, ProjC1 is the projection over the positive orthant,
+that has a simple closed form expression, while ProjC2 is the
+projection over the simplex unit set, that can be computed
+using the fast procedure from [26].
+For simplicity, we set constant stepsizes γs,k ≡ 1.9 and
+γπ,k ≡ 1.9, thus satisfying the required range assumption.
+Moreover, we take θ = 0.5 in the TR update, and a maximum
+of I = 50 of TR trials. We use the same initialization strategy
+for all methods as in [13], namely s0 ∈ C1 is a constant
+positive valued signal and π0 ∈ C2 is a centered Gaussian
+filter with standard deviation of 1. The stopping criterion
+parameters are set as ε =
+√
+N × 10−6 and Kmax = 2000.
+C. Numerical results
+PENDANTSS jointly performs blind deconvolution and
+trend removal, using SPOQ penalty. Let us recall that SOOT
+penalty from [13] is retrieved by setting (p, q) = (1, 2) in
+SPOQ. Another setting will be analyzed, namely (p, q) =
+(0.75, 2), which was shown to be competitive in the problem
+considered in [14]. In the spirit of an ablation study, we com-
+pare: (i) applying the state-of-the-art background estimation
+method backcor [7] to estimate and remove the trend and then
+the blind deconvolution method [13] to estimate the signal �s
+and the kernel �π, (ii) applying our pipeline when using either
+SPOQ (p, q) = (0.75, 2), SPOQ (p, q) = (1, 2) (i.e., SOOT).
+We use signal-to-noise ratios to evaluate our estimations,
+respectively for the for signal (SNRs), kernel (SNRπ) and
+trend (SNRt). For instance, SNRs = 20 log10(∥s∥2/∥s−�s∥2).
+Moreover, TSNR evaluates the SNR only on the support of
+the original sparse signal. While their support are not known
+in general, it reveals how peak-derived quantities (height,
+
+4
+width, area), important for downstream quantitative chemical
+analysis, would be impacted by detrending and deconvolution.
+Hyperparameters, e.g. regularization parameters of back-
+cor [7] and SPOQ/SOOT parameters (λ, α, β, η), are adjusted
+to maximize a weighted sum of SNRs for one completely
+known reference realization, i.e. 2SNRs + SNRπ + SNRt.
+The cutoff frequency of the low-pass filter in (3) is chosen
+as the best performing point over the first ten peak points of
+the modulus of the signal frequency spectrum. To assure the
+kernel is centered, a spatial shift on the estimated kernel and
+the sparse signal is applied as a post-processing step because
+spatially shifted kernels and sparse signals result in the same
+observed signal. A grid search determines the number of inner
+loops to maximize the SNRs of the sparse signal.
+Table I summarizes the results of mean SNR values, and
+standard deviations after the “±” sign, calculated over two
+hundred noise realizations. The highest among the four com-
+pared methods are followed by two asterisks (**); the second
+best are denoted by only one (*). We notice that the best values
+and the second best values are almost always achieved by
+the proposed PENDANTSS approach with (p, q) = (0.75, 2)
+or (1, 2). The difference with the baseline methods is also
+significant for all cases in terms of TSNRs and SNRt. One
+exception lies on SNRπ with dataset B with the noise level
+of 1.0 % of xmax, where the second best is achieved by
+the combination backcor+SPOQ. We stress out that in such
+problems, correct estimations of sparse signal and baseline
+are usually more important than kernel estimation. The per-
+formance of PENDANTSS for the two penalty parameters
+(p, q) = (0.75, 2), (1, 2) is dependent on the datasets and
+the noise level.
+In terms of sparse signal recovery SNRs
+and TSNRs, PENDANTSS with (p, q) = (0.75, 2) achieves
+slightly higher performance than PENDANTSS with (p, q) =
+(1, 2) for dataset A. However, its outcomes are notably lower
+for dataset B, a less sparse signal, while remaining the second
+best method. For dataset A, both PENDANTSS methods have
+similar baseline estimation accuracy, while for dataset B,
+PENDANTSS (p, q) = (0.75, 2) performs better with lower
+noise level and PENDANTSS (p, q) = (1, 2) better with
+greater noise level, with a difference of SNR of about 2
+dB. As for the estimation of SNRπ, PENDANTSS with
+(p, q) = (1, 2) performs the best for all four cases with
+little difference for dataset A but a larger difference for more
+challenging cases with dataset B and higher noise levels.
+Considering various SPOQ parameters is indeed beneficial.
+According to the presented simulation results, PENDANTSS
+with (p, q) = (0.75, 2) is better for datasets with sparser, well-
+separable peaks whereas PENDANTSS with (p, q) = (1, 2) for
+more challenging datasets. Graphical details on the quality of
+estimated peaks are provided as supplementary material.
+V. CONCLUSION AND PERSPECTIVES
+We propose to solve a complicated joint sparse signal blind
+deconvolution and additive trend problem. Our method handles
+smooth trend removal by exploiting their low-pass property
+and simplifies the problem into a blind deconvolution prob-
+lem. The blind deconvolution problem uses the recent SPOQ
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+0
+1
+2
+3
+4
+5
+6
+7
+(a) Dataset A.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+0
+5
+10
+15
+20
+25
+(b) Sparse spike signal for dataset A.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+0
+2
+4
+6
+8
+10
+12
+(c) Dataset B.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+0
+5
+10
+15
+20
+25
+30
+(d) Sparse spike signal for dataset B.
+Fig. 1. Unknown sparse signal s (b) and (d); in (a) and (c) observation y
+(blue) and baseline t (black) (bottom) for datasets A and B. Signal A has 10
+spikes (5.0 % of sparsity) while signal B has 20 spikes (10.0 % of sparsity).
+TABLE I
+NUMERICAL RESULTS ON DATASETS A AND B. SNR QUANTITIES IN DB.
+BEST PERFORMING METHOD FOLLOWED BY **, SECOND BY *.
+Dataset A
+Dataset B
+Noise level σ (% of xmax)
+0.5 %
+1.0 %
+0.5 %
+1.0 %
+SNRs
+backcor+SOOT
+29.2±0.7
+28.5±1.9
+14.9±4.0
+11.5±4.7
+backcor+SPOQ
+29.2±0.7
+29.3±1.3
+12.9±3.5
+11.3±4.4
+PENDANTS (1, 2)
+32.9±1.5*
+30.9±2.2*
+22.3±8.2**
+17.5±8.4**
+PENDANTS (0.75, 2)
+33.2±2.3**
+31.0±4.2**
+15.9±4.5*
+12.9±4.6*
+TSNRs
+backcor+SOOT
+29.2±0.7
+29.3±1.3
+16.6±3.5
+13.4±4.3
+backcor+SPOQ
+29.2±0.7
+29.3±1.3
+15.1±3.0
+13.7±3.7
+PENDANTS (1, 2)
+34.1±1.4*
+32.2±2.1*
+24.9±8.0**
+19.2±7.7**
+PENDANTS (0.75, 2)
+35.4±1.7**
+32.6±3.8**
+17.7±4.0*
+14.5±4.1*
+SNRt
+backcor+SOOT
+20.5±0.2
+20.3±0.4
+15.5±0.5
+14.8±0.8
+backcor+SPOQ
+20.5±0.2
+20.3±0.4
+15.5±0.5
+14.8±0.8
+PENDANTS (1, 2)
+26.9±0.5**
+26.0±0.8**
+22.0±0.4*
+21.6±1.0**
+PENDANTS (0.75, 2)
+26.9±0.6**
+26.0±1.0**
+24.6±0.6**
+19.6±3.9*
+SNRπ
+backcor+SOOT
+36.3±1.3
+33.9±1.7
+30.3±1.3
+28.5±1.8
+backcor+SPOQ
+36.3±1.3
+34.0±1.7
+33.1±1.9
+31.2±2.1*
+PENDANTS (1, 2)
+41.3±2.0**
+34.4±2.4**
+38.3±1.9**
+33.6±2.2**
+PENDANTS (0.75, 2)
+41.3±2.0**
+34.2±2.5*
+35.7±1.5*
+25.4±5.5
+sparse penalty. Simulation results confirm that PENDANTSS
+outperforms comparable methods on typical sparse analytical
+signals. Further works include its validation on a variety of
+other sparse spike signals. The appropriate parameters for the
+sparsity-promoting norm ratio penalty ought to be investigated,
+for instance with respect to the alleged signal sparsity or
+peak separability. PENDANTSS Matlab code is available at
+https://github.com/paulzhengfr/PENDANTSS.
+
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+1
+PENDANTSS: Supplementary Material
+I. PROOF OF THEOREM 1 FOR ALGORITHM 1
+We first provide a useful majorant metric matrix property.
+Lemma 1 There exists (λ, λ) ∈]0, +∞[2 such that for every
+k ∈ N, and for every i ∈ {1, . . . , I},
+�
+λIdN ⪯ A1,ρk,i(sk, πk) ⪯ λIdN,
+λ ≤ Λ2(sk) ≤ λ.
+(A1)
+Proof. Direct consequence of [14, Prop. 2] and [13, Prop. 1].
+We then show that Algorithm 1 satisfies two essential descent
+properties, that are key for the convergence analysis.
+Lemma 2 There exists (µ1, µ2) ∈]0, +∞[ such that, for every
+k ∈ N, the following descent properties hold:
+Ω(sk+1, πk) ≤ Ω(sk, πk) − µ1
+2 ||sk+1 − sk||2,
+(A2)
+Ω(sk+1, πk+1)≤Ω(sk+1, πk) − µ2
+2 ||πk+1 − πk||2.
+(A3)
+Proof. Let k ∈ N. We remind that the objective function Ω
+is defined in (12), with g specified in (11). By construction,
+sk+1 ∈ ¯Bq,ρ ∩ C1 for some i ∈ {1, . . . , I}. Summing the
+majoration (14) and the first inequality in (18) yields:
+Ω(sk+1, πk) ≤ f(sk, πk)−(γ−1
+s,k − 1
+2)∥sk −sk+1∥2
+A1,ρ(sk,πk).
+We notice that f(sk, πk) = Ω(sk, πk) since sk
+∈ C1
+and πk ∈ C2. Using Lemma 1 and the range assumption on
+γs,k allows to show (A2) for µ1 = λγ/(2 − γ). Again by
+construction, πk+1 ∈ C2. Summing inequalities (19) and (20)
+leads to:
+Ω(sk+1, πk+1) ≤ f(sk+1, πk)−
+(γ−1
+π,k − 1
+2)Λ2(sk+1)∥πk+1 − πk∥2.
+Here again, we use f(sk+1, πk) = Ω(sk+1, πk) as sk+1 ∈
+C1 and πk ∈ C2. The descent property (A3) is obtained by
+using Lemma 1, the range constraint on γπ,k, and setting µ2 =
+λ¯γ(2 − ¯γ).
+The rest of the proof of Theorem 1 is obtained by following
+the same lines than the one of [16, Theorem 3.1].
+II. ADDITIONAL RESULTS
+Figures 2 and 3 provide additional insights into PEN-
+DANTSS restoration. Dataset A in Figure 2-(a) presents
+sparse and well-isolated peaks. Accurate peak restoration is
+secured. Peak shapes are well recovered (left-hand zoom),
+and the estimated trend matches well the actual baseline. As
+a consequence, peak features that are computed with respect
+to the trend (height, area) are likely to be well-estimated with
+PENDANTSS. The less sparse Dataset B in Figure 2-(b) shows
+that the separation and the height of close peaks are accurately
+matched. Some overshoot in trend estimation can be noticed. It
+is however not likely to drastically affect relative peak height
+or area computations.
+Retrieved spikes are exposed in Figure 3. For Dataset A,
+well-separated spikes are accurately recovered using PEN-
+DANTSS. Estimated amplitudes and locations are almost
+indistinguishable from the original ones. This is exemplified
+for the less sparse Dataset B in Figure 3-(b). Isolated peaks
+are well-estimated. However, some spikes (for instance around
+index 175) for Dataset B in Figure 3-(b) remain unelucidated.
+Three contiguous spikes are estimated, instead of two. Such an
+ambiguous solution is typical to source separation problems.
+(a) Dataset A reconstruction and trend.
+(b) Dataset B reconstruction and trend.
+Fig. 2: Ground truth (thick black line) and proposed estimation results (thin
+blue line), and the baseline t (dashed dot) and the signal s ∗ p (continuous).
+(a) Dataset A sparse spike signal.
+(b) Dataset B sparse spike signal.
+Fig. 3: Ground truth (black line with circle marker) and proposed estimation
+results (blue line with cross marker).
+arXiv:2301.01514v1  [eess.SP]  4 Jan 2023
+

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@@ -0,0 +1,2053 @@
+Draft version January 11, 2023
+Preprint typeset using LATEX style emulateapj v. 12/16/11
+DETECTING ISOLATED STELLAR-MASS BLACK HOLES BY THE Roman TELESCOPE
+Sedighe Sajadian1
+Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
+Kailash C. Sahu2
+Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+and
+Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
+Draft version January 11, 2023
+Abstract
+Isolated Stellar-Mass BlackHoles (ISMBHs) are potentially discernible through microlensing obser-
+vations because they are expected to be long-duration microlensing events. In this work, we study
+detecting and characterizing ISMBHs with the Roman observations. We simulate a big ensemble of
+these events as seen by Roman and estimate the errors in the physical parameters of the lens objects,
+including their masses, distances, and proper motions through calculating Fisher and Covariance
+matrices. Since the ∼2.3-year time gap between Roman’s first three observing seasons and the others
+may lower the efficiency of realizing microlensing events and characterizing ISMBHs, we additionally
+consider a scenario where we add a small amount of additional observations –one hour of observations
+every 10 days when the Bulge is observable during the large time gap– which is equivalent to a total
+of about one additional day of observations with the Roman telescope.
+These extra observations
+increase Roman’s efficiency for characterizing ISMBHs by ∼ 1-2% and, more importantly, improve
+the robustness of the results by avoiding possible degenerate solutions. By considering uniform, and
+power-law mass functions (dN/dM ∝ M −α, α = 2, 1, 0.5) for ISMBHs in the range of [2, 50]M⊙, we
+conclude that the Roman telescope will determine the physical parameters of the lenses within < 5%
+uncertainty, with efficiencies of 21%, and 16-18%, respectively. By considering these mass functions,
+we expect that the Roman telescope during its mission will detect and characterize 3-4, 15-17 and
+22-24 ISMBHs through astrometric microlensing, with the relative errors for all physical parameters
+less than 1, 5, 10%, respectively. Microlensing events owing to ISMBHs with a mass ≃ 10-25M⊙
+and located close to the observer with Dl ≲ 0.5Ds while the source is inside the Galactic disk can be
+characterized with least errors.
+Subject headings: (cosmology:) gravitational lensing; astrometry; techniques: photometric; methods:
+numerical
+1. INTRODUCTION
+A black hole (BH) refers to a massive object where
+the escape velocity from it exceeds the speed of light.
+Therefore, a BH can not reflect any light. However, it
+radiates what is called the Hawking radiation (Hawking
+1974), which is generally faint (Malyshev et al. 2022;
+Auffinger 2022).
+Their formation mechanisms are as follows: (a) BHs
+can be formed by the death of massive stars with an ini-
+tial mass higher than 20M⊙ (Bailyn et al. 1998; Fryer
+& Kalogera 2001; Bambi 2018). (b) The interstellar gas
+at the centre of massive galaxies can directly collapse to
+form massive BHs (Volonteri 2010; Haiman 2013; Wise
+et al. 2019). (c) Initial spatial fluctuations in the early
+universe (during the first second after the Big Bang)
+could potentially lead to the formation of primordial BHs
+as proposed by S. Hawking (Hawking 1971).
+BHs are generally classified based on their mass
+into three categories:
+(i) Super-massive BHs,
+(ii)
+Intermediate-Mass BHs (IMBHs), and (iii) Stellar-Mass
+BHs.
+1 Email: [email protected]
+2 Email: [email protected]
+The first class—the super-massive BHs—have masses
+M ≥ 105M⊙. These objects can be found at the centers
+of massive galaxies (such as the Milky Way Galaxy, and
+M87), bright quasars, and Active Galactic Nuclei (AGN).
+These massive objects can be detected and characterized
+by tracking stars near massive galaxies’ centre (Volonteri
+et al. 2021).
+The second class—the IMBHs—have masses in the
+range of 100-105 M⊙ and are thought to reside at cen-
+tres of globular clusters (Koliopanos 2017; Greene et al.
+2020). One method to indirectly detect these objects is
+through gravitational waves caused by the merging of
+these massive objects (Abbott et al. 2016, 2017).
+At-
+tempts have also been made to detect IMBHs through
+astrometric microlensing of background stars caused by
+the IMBHs (Kains et al. 2016, 2018).
+The third class—the stellar-mass BHs—form after the
+gravitational collapse of massive stars.
+These objects
+have masses as high as a few tens of solar mass. The num-
+ber of such BHs in our galaxy has been predicted to be
+more than 10 million (Shapiro & Teukolsky 1983; Lam-
+berts et al. 2018).
+The lowest-mass confirmed stellar-
+mass BHs have a mass in the range of 3-4.5 M⊙ (Thomp-
+son et al. 2019; Jayasinghe et al. 2021), whereas the most
+arXiv:2301.03812v1  [astro-ph.GA]  10 Jan 2023
+
+2
+Sajadian and Sahu
+massive neutron stars (NSs) confirmed up to now have
+masses of ≲ 2M⊙ (Fonseca et al. 2021), so there is a mass
+gap between confirmed NSs and stellar-mass BHs (see,
+e.g., Gao et al. 2022).
+Stellar-mass BHs in binary systems can be detected
+either through transient X-rays emitted by the accretion
+of matter (from companions or close objects) onto the
+BHs’ surface, or through observations of Doppler shifts
+in the spectra of stellar companions orbiting the BHs,
+or through both of them (Webster & Murdin 1972).
+In these systems, the Doppler shifts provide radial
+velocity measurements which are used to determine the
+dynamic masses of BHs.
+Up to now, more than 65
+stellar-mass BHs have been discovered in binary systems
+and through X-ray transient observations, mostly in
+our galaxy 3 (Corral-Santana et al. 2016). This method
+is restricted only to cases where the stellar-mass BHs
+are in binary systems with luminous companion objects,
+thus ISMBHs cannot be detected by this method.
+A unique and powerful method for discovering ISMBHs
+is gravitational microlensing, which refers to a temporary
+enhancement in the brightness of a background star while
+passing behind a massive foreground object (the so-called
+gravitational lens) (Einstein 1936; Liebes 1964; Refsdal
+1964). In this phenomenon, the lens could be completely
+dark. Hence, microlensing observations can reveal the
+existence of dark (or faint) and massive compact objects,
+e.g., stellar-mass BHs, even ones located outside of our
+galaxy (Paczynski 1986; Sajadian & Rahvar 2012; Sahu
+et al. 2017).
+The important observing issue is that the photometric
+light curve alone is not sufficient to calculate the physi-
+cal parameters of the lens, such as its mass, distance and
+proper motion. However, by additionally measuring the
+parallax effect and astrometric shift in the source star
+position which is proportional to the angular Einstein
+radius, θE, a length-scale in the lensing formalism (see,
+e.g., Walker 1995; Hog et al. 1995; Miyamoto & Yoshii
+1995; Dominik & Sahu 2000)), the lensing degeneracy can
+be resolved. Instead of measuring the astrometric mo-
+tion of the source star, the interferometry observations
+by even ground-based telescopes can resolve the lensing
+images. This leads to a direct measurement of θE, which
+also resolves the lensing degeneracy (Dong et al. 2019;
+Zang et al. 2020). Measuring finite source effects in tran-
+sit, caustic-crossing and high-magnification microlensing
+events is another method to estimate θE and resolve the
+lensing degeneracy (An et al. 2002).
+The first unambiguous detection of an ISMBH in the
+Galactic disk has been reported recently based on the
+combined observations by the Hubble Space Telescope
+(HST) and ground-based telescopes in the microlensing
+event OGLE-2011-BLG-0462 (Sahu et al. 2022). There
+were some claims that this long-duration microlensing
+event could also be due to lower-mass objects (Lam
+et al. 2022), but recently Mroz et al. (2022) have shown
+that the lower mass estimates come from systematic er-
+rors and the lens mass should be ≃ 7M⊙. There were
+other reports of microlensing events due to ISMBHs, but
+their lensing parameters or the nature of the lens objects
+were not determined uniquely (Mao et al. 2002; Bennett
+3 https://www.astro.puc.cl/BlackCAT/
+et al. 2002; Agol et al. 2002; Poindexter et al. 2005; Lu
+et al. 2016).The Optical Gravitational Lensing Experi-
+ment group (OGLE) (Udalski et al. 2015; Udalski 2003)
+has also found 13 long-duration microlensing events from
+observations in the years 2001-2009 which were due to
+white dwarfs, neutron stars, or black holes (Wyrzykowski
+et al. 2016).
+In this work, we aim to study the possible detection
+and characterization of ISMBHs by the Roman mission.
+The Nancy Grace Roman Telescope will observe the
+Galactic-bulge field during six 62-day seasons in its
+5-year mission (Penny et al. 2019).
+Even though its
+observing strategy is aimed at detecting free-floating
+planets and exoplanets beyond the snow line, we expect
+that the Roman telescope will also detect microlensing
+events due to other lens objects (Sajadian 2021a,b).
+Additionally,
+because of high photometric accuracy
+during microlensing observations, it can resolve some
+second-order perturbations (Bagheri et al. 2019; Sa-
+jadian & Salehi 2020).
+Roman is also expected to
+detect ISMBHs through observations of long-duration
+microlensing events.
+The relatively long lifespan of
+the Roman mission is very appropriate for detecting
+long-duration microlensing events and measuring both
+annual parallax effects and astrometric trajectories of
+source stars.
+The scheme of the paper is as follows. In Section 2,
+we explain all the details for simulating astrometric mi-
+crolensing events as seen by the Roman telescope. In Sec-
+tion 3, we first explain how to calculate Fisher and Co-
+variance matrices for photometry and astrometry mea-
+surements by Roman from microlensing events due to
+ISMBHs. Then, we illustrate the results of our simula-
+tions and statistical calculations. Finally, in Section 4,
+we briefly review our results and conclusions.
+2. FORMALISM
+Here we review the known formalism for astrometric
+microlensing. We start with ignoring the parallax effect
+but add this at a later stage. The temporary enhance-
+ment in the stellar brightness due to the gravitational
+lensing of a point-like and massive object which is called
+the magnification factor versus time, t, is given by (see,
+e.g., Gaudi 2012; Tsapras 2018):
+A(t) =
+u2 + 2
+u
+√
+u2 + 4
+,
+u =
+�
+u2
+0 +
+�t − t0
+tE
+�2,
+(1)
+where, u is the lens-source distance projected on the sky
+plane and normalized to the Einstein radius (i.e., RE the
+radius of the image ring at the complete alignment), u0
+is the lens impact parameter (the smallest lens-source
+distance), and t0 is the time of the closest approach.
+The Einstein crossing time, tE, represents the lensing
+timescale which is given by:
+tE =
+θE
+µrel,⊙
+=
+1
+µrel,⊙
+�
+Ml πrel κ,
+(2)
+Here, Ml is the lens mass, κ = 8.14 mas.M−1
+⊙
+is a con-
+stant, and πrel = au
+�
+1/Dl −1/Ds
+�
+is the relative parallax
+amplitude, and Dl, Ds are the lens and source distances
+
+Detecting stellar-mass black holes by Roman
+3
+Fig. 1.— Two examples of simulated magnification curves. The left panels show the magnification curves with (dashed curves) and
+without (dotted curves) the parallax effect. The right panels show the corresponding astrometric motions of the source stars (blue curves),
+lens objects (magenta curves), and their relative motions (dark red curves) projected on the sky plane. The synthetic data are taken with
+the Roman telescope. The observable parameters used to make them are mentioned at the top of their lightcurves and astrometric plots.
+from the observer. We note that θE = RE
+�
+Dl is an an-
+gular length-scale in the lensing formalism.
+µrel,⊙ is the size of the relative lens-source angular veloc-
+ity. If we ignore the observer’s motion around the Sun,
+the relative velocity vector (with respect to the Sun) is
+given by:
+µrel,⊙ = µs − µl = vs − v⊙
+Ds
+− vl − v⊙
+Dl
+,
+(3)
+where, vs, vl, and v⊙ are the source, lens and the Sun
+velocity vectors projected on the sky plane. In Appendix
+A, we explain how to convert the stellar velocities from
+the Galactic coordinate frame to the observer frame.
+Parallax effect: We know that the observer (here,
+the Roman telescope) rotates around the Sun, so the real
+relative lens-source angular velocity will be a function of
+time and is given by:
+µrel(t) = µrel,⊙ + πrel
+au vo(t),
+(4)
+vo being the velocity vector of the observer with respect
+to the Sun projected on the sky plane as explained in
+Appendix A 4. Hence, the observer’s rotation around the
+Sun, which is a function of time, causes the relative lens-
+source angular velocity to be a function of time, and as
+a result, it makes a periodic perturbation in the magnifi-
+cation curve, the so-called parallax effect (Gould 1994).
+By considering this effect in the lensing formalism, the
+normalized source-lens angular displacement (which de-
+termines the magnification factor) versus time is given
+by:
+u = u0
+�
+− sin ξ
+cos ξ
+�
++ t − t0
+tE
+�
+cos ξ
+sin ξ
+�
++ πE
+au
+� t
+t0
+dt
+�
+vo,n1
+vo,n2
+�
+(5)
+where, πE = πrel/θE which is a dimensionless parameter,
+and ξ is the angle between the relative source-lens
+trajectory and the direction of increasing Galactic
+longitude, i.e. n1 (as defined in Appendix A) which is
+given by tan ξ = µrel,⊙,n2/µrel,⊙,n1.
+4 For projection of the observer orbit on the sky plane, first
+we should project the observer orbit on the Galactic plane by a
+rotation 60◦ around the intersection line of the orbital plane and
+the Galactic plane.
+
+te(days) =134.7, Q(mas) =2.77, TE =0.007
+Magnification
+19.75
+Magnification + parallax
+19.80
+119.95
+149
+W1
+20.00
+20.05
+20.10
+0
+2
+3
+1
+4
+5
+time(yrs)Uo =0.75, mbase(mag) =20.11, to(years) =2.6
+-6
+4
+position(mas)
+2
+0
+2
+source(undeflected) + parallax
+ source(deflected) + parallax
+4
+-lens - source(undeflected) + parallax
+ Lens ＋ parallax
+ Deflection
+6
+-20
+-10
+0
+10
+20
+30
+x position(mas)te(days) =113.9, Qe(mas) =2.64, TE =0.013
+Magnification
+17.8
+Magnification + parallax
+18.0
+18.2
+18.4
+18.6
+18.8
+19.0
+19.2
+0
+1
+2
+3
+4
+5
+time(yrs)Uo =0.16, mbase(mag) =19.25, to(years) =1.0
+source(undeflected) + parallax
+source(deflected) + parallax
+-20
+ lens - source(undeflected) + parallax
+ Lens + parallax
+-15
+-Deflection
+position(mas)
+10
+-5
+y
+0
+5
+10
+-40
+-30
+-20
+-10
+0
+10
+20
+x position(mas)4
+Sajadian and Sahu
+Fig. 2.— Same as Figure 1, but by considering extra observations, one-hour observations of the Galactic bulge every 10 days when the
+Bulge is observable during the ∼2.3-year time gap, with the Roman telescope. These extra data points are depicted in green color.
+
+Uo =0.64, mbase(mag) =15.9, to(years) =2.8
+source(undeflected) + parallax
+source(deflected) + parallax
+-10
+ lens - source(undeflected) + parallax
+Lens + parallax
+ Deflection
+5
+position(mas)
+0
+y
+5
+10
+-7.5
+-5.0
+-2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+x position(mas)te(days) =249.4, Q(mas) =5.19, TE =0.013
+17.4
+Magnification
+Magnification + parallax
+17.6
+ magnitude
+17.8
+18.0
+18.2
+149
+M
+18.4
+18.6
+18.8
+0
+1
+2
+3
+4
+5
+time(yrs)Uo =0.17, mbase(mag) =18.85, to(years) =3.7
+-30
+source(undeflected) + parallax
+source(deflected) + parallax
+lens - source(undeflected) + parallax
+20
+Lens + parallax
+Deflection
+position(mas)
+10
+0
+y
+10
+20
+0
+5
+10
+15
+x position(mas)te(days) =125.2, Qe(mas) =3.03, Te =0.024
+Magnification
+19.5
+Magnification + parallax
+ magnitude
+19.6
+19.7
+W149
+19.8
+19.9
+0
+2
+3
+4
+5
+time(yrs)Uo =0.41, mbase(mag) =19.87, to(years) =2.1
+source(undeflected) + parallax
+6
+source(deflected) + parallax
+ lens - source(undeflected) + parallax
+.4
+: Lens + parallax
+Deflection
+position(mas)
+2
+0
+2
+4
+6
+-20
+-10
+0
+10
+x position(mas)te(days) =275.5, Qe(mas) =1.99, TE =0.009
+15.3
+Magnification
+Magnification + parallax
+15.4
+15.5
+15.6
+15.7
+15.8
+15.9
+0
+1
+2
+3
+4
+5
+time(yrs)Detecting stellar-mass black holes by Roman
+5
+According to the literature, we could define πE as a vec-
+tor which is parallel with the relative lens-source proper
+motion, i.e.,
+πE =
+�
+πn1, πn2
+�
+= πE
+�
+cos ξ, sin ξ
+�
+.
+(6)
+The initial parameters that can be derived from the
+simple form of microlensing lightcurves (Eq. 1) are t0, u0,
+and tE . In observations toward the Galactic bulge, most
+of the source stars are located in the Galactic bulge, at a
+distance Ds = 8 kpc from us. Measuring tE gives us only
+a relation between the lens mass, the lens distance, and
+the relative lens-source angular velocity, even by fixing
+the source distance.
+However, discerning the parallax
+effect in the lightcurve allows us to measure the vector
+of the parallax amplitude, πE, which is still not enough
+to resolve the lensing degeneracy completely.
+Astrometric microlensing: One way to resolve this
+degeneracy and determine these parameters specially for
+long-duration microlensing events due to ISMBHs is re-
+solving the source angular trajectory projected on the
+sky plane:
+θs(t) = θs,0(t) +
+u
+u2 + 2θE,
+(7)
+where, the last term is the astrometric shift in the ap-
+parent brightness center of the source star which is an-
+other result of the lensing effect. In the lensing formal-
+ism where a background star is lensed by a point-like
+and massive lens object, two distorted images are formed
+whose brightness center does not coincide with the source
+center.
+We note that this astrometric shift is propor-
+tional to the Einstein angular radius which is a function
+of the lens mass and its distance (see, e.g., Miyamoto &
+Yoshii 1995; Dominik & Sahu 2000).
+In Equation 7, θs,0(t), is the position vector of the
+source star projected on the sky plane as a function of
+time as seen by the observer, which is:
+θs,0(t) = θs,0(t0) + µs(t − t0) − 1
+Ds
+� t
+t0
+vo(t)dt,
+(8)
+where, the first term, θs,0(t0) = u0 θE
+�
+− sin ξ, cos ξ
+�
+,
+is the source position on the sky plane at the time of
+the closest approach with respect to the lens position
+(i.e., the coordinate center). The second term specifies a
+straight line over the sky plane. The last term, which is
+related to the effect of the observer’s motion around the
+Sun on the source position, is mostly very small because
+of the large source distance from the observer. This can
+be clearly seen by comparing the blue dotted lines (which
+do not take the parallax effect into account) and the blue
+dashed lines (which take the parallax effect into account)
+in the right panels of Figures 1 and 2. This term makes a
+periodic perturbation on the source trajectory projected
+on the sky plane.
+The lens also has a similar angular trajectory projected
+on the sky plane, given by
+θl(t) = µl(t − t0) − 1
+Dl
+� t
+t0
+vo(t)dt.
+(9)
+Here, we have set the lens location at the coordinate
+center at the time of the closest approach. However, in
+most of the gravitational microlensing events the lens
+objects are dark and their angular trajectories cannot be
+determined. We note that
+u(t) = θs(t) − θl(t)
+θE
+Let’s come back to Equation 7, which describes the
+source angular trajectory projected on the sky plane ver-
+sus time. In the case of astrometric observations where
+we discern this source trajectory, the observables that
+we can measure are: (a) θE, which is the angular size
+of the Einstein ring radius, (b) µs, the angular source
+velocity projected on the sky plane with respect to the
+observer, and (c) the sign of the lens impact parameter
+(e.g., Sajadian & Rahvar 2015).
+However, for discerning the second one, observations
+are necessary either long after or long before the lensing
+event. Additionally, the astrometric shift due to lensing
+effect has longer timescale than tE. It tends to zero as
+u−1, while the magnification factor is proportional to
+∝ u−4 for u ≫ 1 (see, e.g.,
+Dominik & Sahu 2000).
+Its long timescale helps to resolve the time dependent
+perturbations, such as the orbital-motion effect in binary
+lensing (Sajadian 2014).
+By measuring both astrometric shift due to microlens-
+ing and the parallax effect in the magnification curve,
+we determine tE, θE, πE, ξ, and µs, which allows us to
+completely resolve the lensing degeneracy and determine
+Dl, Ml, µrel,⊙, and µl.
+We note that u0, and t0 are
+measurable from magnification curve and are necessary
+while modeling the astrometric motion of the source
+star, but they are not directly involved in extracting the
+physical parameters.
+One class of microlensing events that are specially
+interesting are the long-duration events caused by
+ISMBHs. In these events, the astrometric shift in the
+source angular position is considerable, because of the
+large angular Einstein radius.
+Additionally the paral-
+lax effect potentially could be measured, because of long
+duration of such events. We note that in most of the mi-
+crolensing events due to ISMBHs, the finite source effect
+is negligible, unless the lens passes over the source sur-
+face. This is is rare since the impact parameter has to be
+less than the normalized angular source radius, u0 < ρs,
+ρs = θs/θE, where θs is the angular source radius, and
+the large value of θE decreases ρs.
+Using the introduced formalism, we simulate the astro-
+metric microlensing events due to ISMBHs toward the
+Galactic bulge. We also make the synthetic data points
+according to the Roman observing strategy. In this re-
+gard, the observing cadence is fixed at 15.16 min. The
+observations include six 62-day seasons, three of them at
+the first part of the Roman mission with a time interval
+110-day between seasons, and three other seasons at the
+end.
+The photometric observations are mostly done in
+the W149 filter.
+This filter roughly corresponds to
+W149 = (K + H + J)
+�
+3 (Montet et al. 2017).
+Its
+photometric precision, σm, is a function of the apparent
+magnitude (Penny et al. 2019; Johnson et al. 2020). The
+astrometric precision of the Roman observations also
+
+6
+Sajadian and Sahu
+Fig. 3.— The normalized (fractional) distributions of tE, mbase, t0, and u0 for all the detected microlensing events by Roman are depicted
+in green. Also, the normalized distributions of the events for which the physical parameters of the lenses are measurable with ≤ 5% relative
+errors (after considering the extra observations during ∼2.3-year time gap) are shown as black stepped curves. The average values of these
+parameters calculated from related distributions are mentioned in the legends.
+strongly depends on the apparent stellar brightness. S.
+Calchi Novati (private communication) has modelled
+the Roman astrometric precisions for stars of different
+magnitudes through Jitter
+simulations and in this
+work we use his simulations to determine the Roman
+astrometric precision. He has used the Roman observing
+strategy described by Penny et al. (2019), and calculated
+the astrometry precisions through simulations (see, e.g.,
+Monet et al. 2010).
+Two examples of simulated astrometric microlensing
+events are shown in Figure 1.
+The left panels show
+the magnification curves with (dashed curves) and with-
+out (dotted curves) the parallax effect and their cor-
+responding right panels show the related astrometric
+motions of the source stars (blue curves), lens objects
+(magenta curves), and their relative motions (dark red
+curves). The observable parameters which characterize
+these events are specified at the top of the light curve
+and astrometric motion plots.
+There is a large time gap of ∼2.3 years between the first
+three and the last three observing seasons of Roman5,
+5
+https://roman.gsfc.nasa.gov/galactic_bulge_time_
+which lowers the detection efficiency of ISMBHs. If the
+peak of the light curve happens during this large time
+gap (which lasts ∼ 2.3 years), discerning such events will
+have large uncertainties, and several degenerate models
+will fit the data well. For instance, the peak of the first
+lightcurve in the top panel of Figure 1 was not covered by
+Roman data which would have been useful in correctly
+determining the microlensing parameters, including the
+parallax.
+Hence, for a robust determination of the microlensing pa-
+rameters, we additionally consider a case where the Ro-
+man telescope observes the seven Galactic-bulge fields
+for a total of one hour every 10 days when the Galac-
+tic bulge is observable during the ∼2.3-year time gap.
+Although these observations are sparse and use a total
+of ∼1-day of Roman time, they are very helpful in dis-
+cerning the source trajectories during the Roman mission
+(see the first astrometry microlensing event in Figure 1),
+and fully characterizing the microlensing lightcurves with
+high confidence. In Figure 2, we show three more simu-
+lated astrometric microlensing events due to ISMBHs as
+detected by Roman, by assuming additional sparse obser-
+domain_survey.html
+
+0.20
+303.01
+0.17
+556.93
+0.11
+0.08
+0.06
+0.03
+0.00
+1.5
+2.0
+3.0
+3.5
+log1o[te(days)]0.11
+20.06
+0.10
+19.31
+0.08
+Distribution
+0.07
+0.06
+0.04
+0.03
+0.01
+0.00
+16
+17
+18
+19
+20
+21
+22
+23
+24
+mbase(mag)0.05
+2.47
+0.04
+2.45
+0.03
+Distribution
+0.03
+0.02
+Normalized [
+0.02
+0.01
+0.01
+0.00
+0
+3
+2
+4
+5
+to(years)0.05
+0.49
+0.04
+0.36
+0.03
+Normalized Distribution
+0.03
+0.02
+0.02
+0.01
+0.01
+0.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+UoDetecting stellar-mass black holes by Roman
+7
+vations as discussed above. In these plots the extra data
+points are depicted in green. We note that the astrom-
+etry data points during the time gap (green points) can
+jump to the observing seasons (shown by the red points)
+because of the added noise in the simulated data.
+In the next section, we evaluate the expected errors
+in the physical parameters of ISMBHs detected through
+astrometric microlensing by the Roman telescope.
+3. OBSERVATIONS OF ASTROMETRIC MICROLENSING
+To study detection and characterization of the ISMBHs
+by microlensing observations during the Roman mission,
+we extend our simulation and make a big ensemble of
+detectable astrometric microlensing events.
+Since the mass function for ISMBHs are not well de-
+termined, so we consider several different mass functions.
+A simple form for ISMBHs’ mass function is a uniform
+function versus mass in the range of Ml ∈ [2, 50]M⊙.
+Through modeling of black holes, Sicilia et al. (2022)
+have found that the mass function of ISMBHs is almost
+flat up to 50M⊙. Additionally, we examine three more
+mass functions, which are log-uniform (dN/dM ∝ 1/M)
+and power-law (dN/dM ∝ M −0.5, and dN/dM ∝ M −2)
+ones.
+Other parameters are determined according to their
+distribution functions, as explained in the previous pa-
+pers (see, e.g.,
+Sajadian & Poleski 2019; Moniez et al.
+2017). For each mass function, we perform the simula-
+tions two times, i.e., with and without considering sparse
+observations during the ∼2.3-year time gap.
+We choose the discernible events. Our criteria for de-
+tectability are (i) ∆χ2(=
+��χ2
+base − χ2
+real
+��) > 800 for pho-
+tometry data points, and (ii) at least three photome-
+try data points above the baseline by 4σm, where σm
+is the photometric accuracy. In Figure 3, we show the
+normalized (fractional) distributions for four observing
+parameters including tE, mbase, t0, u0 of detectable mi-
+crolensing events due to ISMBHs (by considering a uni-
+form mass function and sparse observations during the
+large time gap) in green color. In order to study for which
+kind of these microlensing events the physical parame-
+ters of their lens objects are measurable with reasonable
+accuracy, we also plot the corresponding normalized dis-
+tributions of events with the relative errors in the lens
+mass, distance, and proper motion ≤ 5% (black stepped
+curves).
+Accordingly, detectable microlensing events due to
+ISMBHs have the average timescale of ⟨tE⟩ = 303 days
+and their average source magnitude at the baseline is
+⟨mbase⟩ = 20.1 mag. Discerning these microlensing light
+curves (by adding extra observations during the large
+time gap) does not highly depend on the time of the
+closest approach and the lens impact parameter.
+The
+events with measurable physical parameters of their lens
+objects have on average smaller lens impact parameters
+(by 0.13), and mostly happen during either three first or
+three last observing seasons of the Roman telescope.
+For each discernible event, we determine the errors in
+the physical parameters of microlenses through calculat-
+ing Fisher and Covariance matrices (see, e.g., Boutreux
+& Gould 1996; Gould & Salim 1999; Sajadian 2015). In
+this regard, we make several simple assumptions which
+are listed here.
+(i) We separate the photometry and
+astrometry measurements completely and calculate two
+Fisher matrices corresponding to these measurements,
+A, and B for each event. (ii) We assume that the lens-
+ing parameters such as t0, u0, tE, and ξ are determined
+through photometry observations well and their real val-
+ues are used for astrometric modeling. In fact, the photo-
+metric accuracy is better than the astrometric accuracy.
+(iii) We ignore the parallax effect on the source trajec-
+tories, which are too small to be measured (compare the
+dotted and dashed blue lines in right panels in Figures
+1, and 2).
+(iv) We ignore the finite source effects on
+both microlensing lightcurves and astrometric shifts in
+the source position. (v) We assume that the source dis-
+tances from the observer, Ds, are determined by other
+observations, and we do not need to tune them through
+microlensing observations. For instance, the Gaia obser-
+vations provide stellar parallax distances for some source
+stars.
+Photometry and astrometry Fisher matrices are:
+Aij =
+N
+�
+k=1
+1
+σ2m(tk)
+∂2ms(tk)
+∂pi∂pj
+,
+Bij =
+N
+�
+k=1
+1
+σ2a(tk)
+�∂2θs,n1(tk)
+∂qi ∂qj
++ ∂2θs,n2(tk)
+∂qi ∂qj
+�
+, (10)
+where, ms(tk) = mbase − 2.5 log10
+�
+fbA(tk) + 1 − fb
+�
+is
+the apparent source magnitude at the given time tk. fb
+is the blending factor in W149 filter, mbase is the base-
+line magnitude without lensing effect in that filter (its
+distribution for detectable events is shown in the second
+panel of Figure 3). pis, and qis are observable parameters
+that affect on photometry and astrometry measurements
+(ms, θs), respectively.
+Observable parameters: A microlensing light curve
+by considering the parallax effect can be modeled with 7
+parameters which are: pi ∈ t0, u0, tE, ξ, fb, mbase, πE.
+The finite source effect can be ignored in long-duration
+microlensing events due to ISMBHs, so we put aside this
+effect while calculating A. The source apparent trajec-
+tory on the sky plane can be modeled with 3 parameters:
+qi ∈ θE, µs,n1, µs,n2.
+We calculate Fisher matrices numerically.
+Their in-
+verses (i.e., covariance matrices, A−1 and B−1) are de-
+rived using the Python module Numpy 6.
+The square
+roots of diagonal elements are the errors in the observ-
+able parameters, e.g., σpi =
+�
+A−1
+ii and σqi =
+�
+B−1
+ii ,
+and non-diagonal elements are the correlation coefficients
+between errors in the parameters.
+Taking these errors into account, we determine the errors
+in the physical parameters of ISMBHs, which is explained
+in the next subsection.
+3.1. Errors in the physical parameters
+According to Equation 2, the lens mass and its error
+as a function of observable parameters are:
+6 https://numpy.org/
+
+8
+Sajadian and Sahu
+Fig. 4.— The fractional distributions of the relative errors in the normalized parallax amplitude, the lens mass, the lens distance, and the
+lens proper motion for a big samples of microlensing events due to ISMBHs detectable by the Roman telescope with (green distributions)
+and without (black step ones) considering sparse observations when the Galactic bulge is observable during the large time gap.
+The
+vertical (solid, dashed and dotted) lines show the thresholds of the relative errors 10%, 5%, and 1%, respectively. The samples due to both
+distributions have the same entrances.
+Ml = θE
+κ πE
+,
+σMl =Ml
+��σθE
+θE
+�2
++
+�σπE
+πE
+�2
+,
+(11)
+where σMl, σθE, and σπE are the error in the lens mass,
+error in the angular Einstein radius, and the error in
+normalized parallax amplitude, respectively.
+We note
+that there is no correlation between σπE and σθE, because
+these two parameters are determined from photometry
+and astrometry Fisher matrices independently. The next
+parameter is the lens distance which is given by:
+1
+Dl
+= 1
+Ds
++ πE θE
+au
+,
+σDl =Dl
+Ds − Dl
+Ds
+σMl
+Ml
+,
+(12)
+Here, we assume that the error in source distance is very
+small and can be ignored. The last parameter is the lens
+angular velocity components which are:
+µl,n1 =µs,n1 − θE
+tE
+cos ξ,
+µl,n2 =µs,n2 − θE
+tE
+sin ξ,
+(13)
+Accordingly, the errors in the lens angular velocity com-
+ponents are given by:
+σ2
+l,n1 = σ2
+s,n1 +µ2
+rel,⊙ cos2 ξ
+��σθ
+θE
+�2 +
+�σt
+tE
+�2
++
+� σξ
+cot ξ
+�2 − 2σt
+tE
+σξ
+cot ξ
+ˆ
+A−1
+ij
+�
+,
+σ2
+l,n2 = σ2
+s,n2 +µ2
+rel,⊙ sin2 ξ
+��σθ
+θE
+�2 +
+�σt
+tE
+�2
++
+� σξ
+tan ξ
+�2 − 2σt
+tE
+σξ
+tan ξ
+ˆ
+A−1
+ij
+�
+.
+(14)
+where, σl,i, σs,i are the errors in ith component of the lens
+and source angular velocity projected on the sky plane,
+and ˆ
+A−1
+ij = A−1
+ij /
+�
+A−1
+ii A−1
+jj is the correlation coefficient
+
+0.11
+0.09
+Distribution
+0.06.
+..
+..........
+ Normalized 
+0.04
+0.02
+0.00
+0
+2
+3
+5
+0g10l0E
+/ TE(%))0.12
+0.11
+0.09
+0.08
+istribu
+20.07
+30.05
+0.04
+0.03
+.................
+0.01
+0.00
+0
+3
+5
+[(%)W / W0]0160l0.13
+0.12
+≤0.10
+ibutior
+0.08-
+Distril
+D
+0.07
+lormalized 
+0.05
+Z0.03
+0.02 -
+0.00
+2
+log10[g D// Di(%)0.15
+0.13
+0.12
+ution
+0.10
+istribu
+-
+20.08
+D
+8
+30.07
+Normalize
+0.05
+0.03
+0.02
+0.00
+0
+3
+5
+l0g10[0μ/μ(%)]Detecting stellar-mass black holes by Roman
+9
+TABLE 1
+Statistical information about simulated microlensing events due to ISMBHs detectable with the Roman telescope
+by assuming different ISMBHs mass functions.
+σtE
+�
+tE
+σπE
+�
+πE
+σθE
+�
+θE
+σMl
+�
+Ml
+σDl
+�
+Dl
+σµs
+�
+µs
+σµl
+�
+µl
+ϵm(%)
+Ne,BHs
+dN/dM = const
+No observations during the time gap
+≤ 1%
+23.60
+7.50
+85.56
+6.11
+21.15
+99.67
+5.16
+4.21
+2
+≤ 5%
+53.26
+24.35
+99.32
+24.08
+50.59
+99.98
+22.32
+19.37
+11
+≤ 10%
+65.91
+34.86
+99.88
+34.77
+64.11
+100.00
+33.00
+29.29
+17
+Sparse observations during the time gap
+≤ 1%
+30.81
+8.32
+83.15
+6.93
+22.99
+99.66
+6.10
+5.15
+4
+≤ 5%
+63.72
+25.66
+98.85
+25.40
+52.37
+99.98
+24.27
+21.48
+17
+≤ 10%
+76.00
+36.14
+99.75
+36.05
+65.26
+99.99
+34.98
+31.54
+24
+dN/dM ∝ M−0.5
+No observations during the time gap
+≤ 1%
+22.20
+7.52
+75.03
+5.34
+19.43
+99.68
+4.38
+3.64
+2
+≤ 5%
+49.88
+22.52
+98.29
+21.98
+45.97
+99.99
+20.34
+17.57
+12
+≤ 10%
+62.02
+31.84
+99.65
+31.66
+59.07
+99.99
+29.94
+26.30
+18
+Sparse observations during the time gap
+≤ 1%
+25.77
+7.70
+71.64
+5.65
+19.49
+99.66
+4.94
+4.22
+3
+≤ 5%
+56.57
+22.29
+97.40
+21.82
+45.21
+99.98
+20.81
+18.25
+15
+≤ 10%
+69.18
+31.33
+99.32
+31.15
+57.54
+99.99
+30.05
+26.75
+22
+dN/dM ∝ M−1
+No observations during the time gap
+≤ 1%
+21.89
+7.52
+71.23
+5.11
+18.85
+99.67
+4.19
+3.51
+3
+≤ 5%
+48.83
+22.00
+97.82
+21.34
+44.75
+99.98
+19.79
+17.00
+14
+≤ 10%
+61.02
+31.20
+99.56
+30.97
+57.68
+99.99
+29.15
+25.56
+21
+Sparse observations during the time gap
+≤ 1%
+24.48
+7.55
+67.89
+5.30
+18.56
+99.67
+4.56
+3.92
+3
+≤ 5%
+54.23
+21.38
+96.75
+20.81
+43.22
+99.99
+19.85
+17.33
+15
+≤ 10%
+66.95
+30.00
+99.17
+29.79
+55.42
+100.00
+28.79
+25.61
+22
+dN/dM ∝ M−2
+No observations during the time gap
+≤ 1%
+21.75
+7.15
+59.45
+4.51
+16.60
+99.69
+3.83
+3.34
+3
+≤ 5%
+49.50
+19.65
+95.20
+18.83
+39.16
+99.99
+17.93
+15.53
+12
+≤ 10%
+62.21
+27.56
+98.69
+27.24
+50.89
+100.00
+26.30
+23.07
+18
+Sparse observations during the time gap
+≤ 1%
+21.00
+7.57
+62.54
+4.46
+17.91
+99.67
+3.71
+3.31
+3
+≤ 5%
+46.86
+21.33
+96.58
+20.35
+42.25
+99.98
+18.81
+16.08
+15
+≤ 10%
+58.57
+29.98
+99.28
+29.61
+54.68
+100.00
+27.93
+24.39
+23
+Note. — Each entry represents the persentage of simulated events with the desired relativel error (specified in its row) be less
+than the given threshold (determined in its column). ϵm is the Roman efficiency for measuing the lens mass, distance, and its proper
+motion with the relative errors less than the given threshold. The last column reports the estimated number of ISMBHs that can
+be detected in the Roman observations by considering different mass functions, as explained in Subsection 3.4.
+between errors in tE, and ξ. The errors in the lens and
+source proper motion can be determined using the errors
+in their components.
+3.2. Results
+The normalized distributions for four relevant param-
+eters (i.e., tE, mbase, t0, and u0) for simulated events
+whose relative errors in the lens mass, distance and
+proper motion are ≤ 5%, are shown in Figure 3 with
+black step lines. Accordingly, longer microlensing events
+from brighter source stars, whose times of the closest ap-
+proach happen during either the first three or the last
+three observing seasons are more favourable for the mea-
+surement of the physical parameters of the lens objects
+with reasonable accuracy.
+In Figure 4, we show the normalized distributions
+of the relative errors in the physical parameters of
+ISMBHs (as microlenses), resulting from Monte Carlo
+simulations, by considering a uniform mass function
+for ISMBHs. Green and black distributions are related
+to detectable events by the Roman telescope with and
+without considering sparse data points during the time
+gap, respectively.
+These parameters are the normalized
+parallax amplitude, the lens mass, the lens distance and
+the lens proper motion. The threshold amounts of the
+relative errors in the given parameters of 10%, 5%, and
+1% are depicted with solid, dashed, and dotted lines.
+Accordingly, adding extra observations during the time
+gap (one hour of observations every 10 days when the
+Galactic bulge is observable) improves the relative errors
+in all physical parameters, especially the lens distance
+from the observer.
+For numerical evaluation, in Table 1 we give the per-
+
+10
+Sajadian and Sahu
+centages of simulated detectable events with the rela-
+tive errors (specified in the first row) less than the given
+thresholds (i.e., 1, 5, 10% as mentioned in the first col-
+umn) are reported. Hence, sparse observations during
+the time gap improve the Roman efficiencies by ∼ 1%,
+∼ 2%, and ∼ 2% for measuring the physical parameters
+by the relative errors less than 1, 5, 10%, respectively.
+In 20-25% detectable events, the lens mass can be de-
+termined with the relative error less than 5%.
+These
+events have smaller relative errors in the lens distance,
+because the factor (Ds − Dl)/Ds is less than one.
+The source proper motion can be determined by
+monitoring the source positions during 6 observing
+seasons (with a 15 min cadence) of the Roman mission
+even without taking sparse data points during the
+∼2.3-year time gap very well.
+Nevertheless, the lens
+proper motion can be determined with the relative error
+less than 5% in 19-24% of these events.
+Even though ISMBHs produce long-duration mi-
+crolensing events,
+which are suitable for discerning
+the annual parallax effects, the normalized parallax
+amplitude, πE, decreases with increasing the lens mass.
+Hence, the parallax effect can be discerned in these
+long-duration microlensing events with the relative
+errors less than 5% only in 21-26% of all detectable
+events.
+In order to determine which kinds of ISMBHs might
+be well characterized through astrometric microlensing
+observations with the Roman telescope, we show the de-
+pendence of the relative errors in the lens mass, the lens
+distance, its proper motion, and the parallax amplitude
+to Ml, xls, Ds, and mbase in Figure 5, in different panels,
+respectively. For these plots, we only use the events with
+the relative errors less than 5%. There are several factors
+which determine their dependencies.
+According to the first panel, the relative error in the
+lens mass minimize when Ml ≃ 10-25M⊙.
+Increasing
+the lens mass has two against effects: (i) The lens mass
+enhances the Einstein crossing time and decreases the
+average photometry errors.
+Because more data points
+are taken while the source is being lensed, and less data
+points are recorded over the baseline. (ii) Enhancing the
+lens mass decreases the normalized parallax amplitude
+πE significantly, and makes hard measure it (see the dot-
+ted red step line in the top panel). This point was also
+expressed by Karolinski & Zhu (2020) and while model-
+ing OGLE-2006-BLG-044 microlensing event. For that
+reason, the optimum value for the lens mass with least
+errors is neither the least (2-3 solar mass), nor the most
+(40-50 solar mass). The relative error in the lens distance
+decreases with the lens mass. In fact, by increasing the
+lens mass xls enhances to keep the Einstein crossing times
+close to reasonable values for detection.
+The relative error in the lens proper motion weakly de-
+pends on the lens mass. In fact, σtE/tE is an increasing
+function versus the lens mass. By fixing the observing
+time and cadence (considering a determined observing
+platform) and increasing tE, its error increases. In to-
+tal, the relative errors in the lens physical parameters
+enhance with the lens mass slowly.
+The second panel of Figure 5 shows the relative errors
+in the lens mass, lens distance, its proper motion, and
+the parallax amplitude versus xls = Dl/Ds. The smaller
+xls make larger πE and θE, with smaller observing errors.
+That increases the relative error in the lens mass versus
+xls. However, this enhancement is slower in the relative
+error in the lens distance, because of the factor (Ds −
+Dl)/Ds in Equation 12.
+In the next panel of Figure 5, we show the depen-
+dence of the relative errors with the source distance from
+the observer. The source distance decreases πE, and θE,
+which increases the relative errors in the lens mass and
+its distance. We note that decreasing the parallax am-
+plitude increases both errors in the parallax amplitude,
+and ξ. Comparing these panels, we find that the effect
+of the source distance and the lens relative position (xls)
+on the errors is higher than the effect of the lens mass.
+In the last panel, the relative errors versus the apparent
+magnitude of the source star at the baseline are depicted.
+As shown here, they enhance with the source magnitude.
+Both Roman photometric and astrometric errors increase
+with the apparent magnitude of source stars. Worse ac-
+curacies cause higher relative errors in the lens physical
+parameters.
+Therefore, long-duration microlensing events due to
+ISMBHs with the mass Ml ≃ 10-25M⊙, close to the ob-
+server (xls ≲ 0.5) while the source is inside the Galactic
+disk (Ds ≲ 6kpc) can be characterized with the least
+errors.
+3.3. Different mass function for ISMBHs
+We know that there is no accurate mass function
+for ISMBHs based on observations yet, so we perform
+the simulation by considering several mass functions for
+ISMBHs, which are given in the following:
+dN
+dM =const.,
+dN
+dM ∝1
+�√
+M,
+dN
+dM ∝M −1,
+dN
+dM ∝M −2.
+(15)
+The results from simulations based on each of these mass
+functions are reported in Table 1. Accordingly, by chang-
+ing ISMBHs mass function, the Roman efficiency to mea-
+sure the lens physical parameters can change up to 2-7%.
+Also, the first mass function makes more ISMBHs with
+mass Ml ∈ [10, 25]M⊙ than other mass functions. So it
+has larger efficiencies to measure the physical parameters
+of lens objects than others.
+In the next subsection, we do some statistical estima-
+tions about detecting and characterizing such events dur-
+ing the Roman mission.
+3.4. Statistical estimations
+The number of microlensing events that the Ro-
+man telescope will detect is Ne,tot = 27000, which were
+estimated in Penny et al. (2019); Johnson et al. (2020).
+Here, we want to evaluate what fraction of this total
+number of microlensing events detectable by the Ro-
+man telescope are due to ISMBHs. In this regard, there
+are two factors: (i) the optical depth, and (ii) the av-
+erage microlensing duration which are discussed in the
+
+Detecting stellar-mass black holes by Roman
+11
+Fig. 5.— The dependence of the average relative errors in the lens mass (solid green lines), the lens distance (dashed blue lines), its
+proper motion (dot-dashed magenta lines), and the normalized parallax amplitude (dotted red lines) versus the lens mass, the ratio of the
+lens distance to the source distance from the observer (xls), the source distance, and the source apparent magnitude at the baseline.
+following.
+(i) The number of detectable microlensing events is pro-
+portional to the optical depth. The microlensing optical
+depth at a given line of sight (l, b) and one specified dis-
+tance from the observer, (D), is proportional to the lens
+mass Ml, because it is given by:
+dτ(l, b, D)
+dD
+= π θ2
+E n(l, b, D) D2,
+(16)
+where, (l, b) are the Galactic longitude and latitude,
+respectively. n(l, b, D) is the number density of stars in
+our galaxy which is the Galactic mass density divided by
+the average stellar mass.
+Accordingly, the ratio of the optical depth (and as a re-
+sult the number of microlensing events) due to ISMBHs
+to the overall optical depth due to all potential lens ob-
+jects can be estimated by:
+F1 =
+� ∞
+20M⊙
+Ml η(Ml) dMl
+� � ∞
+13MJ
+Ml η(Ml) dMl,(17)
+where,MJ is the Jupiter mass, η(Ml) is the initial mass
+function in the Galactic disk.
+In fact, F1 determines
+the contribution of the ISMBHs in producing the effec-
+tive lensing surface in comparison with the total lens-
+ing surfaces covered by all possible Einstein rings.
+In
+Equation 17, we use the fact that stars with the initial
+mass M > 20M⊙ will convert to black holes. We ignore
+the contribution of black holes generated from primordial
+fluctuations in the early universe.
+In order to estimate F1, we take the initial mass
+function from the Besan¸con model (Robin et al. 2003,
+2012), and assume that all lens objects are inside the
+Galactic disk. This mass function is η(Ml) ∝ M −1.6
+l
+for
+0.08 ≤ Ml(M⊙) ≤ 1, and η(Ml) ∝ M −3
+l
+for Ml(M⊙) ≥ 1.
+The stars with Ml > 20M⊙ are converted to ISMBHs.
+For 13MJ < Ml < 0.08M⊙ we take the Brown dwarf
+mass function, i.e., M −0.7
+l
+(Muˇzi´c et al. 2015; Luhman
+2004). We do not include free floating planets, because
+of their negligible contribution. The upper limit should
+in reality be the mass due to the most massive star in
+the Galactic disk.
+We set this upper limit to infinity,
+because the mass function for M > 1M⊙ decreases as
+M −3, so it tends to zero fast.
+Accordingly, we find
+F1 = 0.019.
+(ii) The microlensing event rate is proportional to
+�
+ϵ(tE)
+�
+tE
+�
+, which specifies the inverse of the average du-
+ration of microlensing events.
+Here, ϵ(tE) is the
+Ro-
+
+2.2
+2.1
+Relative Error
+2.0
+1.9
+1.8
+1.7
+1.6
+10
+20
+30
+40
+M[Mo]2.8
+2.6
+2.4
+ Error
+2.2
+2.0
+Relative I
+1.8
+1.6
+1.4
+1.2
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+XIs2.25
+2.00
+1.75
+Error
+1.50
+Relative
+1.25
+(0m, / M[%]
+1.00
+(αD. / Di[%]>
+0.75
+(0μ/ / μi[%])
+(O πe / TE[%])
+0.50
+0
+2
+4
+6
+8
+10
+12
+Ds(kpc)3.5
+(om / M[%])
+<gd. / Di[%])
+(0μ / μi[%])
+3.0
+<Oπ= / TE[%]>
+Relative Error
+2.5
+2.0
+1.5
+1.0
+16
+17
+18
+19
+20
+21
+22
+23
+24
+mbase(mag)12
+Sajadian and Sahu
+man efficiency for detecting a microlensing event with the
+specified time scale tE, and was kindly provided by M.
+Penny. Since ISMBHs make longer microlensing events
+than usual events, we expect this factor for ISMBHs to
+be smaller than that due to all detectable microlensing
+events due to all potential lens objects. We define an-
+other factor:
+F2 =
+�ϵ(tE)
+tE
+�
+BHs
+� �ϵ(tE)
+tE
+�
+Total
+.
+(18)
+To estimate this factor, we simulate the microlensing
+events detectable by the Roman telescope, and by adopt-
+ing a uniform mass function for ISMBHs. However, we
+tune the ratio of the number of ISMBHs to the number
+of total objects ≃ 0.0001, as expected. In the simulation,
+the lens objects can be brown dwarfs, main-sequence
+stars and ISMBHs, and we obtain F2 = 0.15, 0.11 with
+and without considering sparse observations during the
+time gap, respectively. We note that considering extra
+observations enables us to detect ISMBHs in shorter mi-
+crolensing events (the average tE changes from 329 days
+to 303 days).
+Therefore, the Roman telescope roughly will detect
+Ne,BHs = Ne,tot × F1 × F2 ≃ 56-77 microlensing events
+due to ISMBHs (under the assumption that their masses
+are uniformly distributed in the range of [2, 50]M⊙,
+and their contribution with respect to all lens objects
+is 0.0001). In 2-4, 11-17, and 17-24 of these events the
+physical parameters of ISMBHs (including their mass,
+distance and proper motion) can be determined with the
+relative errors less than 1%, 5%, and 10%, respectively,
+as reported in the last column of Table 1.
+For other mass functions, i.e., dN/dM ∝ M −α with
+α = 0.5, 1, 2, we get F2 = 0.16-0.13, 0.17-0.16, 0.18-
+0.0.15 (with and without adding extra observations dur-
+ing the time gap), respectively. The corresponding num-
+ber of ISMBHs that can be detected and characterized
+through the Roman observations are reported in Table
+1.
+4. CONCLUSIONS
+In this work, we studied detection and characterization
+of ISMBHs through astrometric microlensing to be done
+by the upcoming microlensing survey by the Roman tele-
+scope.
+This telescope has been planned to detect mostly short-
+duration microlensing events due to exoplanets beyond
+the snow line of main-sequence stars and free-floating
+exoplanets.
+Nevertheless, the duration of its mission is long enough
+to detect and characterize long-duration microlensing
+events, and its astrometric accuracy is high enough to
+discern the astrometric trajectories (and the dimensional
+lensing-induced shifts) of source stars.
+Here, we have done a comprehensive simulation of as-
+trometric microlensing events due to ISMBHs that can
+be discerned by the Roman telescope. For each simu-
+lated event we have calculated Fisher and Covariance
+matrices for photometry and astrometry measurements
+separately, and estimated the errors in observable param-
+eters, and physical parameters of ISMBHs as well.
+Since the long time gap between Roman’s first three
+observing seasons and the other three seasons would limit
+its efficiency and robustness for discerning and charac-
+terizing ISMBHs, we considered a small amount of ad-
+ditional observations when the Galactic bulge is visible
+during this time gap, by adding one hour of observa-
+tions (4 data points) every 10 days when the Galactic
+bulge is detectable in our simulations. These additional
+observations amount to a total of about one day of obser-
+vations with Roman. We found that this small amount
+of extra observations increases Roman’s efficiency of de-
+tecting and characterizing ISMBHs by ∼ 1 − 2%, and,
+more importantly, improve the robustness of the results
+and help avoiding degenerate solutions.
+We note that photometric follow-up of these microlens-
+ing events with ground-based telescopes such as the Ru-
+bin Observatory during the time gap should also be help-
+ful.The ground-based images may suffer from blending,
+but the higher-resolution images of Roman should help in
+correctly estimating the blending factor, thus providing
+useful data for better characterization of the microlens-
+ing light curves.
+For long-duration microlensing events due to ISMBHs,
+the efficiency of Roman microlensing survey for measur-
+ing the physical parameters of the lens by considering
+different ISMBHs mass functions are summarized in Ta-
+ble 1.
+The efficiencies for measuring with better than 5% un-
+certainty the lens mass, its distance, and its proper mo-
+tion are 20-25%, 42-52%, and 19-24%, respectively, and
+the efficiency of measuring all the three parameters with
+better than 5% uncertainty is 16-21%.
+ISMBHs produce long-duration microlensing events
+which are appropriate for discerning the annual parallax.
+On the other hand, the normalized parallax amplitudes
+decrease with 1/√Ml. Therefore, πE can be measured
+with the relative error less than 5% in only 21-26% of
+these long-duration events.
+The relative errors in the physical parameters of
+ISMBHs increases with the source distance and xls =
+Dl/Ds. The dependence of these relative errors to the
+lens mass is relatively weak and by changing the lens
+mass from 2 to 50 solar mass, these error changes less
+than 1%. On the whole, the least relative errors in the
+lens mass and its distance occurs when Ml ≃ 10-25M⊙,
+xls ≲ 0.5, and Ds ≲ 6 kpc.
+We also statistically estimated the total number
+of microlensing events due to ISMBHs that can be
+detected and characterized with the Roman telescope.
+By assuming different mass functions for ISMBHs (given
+in Equation 15) in the range of [2, 50]M⊙, we concluded
+that this telescope will detect 56-77 long-duration
+microlensing events due to ISMBHs during its mission.
+Additionally, it can measure the physical parameters
+of ISMBHs with the relative errors less than 1%, 5%,
+and 10% in 3-4, 15-17, 22-24 of these events, respectively.
+All simulations that have been done for this paper
+are available at:
+https://github.com/SSajadian54/
+AstrometryMicrolensing
+Research efforts of KCS were supported by NASA
+through grants from STScI, under proposal IDs 14783,
+15318 and 16200. We thank the anonymous referee for
+his/her careful and useful comments, which improved the
+
+Detecting stellar-mass black holes by Roman
+13
+Fig. 6.— Figure shows the Galactic plane and two coordinate systems which are needed to project stellar velocities on the sky plane.
+quality of the paper.
+APPENDIX
+TRANSFORMING COORDINATE SYSTEMS
+In this section, we will review how to transform the stellar velocity from the Galactic coordinate frame to the observer
+one and project them on the sky plane.
+In this Figure, the horizontal and vertical black lines describe the Galactic plane and make a right-hand coordinate
+system. We note that in this Figure the scales are not respected.
+We consider a star in our galaxy with the galactic coordinate (l, b), i.e., the galactic longitude and latitude, respectively.
+Three points of the Galactic center (GC), the star position projected on the Galactic plane (yellow star) and the observer
+position (black filled point) make a triangle with the angles l, α, β, as shown in Figure 6. The length scales: Roc the
+observer distance from the Galactic center, Ros the distance between the star position projected on the Galactic plane
+and the observer, and Rsc which is the distance between the Galactic center and the projected stellar position on the
+Galactic plane. Rsc can be given by:
+Rsc =
+�
+R2oc + R2os − 2RosRoc cos(l).
+(A1)
+where, Ros = D⋆ cos(b), and D⋆ is the star distance from the observer. Using the sinuous law in a triangle, we can
+derive the angle of β, as:
+sin(β) = Ros
+Rsc
+sin(l).
+(A2)
+By having the Galactic longitude, we will calculate the angle of α as α = π − l − β.
+In simulations, we determine the stellar velocities in the Galactic coordinate, i.e., (vU, vV, vW), which are toward the
+Galactic center, in the direction of the Galactic rotation, and toward the Galactic north, respectively. These velocities
+include the global rotational velocity which is a function of the stellar distance from the Galactic center (see, e.g.,
+Rahal et al. 2009), and velocity dispersion components which are functions of the stellar age, weakly mass, and the
+Galactic latitude (Carlberg et al. 1985; Sajadian & Rahvar 2019; Sajadian et al. 2021).
+In the lensing formalism, we need the projected components of stellar velocities on the sky plane. So we introduce
+another coordinate frame, (x, y, z), which z-axis is parallel with W (toward the Galactic north), and (x, y) describes
+the Galactic plane, as shown in Figure 6 with red vectors. We can easily convert the velocity components from Galactic
+coordinate frame to this new coordinate system, (x, y, z), as:
+vx =− cos(α) vU − sin(α) vV,
+vy =+ sin(α) vU − cos(α) vV,
+vz =vW,
+(A3)
+Note that stars are not in the Galactic disk and their line of sight (los) with respect to the Galactic plane make
+the angle b, the Galactic latitude. So, we should apply another rotation around y-axis with −b angle to obtain the
+
+GC
+y
+1- V
+B
+α
+Roc
+-U
+Observer14
+Sajadian and Sahu
+components of stellar velocities projected on the sky plane normal to the line of sight toward the stellar position as:
+vlos =cos(b) vx + sin(b) vz,
+vn1 =vy,
+vn2 =− sin(b) vx + cos(b) vz,
+(A4)
+n1 and n2 are two unit vectors describe the sky plane. For projection of the Sun velocity, α⊙ ≃ π − l, since β⊙ ≃ 0.
+For the observer orbit around the Sun, we easily consider a circular orbit with the radius of the astronomical unit.
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@@ -0,0 +1,586 @@
+arXiv:2301.03818v1  [physics.flu-dyn]  10 Jan 2023
+Extension of Moving Particle Simulation including
+rotational degrees of freedom for dilute fiber
+suspension
+Keigo Enomoto1, Takato Ishida1, Yuya Doi1, Takashi Uneyama1, and Yuichi Masubuchi1
+1Department of Materials Physics, Graduate School of Engineering, Nagoya University,
+Furo-cho, Chikusa, Nagoya 464–8603, Japan
+Abstract
+We develop a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion
+of fibers floating in sheared liquids. In conventional MPS schemes, if a fiber suspended in a liquid is
+represented by a one-dimensional array of MPS particles, it is entirely aligned to the flow direction
+due to the lack of shear stress difference between fiber-liquid interfaces. To address this problem, we
+employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles.
+The translational motion of liquid and solid particles and the rotation of solid particles are calculated
+with the explicit MPS algorithm.
+The fiber is modeled as an array of micropolar fluid particles
+bonded with stretching and bending potentials. The motion of a single rigid fiber is simulated in a
+three-dimensional shear flow generated between two moving solid walls. We show that the proposed
+method is capable of reproducing the fiber motion predicted by Jeffery’s theory being different from
+the conventional MPS simulations.
+1
+Introduction
+Fluid particle methods have been developed for simulations of multi-phase flows [1–3]. In the simulations
+of liquid-solid systems, the particles represent the included liquid and solid to possess local quantities
+such as velocity and pressure. The motion of each particle is calculated according to interactions based
+on its discretized governing equation with neighboring particles within a certain distance. The Moving
+Particle Simulation (MPS) method, developed by Koshizuka et al. [4], is one of such methods along
+with Smoothed Particle Hydrodynamics (SPH) [5, 6] and has been actively developed in recent years.
+Following the original MPS, which employs a semi-implicit scheme [7], high-precision schemes such as
+particle regularization schemes [8] and improvements of the differential operator models [9,10] have been
+proposed. Further developments for MPS have been being attempted for various issues including variable
+resolution schemes, theoretical error analysis, momentum conservation at interfaces, etc [11–13].
+A possible direction for further improvement of MPS is the inclusion of rotational degrees of freedom
+for particles. Such an aspect is necessary for fiber suspensions when the fiber is represented by a one-
+dimensional array of particles. Let us consider a rotational motion of a fiber oriented in the flow direction
+under shear. In conventional MPS schemes, this fiber is trapped in the fully aligned state due to the
+balance of particle interactions. However, in reality, due to the difference of the shear stress between the
+interfaces in the shear gradient direction, the fiber exhibits periodic rotation as theoretically argued by
+Jeffery [14]. Although this problem has been known [15], it has not been properly considered in most of
+the simulations for fiber suspensions with MPS [16,17]. In the conventional fluid particle method, viscous
+torque exerted by the fluid cannot be transferred to the motion of solid particles.
+In this study, we propose a novel MPS method for fiber suspensions to reproduce the rotational motion
+of fibers in a correct manner. To achieve this objective, we employ the micropolar fluid model to introduce
+an angular velocity field through the rotational degrees of freedom of the constituent particles [18]. To
+evaluate our method, we performed simulations of a single fiber suspended in the sheared Newtonian
+liquid. The fiber is represented as an array of micropolar fluid particles connected with each other with
+stretching, bending, and torsional potentials. We compare the fiber motion with Jeffery’s theory [14] to
+confirm that the fiber motion is correctly captured. Details are shown below.
+1
+
+2
+Model and Simulation
+2.1
+Explicit MPS with rotational degrees of freedom
+In the MPS model, the dynamics of fluid velocity obey the continuum Navier-Stokes equation.
+To
+incorporate the rotational degrees of freedom into the dynamics model, we employ the micropolar fluid
+model [18] in which the angular velocity field is incorporated. The conservation laws of linear and angular
+momentum are written as follows:
+Du(r,t)
+Dt
+= −1
+ρ∇P(r,t) + ν∇2u(r,t) + νr∇ × Υ(r,t) + f(r,t),
+(1)
+I DΩ(r,t)
+Dt
+= G(r,t) − νrΥ(r,t),
+(2)
+where D/Dt is the time material derivative, r is the position vector, t is time, u(r,t) is the fluid velocity,
+ρ is the mass density, P(r,t) is the pressure, ν is the kinematic viscosity coefficient, νr is the rotational
+kinematic coefficient, Ω(r,t) is the angular velocity field, Υ(r,t) = 2Ω(r,t) − ∇ × u(r,t), f(r,t) is the
+external volume force, I is the micro-inertia coefficient, and G(r,t) is the torque density due to the
+external field. According to the second law of thermodynamics, νr is a parameter properly chosen in the
+following range [19]:
+0 ≤ νr ≤ (1 + 2
+d)ν.
+(3)
+Here, d is the spatial dimension. For a normal fluid without micropolar degrees of freedom, Ω is given as
+Ω = (∇ × u)/2 which guarantees that Eq. (1) reduces the standard Navier-Stokes equation [19]. In this
+work, we simply set Ω = (∇ × u)/2 for liquid region.
+In this study, we employ the explicit MPS (EMPS) method [20,21] to discretize Eqs. (1) and (2). The
+equations for the constituent particle i are as follows:
+dui(t)
+dt
+= − 1
+ρi
+⟪∇P⟫i(t) + 1
+Re ⟪∇2u⟫i(t) +
+1
+Rer
+⟪∇ × Υ⟫i(t) + fi(t),
+(4)
+dΩi(t)
+dt
+= αGi(t) − 2α
+Rer
+Υi(t),
+(5)
+Υi(t) = 2Ωi(t) − ⟪∇ × u⟫i(t),
+(6)
+where ⟪⟫ indicates the quantity evaluated by the operator model in MPS at the position of particle i
+mentioned in the next paragraph. The equations are non-dimensionalized using the following quantities:
+the fluid mass density ρ0, the reference kinematic viscosity coefficient ν0, and the size of the fluid particle
+l0. ν0 is a reference value, and l0 can be interpreted as the characteristic length scale of the discretized
+system (which may be interpreted as the grid size in the finite difference scheme).
+These quantities
+define units of length, time, and energy, and the quantities discussed below are normalized according to
+these units. Re = ν0/ν is the Reynolds number, Rer = ν0/νr is the rotational Reynolds number, and α is
+defined as α = l2
+0/I. As the case of the integration of the micropolar fluid model to the SPH model [19],
+translational and rotational velocities are mapped onto constituent (liquid and solid) particles. In our
+model, solid particles are micropolar fluid particles, and their motion follows Eqs. (4) and (5). The motion
+of liquid particles follows the standard Navier-Stokes equation plus the reaction force based on the third
+term on the right-hand side of Eq. (4) exerted by the surrounding solid particles.
+To calculate the physical quantities and their differentials at the position of particle i, we need the
+weighting function. We employ the following weighting function:
+w(r) =
+⎧⎪⎪⎨⎪⎪⎩
+lc/r − 1
+(0 < r < lc)
+0
+(r ≥ lc)
+.
+(7)
+Here, lc is the cutoff radius. The local density is evaluated by the local number density of the constituent
+particles defined as
+ni = ∑
+j≠i
+w (∣rj − ri∣) .
+(8)
+2
+
+The differential operators in Eqs. (4) and (5) are calculated by the following operator models:
+⟪∇ψ⟫i = d
+n0 ∑
+j≠i
+[ ψi + ψj
+∣rj − ri∣2 (rj − ri)w (∣rj − ri∣)],
+(9)
+⟪∇ × b⟫i = d
+n0 ∑
+j≠i
+[(bj − bi) × (rj − ri)
+∣rj − ri∣2
+(rj − ri)w (∣rj − ri∣)],
+(10)
+⟪∇2b⟫i = 2d
+λn0 ∑
+j≠i
+[(bj − bi)w (∣rj − ri∣)],
+(11)
+λ = ∑j≠i (rj − ri)2w (∣rj − ri∣)
+∑j≠i w (∣rj − ri∣)
+.
+(12)
+Here, ψi and bi are scalar and vector variables on the particle i, n0 is the initial particle number density,
+and λ is the parameter defined by Eq. (12) [7]. Note that in Eq. (9) we use ψi + ψj instead of ψj − ψi, as
+proposed by Oochi et al. [20], for better momentum conservation.
+2.2
+Fiber model
+Ωi
+ui
+ti
+si
+uj
+Ωj = 1
+2 (∇ × uj)
+!"#$"%&'()*"+!,
+-.!"%&'()*"+!,
+/&0"+).'.!()&1!$"%&'()*"+!,&2
+!"#$%
+&"'(")
+y
+z
+x
+ri
+Fig. 1: Schematic of our method. The fiber is composed of micropolar fluid particles which possess the
+velocity ui and angular velocity Ωi. The liquid particles are represented as a micropolar fluid particle
+with Ωj = (∇ × uj)/2.
+The fiber is modeled as an array of solid particles as shown in Fig. 1. The solid particles are connected
+with stretching, bending, and torsional potential energies, in a similar manner proposed by Yamamoto
+and Matsuoka for the other simulation scheme [22]. These potential forces should be a function of the
+bond vector of neighboring particles and the orientation of each solid particle [23].
+To describe the
+orientation of the solid particles, we introduce two directors si and ti on each solid particle i. si and ti
+are unit vectors for which directions are parallel and perpendicular to the bond vector, as shown in Fig. 1
+The time derivative of directors is related to the angular velocity as follows:
+dsi(t)
+dt
+= (1 − sisi) ⋅ (Ωi × si),
+dti(t)
+dt
+= (1 − titi) ⋅ (Ωi × ti).
+(13)
+Here, 1 is the unit tensor. The projection tensors (1 − sisi) and (1 − titi) maintain si ⋅ ti = 0 within
+numerical errors.
+The stretching potential Us, bending potential Ub, and torsional potential Ut are defined as
+Us ({ri}) = ∑
+⟨i,j⟩
+ks
+2 (∣rj − ri∣ − 1)2,
+(14)
+Ub ({ri},{si}) = ∑
+⟨i,j⟩
+⎡⎢⎢⎢⎢⎣
+kb
+2 (sj − si)2 − kr
+2
+⎧⎪⎪⎨⎪⎪⎩
+(si ⋅ rj − ri
+∣rj − ri∣)
+2
++ (sj ⋅ ri − rj
+∣ri − rj∣)
+2⎫⎪⎪⎬⎪⎪��
+⎤⎥⎥⎥⎥⎦
+,
+(15)
+Ut ({ti}) = ∑
+⟨i,j⟩
+kt
+2 (tj − ti)2.
+(16)
+3
+
+Here, ks,kb,kr,kt are the spring constants and ⟨i,j⟩ represents a pair of two adjacent solid particles. The
+potential force fi and torque Gi are calculated as
+fi = −∂ (Us + Ub)
+∂ri
+,
+Gi = si × (−∂Ub
+∂si
+) + ti × (−∂Ut
+∂ti
+).
+(17)
+According to Eq. (14), if ks is large sufficiently, the fiber length L corresponds to the number of solid
+particles in the fiber. Since the unit length of the system is the size of the fluid particle, the aspect ratio
+of the fiber rp corresponds to L.
+2.3
+Numerical algorithms
+In the EMPS method, the fractional step algorithm is applied for time integration as in the original
+semi-implicit scheme for MPS. Each integration step is divided into prediction and correction steps. In
+the prediction step, predicted velocity u∗
+i is calculated by using terms other than the pressure gradient
+term in Eq. (4), and the angular velocity of the solid particles is also updated according to Eq. (5) as
+follows:
+u∗
+i = uk
+i + ∆t[ 1
+Re ⟪∇2u⟫
+k
+i + 1
+Re r ⟪∇ × Υ⟫k
+i + f k
+i ] ,
+Ωk+1
+i
+= Ωk
+i + ∆tα [Gk
+i −
+2
+Rer
+Υk
+i ],
+Υk
+i = 2Ωk
+i − ⟪∇ × u⟫k
+i .
+(18)
+Here, ∆t is the step size, and the upper indexes k represent the step number: bk
+i = bi(t = k∆t). The
+predicted position r∗
+i and directors are updated as
+r∗
+i = rk
+i + ∆tu∗
+i ,
+sk+1
+i
+= sk
+i + ∆t(1 − sk
+i sk
+i ) ⋅ (Ωk+1
+i
+× sk
+i ),
+t∗
+i = tk
+i + ∆t(1 − tk
+i tk
+i ) ⋅ (Ωk+1
+i
+× tk
+i ),
+(19)
+where t∗
+i is the predicted torsional director. To maintain the relation si ⋅ ti = 0, we adjust t as follows:
+tk+1
+i
+= (1 − sk+1
+i
+sk+1
+i
+) ⋅ t∗
+i .
+(20)
+In the correction step, the velocity and position are calculated as
+uk+1
+i
+= u∗
+i − ∆t
+ρi
+⟪∇P⟫k+1
+i
+,
+rk+1
+i
+= r∗
+i + (uk+1
+i
+− u∗
+i )∆t.
+(21)
+In the EMPS, the pressure is calculated by the following explicit form [20]:
+P k+1
+i
+= ρics
+n0
+(n∗
+i − n0).
+(22)
+Here, cs is the sound speed, and n∗
+i is the number density at r∗. This cs is optimized concerning reasonable
+incompressibility and numerical stability.
+2.4
+Simulations
+We apply shear flows in the following boundary conditions. Hereafter, we refer to flow, shear gradient,
+and vorticity directions as x, y, and z directions. We employ periodic boundary conditions for x and
+z directions, whereas we place solid walls at y = 0 and h perpendicular to the y direction. These walls
+consist of three layers of liquid particles, which are fixed on a squared lattice.
+Following the earlier
+study [17], we move the walls toward the x direction with the speed of uwall = ±˙γh/2, where ˙γ is the
+apparent shear rate. We have confirmed that the actual shear rate is equal to ˙γ and uniform throughout
+the system within a numerical error, in simulations without fibers, as shown in Appendix A.
+Simulations of a single fiber in a simple shear flow were carried out, and the rotational motion of
+the fiber was observed. To describe the fiber motion, we use the orientation angles φ and θ as shown
+in Fig. 2. The number of MPS particles was N = 64000 in total including those for walls and the fiber.
+The simulation box dimension was 40 ×40 ×40 in x-y-z directions, respectively, and the distance between
+the walls was 35. The kinematic viscosity coefficient ν and the strain rate ˙γ were chosen so that the
+4
+
+θ
+φ
+Fig. 2:
+Schematic of a fiber (an array of blue particles) at orientation angles φ and θ in a shear flow.
+The dashed curve shows the orbit of the head of the fiber (Jeffery orbit).
+fiber-based Reynolds number was Ref = L2 ˙γ/ν = 0.1 to realize a viscous dominant condition. The sound
+speed of the fluid cs was set so that the Mach number became Ma = 0.5h˙γ/cs = 0.03. The numerical step
+size ∆t was chosen to 0.01 according to the Courant condition, the viscous constraint, and the relation
+to the spring constant. Other model parameters were set as lc = 3.1, νr = 1.5ν, I = 0.8 unless otherwise
+noted. The mass density of the solid particles is the same as that of liquid particles. The aspect ratio
+of the fiber rp was varied in the range from 2 to 20. The spring constants were chosen at ks = 1000 and
+kb = kt = kr = 200. These values realized a rigid fiber, for which the effect of fiber deformation is negligible
+in the result as shown later. We performed the simulations with a house-made code.
+In the initial condition, we placed the fiber at the center of the simulation box to overlap the center
+of mass of the fiber and the box. The initial fiber orientation angle to x-direction, φ0, was fixed at π/2,
+whereas the initial angle to z-direction, θ0, was chosen at π/6, π/3, or π/2. Surrounding liquid particles
+were randomly arranged by the particle packing algorithms proposed by Colagrossi et al. [24].
+3
+Results and Discussion
+Typical snapshots of a single rigid fiber in a shear flow with θ0 = π/3 are shown in Fig. 3 (a). These
+figures clearly demonstrate that the fiber rotates as expected, even after it experiences the configuration
+aligned to the flow direction. Snapshots of another fiber aligned to the vorticity direction (θ0 = 0) are
+also shown in Fig. 3 (b). The fiber exhibits the rolling motion around the vorticity axis induced by the
+flow velocity difference between shear planes above and below the fiber. This behavior is known as the
+log-rolling motion [25]. In principle, we cannot reproduce this log-rolling motion of the fiber using MPS
+without introducing rotational degrees of freedom.
+To analyze rotational behavior in the vorticity plane quantitatively, we show the time evolution of
+the rotation angle φ in Fig. 4. We observe that the fiber rotates and approaches to φ = 0 in the MPS
+without rotational degrees of freedom. This is not consistent with Jeffery’s theory which predicts the
+periodic motion. In contrast, in our model, we observe the clear periodic motion. This fact demonstrates
+the importance of the rotational degrees of freedom integrated into our model. We compare the time
+evolution of φ with Jeffery’s theory. According to Jeffery’s theory, the periodic orbit depends on the
+aspect ratio of the fiber. The aspect ratio can be defined as the ratio of two axes of hydrodynamically
+equivalent ellipsoid for the fiber [26]. Here, one may argue that the fiber in our simulation model is not
+a rigid body and thus the aspect ratio is not well-defined. We found that with the employed simulation
+parameters, the fiber almost keeps its length and shape under the flow, and thus it can be approximately
+treated as a rigid body. We use the effective aspect ratio ref = 0.36rp to achieve the best agreement
+between our model and Jeffery’s theory.
+We performed the simulation with various aspect ratios to examine its effect on the rotation period.
+5
+
+ux/uwall
+= ˙γy
+y
+z
+x
+! " #
+$%&
+'%&
+(%&
+)%'
+*%*
+$$%+
+$,%*
+y
+z
+x
+! " #
+'%-
+)%'
+$-%(
+!"#
+!$#
+Ω
+!"
+Fig. 3:
+Typical snapshots of a fiber with rp = 10. Light blue spheres represent solid particles that
+compose the fiber, and red arrows show directors. Background colors correspond to the velocity of fluid
+particles. (a) The case of φ0 = π/2 and θ0 = π/3. The red arrows show si. (b) The case of φ0 = π/2 and
+θ0 = 0. The red arrows show ti.
+The result is shown in Fig. 5. According to Jeffery [14], the rotation period of the fiber T is described as
+T = 2π
+˙γ (ref + 1
+ref
+).
+(23)
+As mentioned above, Jeffery’s theory with the effective aspect ratio ref = 0.36rp agrees with our simulation
+data for rp = 10. We use the same relation for other rp values. As observed in Fig. 5, our simulation
+data agree well with Jeffery’s theory with ref = 0.36rp within the examined rp range.
+The ratio ref/rp = 0.36 is not close to unity. Here, we briefly discuss the validity of this value. A
+typical value in experiments is ref/rp = 0.7 [27]. This value is larger than ours. If we calculate the ratio
+of these two values, we have 0.7/0.36 ≈ 1.9. One interpretation of this result is that the fiber width in our
+model is twice larger than the expected value. Intuitively, we expect that the motion of fluid particles
+around the fiber is somewhat synchronized and increases the effective width of the fiber. Note that this
+ratio ref/rp depends on νr as shown in Appendix B.
+We further examine pivoting motion of fibers that tilt from the vorticity plane. Fig. 6 shows typical
+rotation orbits of the head of fibers for (a) θ0 = π/3 and (b) θ0 = π/6 with Re = 0.01. These orbits are
+characterized by Cb defined as
+Cb = ∣CJ∣/(1 + ∣CJ∣),
+(24)
+CJ = 1
+ref
+tanθ0(r2
+ef sin2 φ0 + cos2 φ0)
+1
+2 ,
+(25)
+where CJ is the orbit constant determined only by the initial configuration of the fiber φ0 and θ0. The
+examined cases correspond to Cb = 0.31 and 0.63, respectively. Although there are small fluctuations
+due to discretization errors, the fibers reasonably follow closed trajectories, which are consistent with the
+Jeffery orbits.
+To be fair, we note that the fiber in our method eventually falls out of the Jeffery orbit if we continue
+the simulation for a long time. Such behavior would be attributed to the properties of the Jeffery orbit
+and our model. The Jeffery orbit is not stable against a perturbation [28]. If a fiber motion or flow field is
+slightly perturbed, the orbit moves to others. In our model, due to the discretization by using particles,
+6
+
+0.40
+0.35
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+0.00Fig. 4:
+Time evolution of φ by our model (circle) and the MPS without rotational degrees of freedom
+(triangle) in dilute regime.
+(rp = 10,φ0 = π/2,θ0 = π/2) Solid curves represent Jeffery’s theory with
+ref = 0.36rp.
+both the fiber motion and flow field contain fluctuations. These fluctuations drive the orbit away from
+the original Jeffery orbit. We also note that the solid walls in our system and fluid inertia may probably
+play some roles. Nevertheless, as shown in Fig. 6, our scheme reasonably reproduces the Jeffery orbit
+in a similar manner to the other numerical studies [29–31]. Since our method is capable of reproducing
+the motion of single fibers in the dilute regime, extensions to the concentrated regime or real industrial
+application would be readily achievable.
+4
+Conclusion
+We have developed a new MPS method to accurately reproduce fiber motion in shear flows. We employ
+the micropolar fluid model to introduce rotational degrees of freedom into constituent particles.
+To
+validate our method, we simulated the single fiber motion suspended in the sheared liquid. The fiber is
+represented by a single array of micropolar fluid particles bonded with stretching, bending, and torsional
+potentials. We demonstrated that the simulated rotation period and rotation orbits of the fiber are in
+good agreement with Jeffery’s theory given that the effective aspect ratio is tuned as a fitting parameter
+of the theory.
+As an application of the proposed method, we are conducting simulations for dense fiber suspensions
+since fiber rotation possibly plays some roles as argued by Lindstr¨om and Uesaka [32]. The proposed
+method is also capable of representing solids of arbitrary shape such as plate-shaped particles [33], not just
+fibers. We are aware that the micropolar fluid model can be implemented to other fluid particle methods
+such as SPH. Studies toward such directions are ongoing and the results will be reported elsewhere.
+Acknowledgement
+The authors would like to express their gratitude to Dr. Satoru Yamamoto at Center for Polymer Interface
+and Molecular Adhesion Science, Kyushu University for helpful discussions.
+Appendix A
+Calculation of a simple shear flow using EMPS
+We have conducted EMPS simulations without solid particles to test the method and the code. The
+system settings are the same as simulations in Sec. 3 except for the gap size h and the absence of a fiber.
+7
+
+Fig. 5:
+The aspect ratio dependence of the rotation period. Symbols show our simulation data and the
+dashed curve shows the prediction by Jeffery’s theory (Eq. (23)) with ref = 0.36rp.
+An example of the steady-state flow profile of a shear flow is shown in Fig. 7 (a). Here, u∗
+x = ux/uwall is the
+normalized fluid velocity in the flow direction (x– direction) where the wall velocity uwall, and y∗ = y/h is
+the normalized distance from the moving wall. The Reynolds number of the flow is Reh = huwall/ν = 1.5
+for h = 55. The result is in good agreement with the analytical solution u∗
+x = 2(y/h − 0.5). The gap size
+dependence of the shear rate is shown in Fig. 7 (b). Here, ˙γ∗ is the average slope of the velocity profile
+divided by the shear rate expected from the wall velocity. This result shows that the numerical error of
+the shear rate is less than 1% for h > 40. These results are consistent with the earlier study [17].
+Appendix B
+νr dependence of the effective aspect ratio
+As mentinoed in Sec. 3, the ratio ref/rp depends on νr. The results are shown in Fig. 8. As νr increases,
+ref/rp monotonically decreases. Thus, one may optimize νr to have ref/rp that is consistent with a specific
+experimental system. To be fair, we note that the conditions νr < 0.5 are not suitable to our numerical
+method, and we cannot attain ref/rp value larger than 0.7, because the torque exerted to solid particles
+becomes comparable to the discretization error.
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+Fig. 6:
+Rotation orbits of a single fiber. Solid curves show our simulation results of (a) θ0 = π/6 for
+12 ≤ γ ≤ 50 and (b) θ0 = π/3 for 0 ≤ γ ≤ 50. Other parameters are the same as Fig. 3 (a). Dashed curves
+are the Jeffery orbits with ref = 0.36rp.
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+jun 2015.
+9
+
+Fig. 7:
+(a) Flow velocity profile generated by moving walls in our numerical method for a fluid without
+a fiber. The gap length is 55. (b) The gap length dependence of the average shear rate.
+Fig. 8:
+The rotational kinematic viscosity dependence of the effective aspect ratio in our model. ref and
+νr are normalized by rp and ν, respectively.
+[26] F. P. Bretherton. J. Fluid Mech., 14(2):284–304, 1962.
+[27] B. J. Trevelyan and S. G. Mason. J. Colloid Sci., 6(4):354–367, aug 1951.
+[28] P. G. Saffman. J. Fluid Mech., 1(5):540–553, 1956.
+[29] Xijun Fan, N. Phan-Thien, and Rong Zheng. J. Nonnewton. Fluid Mech., 74(1-3):113–135, jan 1998.
+[30] Satoru Yamamoto and Takaaki Matsuoka. J. Chem. Phys., 100(4):3317–3324, 1994.
+[31] Paal Skjetne, Russell F. Ross, and Daniel J. Klingenberg. J. Chem. Phys., 107(6):2108–2121, aug
+1997.
+[32] Stefan B. Lindstr¨om and Tetsu Uesaka. Phys. Fluids, 21(8):083301, aug 2009.
+[33] Toshiki Sasayama, Hirotaka Okamoto, Norikazu Sato, and Jumpei Kawada.
+Powder Technol.,
+404:117481, may 2022.
+10
+

KdE2T4oBgHgl3EQfVAc5/content/tmp_files/load_file.txt

@@ -0,0 +1,522 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf,len=521
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='03818v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='flu-dyn]  10 Jan 2023  Extension of Moving Particle Simulation including  rotational degrees of freedom for dilute fiber  suspension  Keigo Enomoto1, Takato Ishida1, Yuya Doi1, Takashi Uneyama1, and Yuichi Masubuchi1  1Department of Materials Physics, Graduate School of Engineering, Nagoya University,  Furo-cho, Chikusa, Nagoya 464–8603, Japan  Abstract  We develop a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion  of fibers floating in sheared liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In conventional MPS schemes, if a fiber suspended in a liquid is  represented by a one-dimensional array of MPS particles, it is entirely aligned to the flow direction  due to the lack of shear stress difference between fiber-liquid interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To address this problem, we  employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The translational motion of liquid and solid particles and the rotation of solid particles are calculated  with the explicit MPS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The fiber is modeled as an array of micropolar fluid particles  bonded with stretching and bending potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The motion of a single rigid fiber is simulated in a  three-dimensional shear flow generated between two moving solid walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We show that the proposed  method is capable of reproducing the fiber motion predicted by Jeffery’s theory being different from  the conventional MPS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  1  Introduction  Fluid particle methods have been developed for simulations of multi-phase flows [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In the simulations  of liquid-solid systems, the particles represent the included liquid and solid to possess local quantities  such as velocity and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The motion of each particle is calculated according to interactions based  on its discretized governing equation with neighboring particles within a certain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The Moving  Particle Simulation (MPS) method, developed by Koshizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' [4], is one of such methods along  with Smoothed Particle Hydrodynamics (SPH) [5, 6] and has been actively developed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Following the original MPS, which employs a semi-implicit scheme [7], high-precision schemes such as  particle regularization schemes [8] and improvements of the differential operator models [9,10] have been  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Further developments for MPS have been being attempted for various issues including variable  resolution schemes, theoretical error analysis, momentum conservation at interfaces, etc [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  A possible direction for further improvement of MPS is the inclusion of rotational degrees of freedom  for particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Such an aspect is necessary for fiber suspensions when the fiber is represented by a one-  dimensional array of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Let us consider a rotational motion of a fiber oriented in the flow direction  under shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In conventional MPS schemes, this fiber is trapped in the fully aligned state due to the  balance of particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' However, in reality, due to the difference of the shear stress between the  interfaces in the shear gradient direction, the fiber exhibits periodic rotation as theoretically argued by  Jeffery [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Although this problem has been known [15], it has not been properly considered in most of  the simulations for fiber suspensions with MPS [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In the conventional fluid particle method, viscous  torque exerted by the fluid cannot be transferred to the motion of solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  In this study, we propose a novel MPS method for fiber suspensions to reproduce the rotational motion  of fibers in a correct manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To achieve this objective, we employ the micropolar fluid model to introduce  an angular velocity field through the rotational degrees of freedom of the constituent particles [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To  evaluate our method, we performed simulations of a single fiber suspended in the sheared Newtonian  liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The fiber is represented as an array of micropolar fluid particles connected with each other with  stretching, bending, and torsional potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We compare the fiber motion with Jeffery’s theory [14] to  confirm that the fiber motion is correctly captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Details are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  1  2  Model and Simulation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='1  Explicit MPS with rotational degrees of freedom  In the MPS model, the dynamics of fluid velocity obey the continuum Navier-Stokes equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To  incorporate the rotational degrees of freedom into the dynamics model, we employ the micropolar fluid  model [18] in which the angular velocity field is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The conservation laws of linear and angular  momentum are written as follows:  Du(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t)  Dt  = −1  ρ∇P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) + ν∇2u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) + νr∇ × Υ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) + f(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (1)  I DΩ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t)  Dt  = G(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) − νrΥ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (2)  where D/Dt is the time material derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' r is the position vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' t is time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) is the fluid velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  ρ is the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) is the pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' ν is the kinematic viscosity coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' νr is the rotational  kinematic coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Ω(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) is the angular velocity field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Υ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) = 2Ω(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) − ∇ × u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' f(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) is the  external volume force,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' I is the micro-inertia coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' and G(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='t) is the torque density due to the  external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' According to the second law of thermodynamics, νr is a parameter properly chosen in the  following range [19]:  0 ≤ νr ≤ (1 + 2  d)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (3)  Here, d is the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' For a normal fluid without micropolar degrees of freedom, Ω is given as  Ω = (∇ × u)/2 which guarantees that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (1) reduces the standard Navier-Stokes equation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In this  work, we simply set Ω = (∇ × u)/2 for liquid region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  In this study, we employ the explicit MPS (EMPS) method [20,21] to discretize Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The  equations for the constituent particle i are as follows:  dui(t)  dt  = − 1  ρi  ⟪∇P⟫i(t) + 1  Re ⟪∇2u⟫i(t) +  1  Rer  ⟪∇ × Υ⟫i(t) + fi(t),  (4)  dΩi(t)  dt  = αGi(t) − 2α  Rer  Υi(t),  (5)  Υi(t) = 2Ωi(t) − ⟪∇ × u⟫i(t),  (6)  where ⟪⟫ indicates the quantity evaluated by the operator model in MPS at the position of particle i  mentioned in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The equations are non-dimensionalized using the following quantities:  the fluid mass density ρ0, the reference kinematic viscosity coefficient ν0, and the size of the fluid particle  l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' ν0 is a reference value, and l0 can be interpreted as the characteristic length scale of the discretized  system (which may be interpreted as the grid size in the finite difference scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  These quantities  define units of length, time, and energy, and the quantities discussed below are normalized according to  these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Re = ν0/ν is the Reynolds number, Rer = ν0/νr is the rotational Reynolds number, and α is  defined as α = l2  0/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' As the case of the integration of the micropolar fluid model to the SPH model [19],  translational and rotational velocities are mapped onto constituent (liquid and solid) particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In our  model, solid particles are micropolar fluid particles, and their motion follows Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The motion  of liquid particles follows the standard Navier-Stokes equation plus the reaction force based on the third  term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (4) exerted by the surrounding solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To calculate the physical quantities and their differentials at the position of particle i, we need the  weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We employ the following weighting function:  w(r) =  ⎧⎪⎪⎨⎪⎪⎩  lc/r − 1  (0 < r < lc)  0  (r ≥ lc)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (7)  Here, lc is the cutoff radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The local density is evaluated by the local number density of the constituent  particles defined as  ni = ∑  j≠i  w (∣rj − ri∣) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (8)  2  The differential operators in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (4) and (5) are calculated by the following operator models:  ⟪∇ψ⟫i = d  n0 ∑  j≠i  [ ψi + ψj  ∣rj − ri∣2 (rj − ri)w (∣rj − ri∣)],  (9)  ⟪∇ × b⟫i = d  n0 ∑  j≠i  [(bj − bi) × (rj − ri)  ∣rj − ri∣2  (rj − ri)w (∣rj − ri∣)],  (10)  ⟪∇2b⟫i = 2d  λn0 ∑  j≠i  [(bj − bi)w (∣rj − ri∣)],  (11)  λ = ∑j≠i (rj − ri)2w (∣rj − ri∣)  ∑j≠i w (∣rj − ri∣)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (12)  Here, ψi and bi are scalar and vector variables on the particle i, n0 is the initial particle number density,  and λ is the parameter defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (12) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (9) we use ψi + ψj instead of ψj − ψi, as  proposed by Oochi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' [20], for better momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='2  Fiber model  Ωi  ui  ti  si  uj  Ωj = 1  2 (∇ × uj)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' "#$"%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=',  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' "%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=',  /&0"+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content="'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' ()&1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='$"%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=',&2  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' "#$%  &"\'(")  y  z  x  ri  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 1: Schematic of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The fiber is composed of micropolar fluid particles which possess the  velocity ui and angular velocity Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The liquid particles are represented as a micropolar fluid particle  with Ωj = (∇ × uj)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The fiber is modeled as an array of solid particles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The solid particles are connected  with stretching, bending, and torsional potential energies, in a similar manner proposed by Yamamoto  and Matsuoka for the other simulation scheme [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These potential forces should be a function of the  bond vector of neighboring particles and the orientation of each solid particle [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To describe the  orientation of the solid particles, we introduce two directors si and ti on each solid particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' si and ti  are unit vectors for which directions are parallel and perpendicular to the bond vector, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 1  The time derivative of directors is related to the angular velocity as follows:  dsi(t)  dt  = (1 − sisi) ⋅ (Ωi × si),  dti(t)  dt  = (1 − titi) ⋅ (Ωi × ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (13)  Here, 1 is the unit tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The projection tensors (1 − sisi) and (1 − titi) maintain si ⋅ ti = 0 within  numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The stretching potential Us, bending potential Ub, and torsional potential Ut are defined as  Us ({ri}) = ∑  ⟨i,j⟩  ks  2 (∣rj − ri∣ − 1)2,  (14)  Ub ({ri},{si}) = ∑  ⟨i,j⟩  ⎡⎢⎢⎢⎢⎣  kb  2 (sj − si)2 − kr  2  ⎧⎪⎪⎨⎪⎪⎩  (si ⋅ rj − ri  ∣rj − ri∣)  2  + (sj ⋅ ri − rj  ∣ri − rj∣)  2⎫⎪⎪⎬⎪⎪⎭  ⎤⎥⎥⎥⎥⎦  ,  (15)  Ut ({ti}) = ∑  ⟨i,j⟩  kt  2 (tj − ti)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (16)  3  Here, ks,kb,kr,kt are the spring constants and ⟨i,j⟩ represents a pair of two adjacent solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The  potential force fi and torque Gi are calculated as  fi = −∂ (Us + Ub)  ∂ri  ,  Gi = si × (−∂Ub  ∂si  ) + ti × (−∂Ut  ∂ti  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (17)  According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (14), if ks is large sufficiently, the fiber length L corresponds to the number of solid  particles in the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Since the unit length of the system is the size of the fluid particle, the aspect ratio  of the fiber rp corresponds to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='3  Numerical algorithms  In the EMPS method, the fractional step algorithm is applied for time integration as in the original  semi-implicit scheme for MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Each integration step is divided into prediction and correction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In  the prediction step, predicted velocity u∗  i is calculated by using terms other than the pressure gradient  term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (4), and the angular velocity of the solid particles is also updated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (5) as  follows:  u∗  i = uk  i + ∆t[ 1  Re ⟪∇2u⟫  k  i + 1  Re r ⟪∇ × Υ⟫k  i + f k  i ] ,  Ωk+1  i  = Ωk  i + ∆tα [Gk  i −  2  Rer  Υk  i ],  Υk  i = 2Ωk  i − ⟪∇ × u⟫k  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (18)  Here, ∆t is the step size, and the upper indexes k represent the step number: bk  i = bi(t = k∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The  predicted position r∗  i and directors are updated as  r∗  i = rk  i + ∆tu∗  i ,  sk+1  i  = sk  i + ∆t(1 − sk  i sk  i ) ⋅ (Ωk+1  i  × sk  i ),  t∗  i = tk  i + ∆t(1 − tk  i tk  i ) ⋅ (Ωk+1  i  × tk  i ),  (19)  where t∗  i is the predicted torsional director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To maintain the relation si ⋅ ti = 0, we adjust t as follows:  tk+1  i  = (1 − sk+1  i  sk+1  i  ) ⋅ t∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (20)  In the correction step, the velocity and position are calculated as  uk+1  i  = u∗  i − ∆t  ρi  ⟪∇P⟫k+1  i  ,  rk+1  i  = r∗  i + (uk+1  i  − u∗  i )∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (21)  In the EMPS, the pressure is calculated by the following explicit form [20]:  P k+1  i  = ρics  n0  (n∗  i − n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (22)  Here, cs is the sound speed, and n∗  i is the number density at r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This cs is optimized concerning reasonable  incompressibility and numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='4  Simulations  We apply shear flows in the following boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Hereafter, we refer to flow, shear gradient,  and vorticity directions as x, y, and z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We employ periodic boundary conditions for x and  z directions, whereas we place solid walls at y = 0 and h perpendicular to the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These walls  consist of three layers of liquid particles, which are fixed on a squared lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Following the earlier  study [17], we move the walls toward the x direction with the speed of uwall = ±˙γh/2, where ˙γ is the  apparent shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We have confirmed that the actual shear rate is equal to ˙γ and uniform throughout  the system within a numerical error, in simulations without fibers, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Simulations of a single fiber in a simple shear flow were carried out, and the rotational motion of  the fiber was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To describe the fiber motion, we use the orientation angles φ and θ as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The number of MPS particles was N = 64000 in total including those for walls and the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The simulation box dimension was 40 ×40 ×40 in x-y-z directions, respectively, and the distance between  the walls was 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The kinematic viscosity coefficient ν and the strain rate ˙γ were chosen so that the  4  θ  φ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 2:  Schematic of a fiber (an array of blue particles) at orientation angles φ and θ in a shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The dashed curve shows the orbit of the head of the fiber (Jeffery orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  fiber-based Reynolds number was Ref = L2 ˙γ/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='1 to realize a viscous dominant condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The sound  speed of the fluid cs was set so that the Mach number became Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='5h˙γ/cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The numerical step  size ∆t was chosen to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='01 according to the Courant condition, the viscous constraint, and the relation  to the spring constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Other model parameters were set as lc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='1, νr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='5ν, I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='8 unless otherwise  noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The mass density of the solid particles is the same as that of liquid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The aspect ratio  of the fiber rp was varied in the range from 2 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The spring constants were chosen at ks = 1000 and  kb = kt = kr = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These values realized a rigid fiber, for which the effect of fiber deformation is negligible  in the result as shown later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We performed the simulations with a house-made code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  In the initial condition, we placed the fiber at the center of the simulation box to overlap the center  of mass of the fiber and the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The initial fiber orientation angle to x-direction, φ0, was fixed at π/2,  whereas the initial angle to z-direction, θ0, was chosen at π/6, π/3, or π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Surrounding liquid particles  were randomly arranged by the particle packing algorithms proposed by Colagrossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  3  Results and Discussion  Typical snapshots of a single rigid fiber in a shear flow with θ0 = π/3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These  figures clearly demonstrate that the fiber rotates as expected, even after it experiences the configuration  aligned to the flow direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Snapshots of another fiber aligned to the vorticity direction (θ0 = 0) are  also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The fiber exhibits the rolling motion around the vorticity axis induced by the  flow velocity difference between shear planes above and below the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This behavior is known as the  log-rolling motion [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In principle, we cannot reproduce this log-rolling motion of the fiber using MPS  without introducing rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To analyze rotational behavior in the vorticity plane quantitatively, we show the time evolution of  the rotation angle φ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We observe that the fiber rotates and approaches to φ = 0 in the MPS  without rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This is not consistent with Jeffery’s theory which predicts the  periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In contrast, in our model, we observe the clear periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This fact demonstrates  the importance of the rotational degrees of freedom integrated into our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We compare the time  evolution of φ with Jeffery’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' According to Jeffery’s theory, the periodic orbit depends on the  aspect ratio of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The aspect ratio can be defined as the ratio of two axes of hydrodynamically  equivalent ellipsoid for the fiber [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Here, one may argue that the fiber in our simulation model is not  a rigid body and thus the aspect ratio is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We found that with the employed simulation  parameters, the fiber almost keeps its length and shape under the flow, and thus it can be approximately  treated as a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We use the effective aspect ratio ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp to achieve the best agreement  between our model and Jeffery’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  We performed the simulation with various aspect ratios to examine its effect on the rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  5  ux/uwall  = ˙γy  y  z  x  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' " #  $%&  \'%&  (%&  )%\'  %*  $$%+  $,%*  y  z  x  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' " #  \'%-  )%\'  $-%(  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' "#  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='$#  Ω  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='"  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3:  Typical snapshots of a fiber with rp = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Light blue spheres represent solid particles that  compose the fiber, and red arrows show directors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Background colors correspond to the velocity of fluid  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (a) The case of φ0 = π/2 and θ0 = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The red arrows show si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (b) The case of φ0 = π/2 and  θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The red arrows show ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' According to Jeffery [14], the rotation period of the fiber T is described as  T = 2π  ˙γ (ref + 1  ref  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (23)  As mentioned above, Jeffery’s theory with the effective aspect ratio ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp agrees with our simulation  data for rp = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We use the same relation for other rp values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 5, our simulation  data agree well with Jeffery’s theory with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp within the examined rp range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  The ratio ref/rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36 is not close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Here, we briefly discuss the validity of this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' A  typical value in experiments is ref/rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='7 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This value is larger than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' If we calculate the ratio  of these two values, we have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' One interpretation of this result is that the fiber width in our  model is twice larger than the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Intuitively, we expect that the motion of fluid particles  around the fiber is somewhat synchronized and increases the effective width of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Note that this  ratio ref/rp depends on νr as shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  We further examine pivoting motion of fibers that tilt from the vorticity plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 6 shows typical  rotation orbits of the head of fibers for (a) θ0 = π/3 and (b) θ0 = π/6 with Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These orbits are  characterized by Cb defined as  Cb = ∣CJ∣/(1 + ∣CJ∣),  (24)  CJ = 1  ref  tanθ0(r2  ef sin2 φ0 + cos2 φ0)  1  2 ,  (25)  where CJ is the orbit constant determined only by the initial configuration of the fiber φ0 and θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The  examined cases correspond to Cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='31 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='63, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Although there are small fluctuations  due to discretization errors, the fibers reasonably follow closed trajectories, which are consistent with the  Jeffery orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To be fair, we note that the fiber in our method eventually falls out of the Jeffery orbit if we continue  the simulation for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Such behavior would be attributed to the properties of the Jeffery orbit  and our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The Jeffery orbit is not stable against a perturbation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' If a fiber motion or flow field is  slightly perturbed, the orbit moves to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' In our model, due to the discretization by using particles,  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='00Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 4:  Time evolution of φ by our model (circle) and the MPS without rotational degrees of freedom  (triangle) in dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  (rp = 10,φ0 = π/2,θ0 = π/2) Solid curves represent Jeffery’s theory with  ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  both the fiber motion and flow field contain fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These fluctuations drive the orbit away from  the original Jeffery orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We also note that the solid walls in our system and fluid inertia may probably  play some roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Nevertheless, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 6, our scheme reasonably reproduces the Jeffery orbit  in a similar manner to the other numerical studies [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Since our method is capable of reproducing  the motion of single fibers in the dilute regime, extensions to the concentrated regime or real industrial  application would be readily achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  4  Conclusion  We have developed a new MPS method to accurately reproduce fiber motion in shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We employ  the micropolar fluid model to introduce rotational degrees of freedom into constituent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  To  validate our method, we simulated the single fiber motion suspended in the sheared liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The fiber is  represented by a single array of micropolar fluid particles bonded with stretching, bending, and torsional  potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We demonstrated that the simulated rotation period and rotation orbits of the fiber are in  good agreement with Jeffery’s theory given that the effective aspect ratio is tuned as a fitting parameter  of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  As an application of the proposed method, we are conducting simulations for dense fiber suspensions  since fiber rotation possibly plays some roles as argued by Lindstr¨om and Uesaka [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The proposed  method is also capable of representing solids of arbitrary shape such as plate-shaped particles [33], not just  fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' We are aware that the micropolar fluid model can be implemented to other fluid particle methods  such as SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Studies toward such directions are ongoing and the results will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Acknowledgement  The authors would like to express their gratitude to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Satoru Yamamoto at Center for Polymer Interface  and Molecular Adhesion Science, Kyushu University for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Appendix A  Calculation of a simple shear flow using EMPS  We have conducted EMPS simulations without solid particles to test the method and the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The  system settings are the same as simulations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3 except for the gap size h and the absence of a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 5:  The aspect ratio dependence of the rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Symbols show our simulation data and the  dashed curve shows the prediction by Jeffery’s theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (23)) with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  An example of the steady-state flow profile of a shear flow is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Here, u∗  x = ux/uwall is the  normalized fluid velocity in the flow direction (x– direction) where the wall velocity uwall, and y∗ = y/h is  the normalized distance from the moving wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The Reynolds number of the flow is Reh = huwall/ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='5  for h = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The result is in good agreement with the analytical solution u∗  x = 2(y/h − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The gap size  dependence of the shear rate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Here, ˙γ∗ is the average slope of the velocity profile  divided by the shear rate expected from the wall velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' This result shows that the numerical error of  the shear rate is less than 1% for h > 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' These results are consistent with the earlier study [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Appendix B  νr dependence of the effective aspect ratio  As mentinoed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3, the ratio ref/rp depends on νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' As νr increases,  ref/rp monotonically decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Thus, one may optimize νr to have ref/rp that is consistent with a specific  experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' To be fair, we note that the conditions νr < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='5 are not suitable to our numerical  method, and we cannot attain ref/rp value larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='7, because the torque exerted to solid particles  becomes comparable to the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Gonz´alez, and Jose L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content='  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 6:  Rotation orbits of a single fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Solid curves show our simulation results of (a) θ0 = π/6 for  12 ≤ γ ≤ 50 and (b) θ0 = π/3 for 0 ≤ γ ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Dashed curves  are the Jeffery orbits with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Fluids, 190:254–273,  aug 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  [13] Gen Li, Jinchen Gao, Panpan Wen, Quanbin Zhao, Jinshi Wang, Junjie Yan, and Akifumi Yamaji,  aug 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Jeffery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' London, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' A, 102(715):161–179, nov 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  [15] Nils Meyer, Oleg Saburow, Martin Hohberg, Andrew N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Hrymak, Frank Henning, and Luise K¨arger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Okabe, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content='  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Candelier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' 7:  (a) Flow velocity profile generated by moving walls in our numerical method for a fluid without  a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' The gap length is 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=' (b) The gap length dependence of the average shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
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+page_content=' Fluids, 21(8):083301, aug 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  [33] Toshiki Sasayama, Hirotaka Okamoto, Norikazu Sato, and Jumpei Kawada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  Powder Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content=',  404:117481, may 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'}

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+Statistical Power Analysis for Designing Bulk, 
+Single-Cell, and Spatial Transcriptomics 
+Experiments: Review, Tutorial, and Perspectives 
+ 
+Hyeongseon Jeon1,2,*, Juan Xie1,2,3,*, Yeseul Jeon1,4,5,*, Kyeong Joo Jung6, Arkobrato Gupta1,2,3, 
+Won Chang7, Dongjun Chung1,2,# 
+1: Department of Biomedical Informatics, The Ohio State University, Columbus, OH, U.S.A. 
+2: Pelotonia Institute for Immuno-Oncology, The James Comprehensive Cancer Center, The 
+Ohio State University, Columbus, OH 43210, USA. 
+3: The Interdisciplinary PhD program in Biostatistics, The Ohio State University, Columbus, 
+Ohio, U.S.A. 
+4: Department of Statistics and Data Science, Yonsei University, Seoul, South Korea 
+5: Department of Applied Statistics, Yonsei University, Seoul, South Korea 
+6: Department of Computer Science and Engineering, The Ohio State University, Columbus, 
+Ohio, U.S.A. 
+7: Division of Statistics and Data Science, University of Cincinnati, Cincinnati, Ohio, U.S.A. 
+*: Joint first authors 
+#: Correspondence ([email protected]) 
+ 
+Abstract 
+ 
+Gene expression profiling technologies have been used in various applications such as cancer 
+biology. The development of gene expression profiling has expanded the scope of target 
+discovery in transcriptomic studies, and each technology produces data with distinct 
+characteristics. In order to guarantee biologically meaningful findings using transcriptomic 
+experiments, it is important to consider various experimental factors in a systematic way through 
+statistical power analysis. In this paper, we review and discuss the power analysis for three types 
+of gene expression profiling technologies from a practical standpoint, including bulk RNA-seq, 
+single-cell RNA-seq, and high-throughput spatial transcriptomics. Specifically, we describe the 
+existing power analysis tools for each research objective for each of the bulk RNA-seq and 
+scRNA-seq experiments, along with recommendations. On the other hand, since there are no 
+power analysis tools for high-throughput spatial transcriptomics at this point, we instead 
+investigate the factors that can influence power analysis. 
+ 
+
+Keywords 
+Transcriptomics, gene expression analysis, power analysis, RNA-seq, scRNA-seq, high-
+throughput spatial transcriptomics 
+ 
+1. Introduction 
+ 
+Transcriptomics refers to either gene expression profiling or the study of the transcriptome 
+using gene expression profiling technologies, where transcriptome refers to the collection of all 
+the ribonucleic acid (RNA) molecules expressed in a cell, cell type, or organism [1]. According to 
+the central dogma, RNA transcripts are generated by the cellular transcription process, play a role 
+in protein-coding, and connect the genome, proteome, and cellular phenotype [2]. Therefore, as 
+a proxy for proteome analysis, numerous transcriptomic studies have analyzed messenger RNA 
+(mRNA) molecules encoding proteins [3]. In addition, transcriptomic approaches have contributed 
+to the advancement of various biological and medical studies, such as cancer biology by 
+identifying possible prognostic biomarkers [4]. 
+Transcriptomic studies can be categorized by underlying gene expression profiling technology, 
+and technological advancements have increased the scope of target discovery. Figure 1 provides 
+a summary of three types of gene expression profiling technologies in terms of their profiling 
+resolution, data structure, and potential target discoveries. Hong et al. [4] illustrate the evolution 
+of RNA sequencing technology. Unlike microarrays, which profile predefined transcript through 
+hybridization, bulk RNA sequencing (bulk RNA-seq) allows genome-wide analysis across the 
+whole transcriptome within a cell population by employing next-generation sequencing (NGS) 
+technology [5]. In contrast to bulk RNA-seq, single-cell RNA sequencing (scRNA-seq) enables 
+the comparison of the transcriptomes of individual cells and the analysis of heterogeneity within 
+a cell population [3]. The high-throughput spatial transcriptomics (HST) technology permits gene 
+expression profiles at the cell or close-to-cell level while also preserving spatial tissue context 
+information [6]. We note that the characteristics of the transcriptomic data are contingent on the 
+underlying technology. Bulk RNA-seq data are highly reproducible, indicating that technical 
+replicates display minimal systemic changes and are thus unnecessary [7]. Bacher and 
+Kendziorski [8] demonstrate that scRNA-seq data has a greater proportion of zeros, more 
+variability, and a more complex distribution than bulk RNA-seq data. 
+ 
+
+ 
+Figure 1: Comparison of bulk RNA-seq, single-cell RNA-seq, and high-throughput spatial transcriptomics 
+technologies in terms of the profiling resolution (level), data structure, and target discoveries. 
+ 
+When designing a transcriptomic experiment, it is crucial to determine the experimental 
+factors, such as the number of biological replicates, the number of cells and sequencing depth, 
+to guarantee sufficient power. In the statistical framework, power refers to the probability of 
+detecting target discoveries, also known as sensitivity. In bulk RNA-seq analysis, Schurch et al. 
+[9] provided an empirical guideline for the number of biological replicates to guarantee sufficient 
+power, and Liu et al. [10] demonstrated that the number of biological replicates has a greater 
+influence on power than sequencing depth. Pollen, et al. [11] demonstrated that low-coverage 
+scRNA-seq is sufficient for cell-type classification. Despite the existence of basic guidelines, there 
+exists no unifying rule due to the complexity of power. For example, biological factors of the 
+experimental unit, such as sex and breeding type, may impact power and should be considered 
+when selecting experimental parameters more systematically. 
+Therefore, to determine experimental factors in transcriptomic experiments in a systematic 
+way, a power analysis can be conducted. Cohen [12] pioneered the concept of power analysis, 
+which refers to the examination of the relationship between power and all parameters influencing 
+power. The parameters include desired error rate and size of the experimental effect of interest 
+(effect size). In practice, power analysis aims to identify a parameter under the assumption that 
+all other parameters remain constant, with power itself being considered a parameter. In power 
+analysis, sample size or power itself is a common target parameter [13]. In this review paper, the 
+sample size refers to either the number of biological replicates or the number of cells. In an 
+experimental study, power analysis provides crucial information at each stage of the experiment. 
+
+Bulk RNA-Seq
+Single-cell RNA-seq
+High-throughputSpatial Transcriptomics
+Samples
+Bulk Expression Profile
+Single-cell
+Single-cell/spot
+Cell/Spot Coordinates
+Level
+Cell/Spot
+Sample
+Data
+Cell/SpotxGeneExpressionCountData
+SubjectxGeneExpression
+CellxGeneExpressionCountData
+Structure
+CountData
+Cell/Spot2-dimensionalCoordinates
+SpatiallyVariableGenes
+DifferentiallyExpressedGenes
+Detection
+DifferentiallyExpressedGenes
+TissueArchitecture
+Target
+Cell Sub-populations
+Cell-CellCommunicationBefore the study, prospective power analysis helps determine the experimental factors that will 
+provide sufficient power for detecting target discoveries. Researchers can conduct a retrospective 
+power analysis to evaluate the experiment, despite differing opinions regarding how to use the 
+collected data for the power analysis, as discussed in Thomas [14]. 
+Power analysis varies according to the underlying objectives of the study and how the 
+data will be analyzed to achieve the research objective [15]. As previously discussed, the 
+employed technology affects the scope of target discoveries and transcriptomic data 
+characteristics. In this context, the power analysis for three distinct transcriptomic technologies 
+will be examined, including bulk RNA-seq, scRNA-seq, and HST technologies. From Sections 2 
+through 4, each transcriptomic technology is covered in a separate section. For a given 
+technology, we examine the power analysis for transcriptomic experiments with respect to 
+experimental factors, research objectives, and explanations of existing power analysis tools. If 
+there are power analysis tools for a particular technology and research objective, we provide 
+recommendations. 
+ 
+2. Power analysis for bulk RNA-seq experiments 
+ 
+2.1 Bulk RNA-seq experiment 
+ 
+Sequencing technologies originate from Sanger sequencing, first introduced by Sanger et 
+al. [16]. In 2005, the introduction of Next-Generation Sequencing (NGS), also known as massively 
+parallel sequencing, improved sequencing in terms of high throughput, scalability, and speed. 
+Especially, NGS technology enables the bulk RNA-seq profiling of gene expression levels in over 
+ten thousand genes simultaneously in a specific tissue or cell population, where the gene 
+expression is characterized by an abundance of messenger RNA (mRNA). Typical bulk RNA-seq 
+protocol includes sample preparation, mRNA fragmentation, reverse transcription to 
+complementary DNA (cDNA), and mapping of cDNA fragments to a reference genome. A gene's 
+expression level is ultimately determined by counting the cDNA fragments, called reads, that are 
+mapped to the gene. See Stark et al. [17] and Van den Berge et al. [18] for more details. 
+Sequencing depth is defined as the total number of reads, influencing the sequencing's technical 
+precision [19]. The bulk RNA-seq profiling platforms include Illumina's HiSeq and MiSeq and ABI's 
+SOLID. Hong et al. [4] illustrate the RNA sequencing technological evolution over time and in-
+depth explanations of the related platforms. 
+Bulk RNA-seq transcriptomic experiments typically aim to identify differentially expressed 
+genes (DEGs) across various experimental conditions, where multiple biological replicates are 
+
+expected in each condition. DEGs are the bulk RNA-seq experiment’s detection target, with their 
+detection probability determining the associated power. Specifically, the power of the bulk RNA-
+seq gene expression analysis is defined by the expected proportion of DEGs detected among all 
+DEGs, following a prespecified statistical procedure. Unlike conventional microarray technology 
+that generates continuous data, bulk RNA-seq generates count data. Due to the discrete nature, 
+the Poisson distribution was originally employed to model the bulk RNA-seq data. However, due 
+to its one-parameter nature, the Poisson distribution cannot account for extra-biological variation 
+in bulk RNA-seq data. Therefore, the negative binomial (NB) distribution, which can be viewed as 
+a Poisson-gamma mixture, has gained popularity. Under a model assumption, a DEG is 
+characterized as a gene whose mean expression ratio (i.e., fold change) deviates from 1 for any 
+pair of experimental conditions. The difference or ratio can be understood as a measure of the 
+effect size that characterizes DEGs. Bioconductor packages of edgeR [20], DESeq [21], DESeq2 
+[22], and baySeq [23] employ the NB model to identify DEGs. While NB-based methods generally 
+have a higher detection power, there are also reports indicating its FDR inflation [24,25] due to 
+ignoring the uncertainty of the estimated dispersion parameters [26]. Alternatively, the voom 
+method [27] can be used to detect DEGs by applying normal-based theory to the log-transformed 
+count data, which is implemented in the limma Bioconductor package. Even though count data is 
+not directly modeled, the voom method adjusts heterogeneous variances across all observations 
+concurrently by utilizing an adequate mean and variance relationship. Additional software tools 
+for DEG analysis are described in Schurch et al. [9] and Stark et al. [17]. 
+In the case of a bulk RNA-seq experiment, it is essential to determine the number of 
+biological replicates that will provide sufficient DEG detection power, a type of power analysis. 
+Consider the factors that may affect the power. Note that the power depends on the assumed 
+model's parameters and the software tools that provide the p-value for each gene under 
+consideration. Additionally, the power is affected by the considered error rate and the target level. 
+Bulk RNA-seq gene expression analysis typically considers multiple genes. When multiple genes 
+are simultaneously inferred, it is common to control the false discovery rate (FDR) rather than the 
+type 1 error rate, which is appropriate for inferring a single gene. By controlling FDR, it is possible 
+to regulate the proportion of non-DEGs among genes declared to be DEGs on average. 
+Consequently, when inferring multiple genes and conducting power analysis, it is necessary to 
+consider the target FDR level. 
+ 
+ 
+ 
+
+2.2 Bulk RNA-seq power analysis tools 
+ 
+Numerous power analysis software tools calculating the number of biological replicates, 
+alternatively sample size, for bulk RNA-seq experiments have been developed according to the 
+factors affecting the power: model assumptions, the testing type employed for each gene, and 
+desired error rates to be controlled. Model parameters are often estimated using pilot data, and 
+some tools provide stored data for this purpose. As demonstrated by data analysis in Poplawski 
+and Binder [28], if the stored data are utilized carelessly, a highly inappropriate sample size can 
+be suggested. In addition to sample size, some software tools consider sequencing depth to be 
+an experimental factor that influences the power to be chosen during experimental design. Liu et 
+al. [10] demonstrated the tradeoff between biological replicates and sequencing depth in the 
+context of statistical power. 
+Hart et al. [19] suggested a flexible power analysis approach that calculates the sample 
+size for a single gene expression analysis using the NB model, which is implemented in the 
+‘RNASeqPower’ Bioconductor package. Due to the asymptotic normality of the score test statistic, 
+a closed-form power function is obtained as a function of all possible parameters, including sample 
+size, fold change, average sequencing depth, target type 1 error rate, and coefficient of variation. 
+Due to the simplicity of the inference situation and the closed-form power function, it is possible 
+to perceive the relationship between all parameters affecting the detection power. Hart et al. [19] 
+also suggested a sequencing depth motivated by the parameters' relationship and demonstrated 
+that although the method does not assume FDR control, it can be extended to multiple gene 
+inference by setting the p-value threshold α to a small value, such as 0.001. 
+Li et al. [29] proposed a tool for calculating sample size based on the NB model and FDR 
+control via a gene-specific power function. The approach is effectively implemented in the 
+‘RnaSeqSampleSize’ Bioconductor package, with an additional parameter estimation procedure 
+supported by data. However, the ‘RnaSeqSampleSize’ tool tends to overestimate sample size in 
+the data analysis and data-based simulation study of Poplawski and Binder [28]. To overcome 
+this overestimation, Bi and Liu [30] suggested a method that assumes the NB model but uses the 
+normal-based test statistic via the voom method to assess the power function partially analytically, 
+implemented in the ‘ssizeRNA’ R package. According to the data-driven simulation study of 
+Poplawski and Binder [28], this approach is faster and provides the sample size closer to the 
+actual number required to achieve the desired power, compared to other approaches. Additionally, 
+Wu et al. [31] proposed a simulation-based FDR controlling approach, implemented in the 
+‘PROPER’ tool. Table 1 provides a summary of the information from different power analysis tools. 
+
+The tools are chosen from the methods with relevant literature described in Poplawski and Binder 
+[28]. 
+ 
+Table 1: A table shows six software tools for statistical power analysis for bulk RNA-seq 
+experiments. Each tool is presented along with the citation and the software environments that 
+have been implemented.  
+ 
+Tool Name [Citation] (Implementation) 
+Pilot Data 
+Pilot Data with Stored Data 
+Type 1 
+Error 
+Poisson 
+Lognormal 
+- 
+‘Scotty’  
+ [32] (Web Interface) 
+ 
+ 
+ 
+ 
+Negative 
+Binomial 
+‘RNASeqPower’  
+ [19] (R package)  
+ 
+- 
+FDR 
+‘ssizeRNA’  
+ [30] (R package) 
+‘RnaSeqSampleSize’  
+ [33] (R package) 
+ 
+‘RNASeqPowerCalculator’  
+ [34] (R package) 
+‘PROPER’ [31] (R package) 
+ 
+2.3 Bulk RNA-seq power analysis tool recommendation 
+ 
+The ‘ssizeRNA’ R package was chosen based on the outcomes of two simulation studies 
+of Poplawski and Binder [28] and Bi and Liu [30]. From the six power analysis tools mentioned in 
+Table 1, we first considered ‘RnaSeqSampleSize’, ‘ssizeRNA’, and ‘PROPER’ based on their 
+FDR-targeting nature and focus on a single DEG analysis tool. However, depending on the 
+performance of the simulation studies, we decided to exclude ‘RnaSeqSampleSize’ from 
+consideration. Specifically, according to Poplawski and Binder [28], ‘RnaSeqSampleSize’ typically 
+recommends a very large sample size. ‘RnaSeqSampleSize’ performs well in Bi and Liu [30] when 
+the model is simple, and gene-specific parameters are absent. When the simulation model 
+became realistic, the sample size suggested by ‘RnaSeqSampleSize’ was either too large to 
+significantly exceed the desired power or too small to adequately regulate power. The subsequent 
+selection was based on speed. The simulation results presented in both papers indicate that both 
+the ‘PROPER’ and ‘ssizeRNA’ tools recommend sample sizes with target power levels. Due to 
+the conservative nature of the voom method, the ‘ssizeRNA’ tool typically recommends a few 
+
+more samples. In terms of usability, however, we recommend the ‘ssizeRNA’ tool, which is faster 
+due to its analytical nature.  
+ 
+3. Power analysis for single-cell RNA-seq (scRNA-seq) experiments 
+ 
+scRNA-seq technologies have revolutionized the study of transcriptomics by profiling 
+genome-wide gene expression at the individual cell level. The cell-level information provides 
+unprecedented opportunities for studying cellular heterogeneity and expands our understanding 
+of developmental biology [3]. Even though the context of a single-cell transcriptomic study differs 
+from that of a bulk transcriptomic study, DEG detection remains a fascinating study area. In 
+addition, the cell information enables researchers to answer questions about cell subpopulations. 
+The relevant power analysis has been developed in response to the distinct research questions. 
+In general, the sample size in scRNA-seq experiments refers to the number of cells. Due to the 
+additional technical steps required to distinguish cells, scRNA-seq data contain more zeros than 
+bulk RNA-seq data [11], and a zero-inflated model is frequently employed when developing 
+statistical approaches [35]. Sections 3.1 and 3.2 discuss power analysis for identifying cell sub-
+populations and detecting DEGs, respectively, for scRNA-seq experiments. The information 
+presented in Table 2 outlines a variety of power analysis tools applicable to single-cell 
+transcriptomic experiments with distinct research questions. 
+ 
+3.1 Power analysis for cell subpopulation detection 
+ 
+Unlike bulk RNA-seq experiments, scRNA-seq experiments frequently attempt to identify 
+the characteristics underlying cell subpopulations. A cell subpopulation refers to a group of cells 
+determined by various cell types, states, or subclones. Bulk RNA-seq data does not allow cell 
+subpopulation-level investigation, especially for rare cell subpopulations. In contrast, the scRNA-
+seq data provides the cell subpopulation-level resolution [36]. The research questions and 
+associated power analysis can be further divided into two categories, depending on whether 
+scRNA-seq experiments examine the proportion of cell subpopulations within a single tissue 
+(Section 3.1.1) or the proportional differences across experimental conditions for a given cell 
+subpopulation (Section 3.1.2). 
+
+Table 2: A table with information about different software tools for scRNA-seq power analysis with two distinct detection targets. 
+Experimental Factors: Number of cells (1), Number of individuals (2), Sequencing depth (3). 
+ 
+ 
+Detection 
+Target 
+# of 
+Samples 
+Tool Name 
+Experimental 
+Factor 
+Software 
+Model 
+Power 
+Assessment 
+Cell sub- 
+population 
+Single sample 
+‘SCOPIT’ [37] 
+(1) 
+R package & 
+Web application 
+Multinomial 
+Analytical 
+'howmanycells' 
+Web application 
+Negative Binomial 
+Multi sample 
+‘Sensei‘ [38] 
+(1) , (2) 
+Beta Binomial 
+‘scPOST’ [39] 
+R package 
+Linear mixed model 
+Simulation- 
+based 
+DEG 
+‘scPower’ [43] 
+(1), (2), (3) 
+R package & 
+Web server 
+Negative Binomial 
+Pseudobulk 
+‘hierarchicell’ [41] 
+R package 
+Simulation- 
+based 
+Single sample 
+‘powsimR’ [40] 
+(1) 
+‘POWSC’ [42] 
+(1), (3) 
+A mixture of zero-inflated 
+Poisson and log-normal 
+Poisson distributions 
+‘scDesign’ [44] 
+Gamma-Normal 
+mixture model 
+
+3.1.1 Ascertaining cell subpopulation proportions in a single tissue 
+ 
+Multiple cell types in varying proportions compose a biological tissue. In the experimental 
+design phase, power analysis is indispensable for ensuring that enough cells are sampled to 
+adequately represent both normal and rare cell types. The following sections discuss the power 
+analysis for sufficient cell numbers (sample size) in a single tissue.  
+Two software tools, 'howmanycells' (https://satijalab.org/howmanycells) and ‘SCOPIT’ 
+[37], were developed specifically for cell number calculation. Using statistical models, they both 
+approached the problem by calculating the probability of sampling at least a predetermined 
+number of cells from each subpopulation. The 'howmanycells' function uses the NB distribution 
+to estimate the total number of cells required for adequate representation of a given cell 
+subpopulation under the assumption that the number of cells in each cell type is statistically 
+independent. This assumption may not hold in practice, but the results can be used to determine 
+the required minimum sample size. On the other hand, 'SCOPIT' employs the Dirichlet-
+multinomial model for the distribution of the number of cells from each subpopulation, which more 
+accurately reflects the constraint on the proportion of cell subpopulation (i.e., proportions sum to 
+one).  
+Both 'howmanycells' and 'SCOPIT' are comparable in that they use analytical approaches, 
+identify the proportion of the rarest cell type as the most significant statistical factor affecting power, 
+and offer lightweight web applications to facilitate quick and intuitive power calculation. Above all, 
+their estimates of the required sample size are comparable in general. An important distinction is 
+that 'SCOPIT' permits retrospective analysis for hypothetical experiments, i.e., determining how 
+many cells would be required based on the number of sequenced cells, the number of 
+subpopulations detected, and their frequencies. In addition, 'SCOPIT' reports Bayesian credible 
+intervals for the estimated probability and number of cells to account for the uncertainty 
+associated with the observed empirical subpopulation frequencies.  
+The methods mentioned above only consider the effects of cell subpopulation proportions 
+and total cell number but do not account for technical factors such as sequencing depth. This is 
+partially due to the difficulties in obtaining an analytical solution when other factors are considered. 
+Note that these methods are intended to estimate the total number of cells in a single biological 
+sample to identify subpopulations. When dealing with multiple samples in scRNA-seq 
+experiments, the detection target may change, and different approaches are needed. The 
+following section describes the problem. 
+ 
+
+3.1.2 Ascertaining differential cell subpopulation proportions between distinct 
+experimental conditions 
+ 
+In a multi-sample scRNA-seq experiment, researchers are primarily interested in 
+determining whether a specific cell subpopulation has differential abundances between 
+experimental conditions (e.g., diseased vs. healthy). In this case, the difference in cell 
+subpopulation proportion represents the effect size, and the number of biological samples (such 
+as patients and mice) represents the sample size. The proportion of cell subpopulation, the 
+number of cells, and the number of (biological) samples influence the power. Since cell 
+subpopulations are often identified by comparing marker gene expression levels, sequencing 
+depth may affect power since it influences technical variation. Moreover, since this is a multiple-
+sample experiment, the batch effect may be significant, and the experimental design may be 
+unbalanced. Consequently, batch effect and experimental design (balanced or unbalanced, 
+paired or unpaired) may also impact the power.  
+Two approaches, ‘Sensei‘ [38] and ‘scPOST’ [39], have been developed for the power 
+analysis of distinguishing proportional differences within a cell subpopulation. The former provides 
+an analytical solution after a reasonable approximation, whereas the latter relies on simulation. 
+They both consider the potential impact of the proportion of cell subpopulation (biological factor), 
+the number of cells, and the number of samples (experimental factors), but only 'scPOST' 
+considers the effect of gene expression variation. Both works attempt to explain how to balance 
+the number of biological samples and the number of cells within a limited budget, and both 
+suggest that increasing the sample size yields greater power than increasing the number of cells 
+per sample. In addition, ‘scPOST’ indicates that modest reductions in sequencing depth have 
+negligible effects on power.  
+Specifically, ‘Sensei’ integrates the impacts of the number of cells and the number of 
+biological replicates in a mathematical framework. It models the abundance of cell types using a 
+beta-binomial distribution and estimates the sample size based on Welch's t-test. Under this 
+framework, beta distribution captures the biological difference in cell type abundance between 
+groups, as well as variance among samples within a group, while binomial distribution models the 
+technical variation caused by a limited number of cells. ‘Sensei’ provides a closed-form 
+representation for the statistical power upon reasonable approximation, which makes a 
+lightweight web application possible. As an output, ‘Sensei’ shows a table of false negative rates 
+for each feasible sample size combination.  
+Although 'Sensei' attempted to account for some biological and technical variations, the 
+pursuit of an analytical representation of power necessitates the adoption of assumptions and 
+
+simplifications that may not apply to real data (e.g., assume no batch effect). In contrast, ‘scPOST’ 
+employs a simulation-based method to account for the effects of more factors. It begins by 
+estimating key parameters based on the prototype or pilot data supplied by the user. Specifically, 
+it assumes gene expression variation in principal components (PCs) space that arises from three 
+sources (batch, sample, and residual), and employs linear mixed effects models to decompose 
+the total variance for each PC and each cluster. Both fixed and random effects are extracted from 
+the fitted models, and cluster frequency mean and covariance is estimated from the prototype 
+dataset. In the second step, the previously estimated parameters and user-specified batch and 
+sample effect scale parameters are used in linear mixed effects models to simulate PC 
+coordinates for cells. In the final step, ‘scPOST’ employs a test based on logistic mixed effects 
+models to determine whether the mean frequency of a cluster differs significantly between two 
+conditions. The power is computed as the proportion of simulation runs in which at least one 
+cluster represented differential abundance. 
+ 
+3.2 Power analysis for DEG detection 
+ 
+Identifying DEGs is another important goal of scRNA-seq data analysis. DEG analysis can 
+also be divided into two categories, depending on whether the goal is to identify (i) DEGs across 
+different conditions (e.g., treatment vs. control) for a specific cell type or (ii) DEGs that are 
+differentially expressed across cell types for a given biological sample. Numerous factors can 
+influence power, such as effect size, number of cells, number of biological replicates, sequencing 
+depth, dropout rates, cell subpopulation proportion, and multiple testing methods. Given that so 
+many factors may affect power, it is hard to provide an analytical framework to assess power. 
+Therefore, most of the existing work employs simulation-based approaches, which consist of 
+three key steps: parameter estimation, data simulation, and power evaluation. In the parameter 
+estimation step, important parameters like gene-wise mean and standard deviation are estimated 
+from user-provided data or representative example data based on a data model. In the simulation 
+step, gene expression values are simulated based on the estimated parameters. Finally, in the 
+power evaluation step, existing DEG analysis or detection methods are applied to the simulated 
+data to assess power. The subsequent sections discuss the approaches in detail. 
+ 
+3.2.1 DEGs across different conditions for a cell type 
+ 
+Similar to bulk RNA-seq experiments, a DEG analysis can be performed to identify genes 
+whose expression levels vary significantly between experimental conditions. In scRNA-seq 
+
+experiments, such DEG analysis is often performed for a specific cell type. Four software tools 
+are available for this type of power analysis: ‘powsimR’ [40], ‘hierarchicell’ [41], ‘POWSC’ [42], 
+and ‘scPower’ [43]. ‘powsimR’ and ‘POWSC’ are more suitable for single-sample experiments, 
+while ‘hierarchicell’ and ‘scPower’ are designed for multi-sample experiments. ‘powsimR’ 
+assumes an NB distribution for the count data and emphasizes the mean-dispersion relationship 
+during simulation. The existing package is used for DEG detection, and power-related statistics 
+including FDR and true positive rate (TPR) are calculated to evaluate power based on estimated 
+and simulated expression differences. The ‘hierarchicell’ also assumes an NB distribution for gene 
+expression value, and it highlights the hierarchical structure of scRNA-seq data from multiple 
+individuals. For power evaluation, it implements a two-part hurdle model.  
+‘scPower’ uses an analytical-based approach for this task. The fundamental idea behind 
+‘scPower’ is that a gene needs to be expressed and exceed a significance cutoff to be identified 
+as DEG. Therefore, it decomposes the power as the product of the expression probability 
+(probability of detecting an expressed gene) and the DE power (probability of significantly 
+expressed). For the expression probability, a pseudobulk approach is adopted. Specifically, it 
+sums the expression of a gene over all cells of the cell type of interest within an individual to get 
+the pseudobulk count for that gene. Then it calculates the probability of this pseudobulk count 
+greater than a threshold based on an NB distribution. Based on this probability, the probability 
+that the gene is expressed is obtained from a cumulative binomial distribution. The DE power is 
+calculated analytically based on an NB model using existing tools. 
+ 
+3.2.2 DEGs across different cell types 
+ 
+Identifying genes that are differentially expressed across different cell types under the 
+same experimental condition is another common DEG analysis, aiming to identify genes that 
+could distinguish from one cell type to another. ‘scDesign’ [44] and ‘POWSC’ [42] were developed 
+for the power analysis, and both are simulation-based approaches designed for studies involving 
+a single biological sample. ‘scDesign’ assumes gamma-normal distribution for log-transformed 
+count data. ‘POWSC’ assumes a mixture of zero-inflated Poisson and lognormal-Poisson 
+distributions for the count data. ‘scDesign’ and ‘POWSC’ allow user-supplied data for parameter 
+estimation, while ‘POWSC’ also provides precalculated parameter estimates from various tissue 
+types. The parameters to be estimated for ‘scDesign’ include the cell library size and cell-wise 
+dropout rate, as well as the gene-wise mean, standard deviation, and dropout rate. The 
+parameters to be estimated for ‘POWSC’ include the cell-wise zero inflation point mass and 
+Poisson rate, as well as gene-wise mixture proportion, mean, and variance. In the data simulation 
+
+step, both approaches consider the constraint on total reads and allow users to choose the 
+number of cells, and sequencing depths under the constraint. Therefore, they can provide insights 
+regarding how to optimize the tradeoffs between these two experimental factors. ‘scDesign’ 
+performs DEG analysis using a two-sample t-test and reports five power-related measures. On 
+the other hand, ‘POWSC’ utilizes existing DEG analysis tools and reports both stratified and 
+marginal power. 
+ 
+3.3 scRNA-seq power analysis tool recommendations 
+ 
+As illustrated in Table 2, for the scRNA-seq experiments, a unique set of software tools 
+for power analysis has been developed for a specific research objective. Specifically, the tools' 
+distinctive features include the factors considered and the data models. Therefore, users should 
+consider the previously stated distinctive features when selecting an appropriate power analysis 
+tool. Here, we make recommendations based on these considerations.  
+First, the 'SCOPIT' tool is recommended when detecting cell subpopulations is the 
+purpose of the research. In this case, one can choose between the 'howmanycells' and 'SCOPIT'. 
+Both offer lightweight web applications to facilitate fast and intuitive power calculations, and their 
+estimates for the required number of cells are nearly identical. However, we recommend 'SCOPIT' 
+for this research purpose given its more comprehensive and kinder documentation.  
+Second, when the differential proportion of cell subpopulations is the main goal of the 
+research, one can choose between 'Sensei' and 'scPOST'. 'Sensei' provides a lightweight web 
+application that is quick and intuitive. However, 'scPOST' allows considering more factors because 
+it is a simulation-based method. If users desire a quick and approximate estimate of the number 
+of cells, 'Sensei' is a suitable option. On the other hand, 'scPOST' may be preferred if users wish 
+to consider various experimental and biological factors, such as the batch effect and gene 
+expression variation, in the statistical power analysis.  
+Third, 'scPower' and 'hierarchicell' are available tools for power analysis if researchers 
+wish to identify the genes whose expression levels differ under different experimental conditions 
+within a particular cell type, and multiple biological samples are involved. Between these two tools, 
+we recommend 'scPower' over 'hierarchicell' due to its user-friendly web application. Likewise, 
+'POWSC' and 'powsimR' can accomplish the task with a single sample. Between these two tools, 
+we recommend 'POWSC' over 'powsimR' because of the richer documentation for 'POWSC'.  
+Finally, if the genes characterizing one cell type from another are the primary objective, then 
+'scDesign' and 'POWSC' can assist. They address the restriction on total sequencing depth and 
+the zero-inflation issue, although they employ different data models. Between these two tools, we 
+
+recommend 'POWSC' over 'scDesign' because 'POWSC' also reports the stratified power, i.e., 
+stratified based on gene expression level or zero fractions, which makes more sense given that 
+power depends on these two factors.  
+ 
+4. Power analysis for spatial transcriptomic experiments 
+ 
+4.1 Introduction of high-throughput spatial transcriptomics (HST) technology 
+ 
+The lack of spatial information has limited the scope of scRNA-seq data analysis. 
+Technological advancements in HST have made it possible to collect gene expression data along 
+with spatial coordinates. HST technology enables gene expression profiling while preserving the 
+spatial location (coordinate) of each observational unit, depicted in Figure 1. The observational 
+unit can be a cell or a group of cells (spot). There are two main categories of technological 
+variations of HST technology: imaging-based and sequencing-based. seqFISH+ [45] and 
+MERFISH [46] are representative technologies for generating imaging-based HST data with a cell 
+as the observational unit. Due to its probe hybridization-based gene detection, imaging-based 
+HST data can only observe a limited number of genes. 10X Visium [47] is a standard technology 
+for generating sequencing-based HST data with a spot as the observational unit. Since 
+sequencing-based technology employs NGS technology, there are fewer restrictions on the 
+number of genes compared to imaging-based technology. Accordingly, there is currently a 
+technological trade-off between cell resolution and the number (dimension) of genes. For 
+instance, imaging-based HST data can be described as high-resolution and low-dimensional data, 
+while sequencing-based HST data can be considered as low-resolution and high-dimensional 
+data. Note that the spatial information from various HST data types is derived from distinct 
+observational units (cells and spots), which affects the type of inferences we can make. For 
+example, image-based HST data would be more suitable for statistical inferences requiring cell-
+level resolution. 
+As illustrated in Figure 2, researchers can answer multiple research questions using the 
+HST data, including spatially variable gene (SVG) detection, tissue architecture identification, and 
+cell-cell communication prediction. Answering these research questions requires understanding 
+how to incorporate spatial information into a model to define the SVGs, tissue architecture, and 
+cellular phenotype. First, the SVG detection method determines which genes exhibit spatial 
+patterns within the target tissue, where examples include spatialDE [48], SPARK [49], and 
+Trendsceek [50]. spatialDE and SPARK utilize the Gaussian random effect model and the 
+Poisson log-normal model, respectively, with distinct normalization strategies. On the other hand, 
+
+the Trendsceek approach detects spatial variation using a nonparametric approach. Second, the 
+main goal of tissue architecture identification is to group the observational units (i.e., cells or spots) 
+into biologically distinct clusters. Before the advent of HST technologies, previous studies 
+employed clustering based only on the gene expression data [51,52]. Now, additional spatial 
+information available in the HST data allows one to also consider the proximity between cells to 
+improve such clustering. Gitto [53], BayesSpace [54], and SPRUCE [55] are examples of models 
+employing spatial associations between observational units to identify clustering patterns. Third, 
+cell-cell communication analysis is to predict interactions between cells. The spatial closeness or 
+adjacency can provide important information to improve this type of analysis because spatially 
+closer cells are more likely to interact with each other. Previously, with the absence of spatial 
+information, interactions between ligands and receptors were predicted only based on their gene 
+expression patterns [56,57]. For example, CellChat [58] estimates the interaction between ligands 
+and receptors based on the latent distance between cells, which is calculated solely based on 
+gene expression data. This does not reflect the fact that cells located nearby are more likely to 
+interact with each other; incorporating such information can lead to higher accuracy. 
+ 
+Figure 2: The figure depicts three representative research questions for the analysis of HST data. SVG 
+denotes the identification of a gene with a spatial pattern of gene expression. Tissue architecture refers to 
+the identification of a tissue's structure through the clustering of similar gene expression patterns. Cell-cell 
+communication, on the other hand, detects the interaction between cells using their spatial information and 
+gene expression data. 
+
+SpatiallyVariableGene(SVG)
+Single-cell/spot
+20
+Expression
+Gene
+10
+TissueArchitecture
+5
+A
+BC
+D
+Cell/Spot
+sub-population2
+Cell/SpotCoordinates
+uojiendod-qns
+Cell-CellCommunication 
+Given the coordinates from each observation in HST data, the spatial patterns are 
+modeled through the distances among observations. We note that the optimal approach to 
+calculate the distances among observations can be different for different data type. Figure 3 
+illustrates how the imaging-based and sequence-based HST data can be regarded as different 
+types of spatial data. First, one can consider the imaging-based HST data as geostatistical data 
+or spatial point process data. Here, geostatistical data follows a spatial process that varies 
+continuously, but observed only at discrete points (coordinates). By using the coordinate 
+information, we can define the distance (e.g., Euclidean distance) among cells. The existing 
+models, including spatialDE and SPARK, define the spatial closeness by calculating the distances 
+among cell coordinates. On the other hand, the sequencing-based HST data can be thought of 
+as lattice or areal data observed at the discrete points or spots on a regular or irregular grid. In 
+the lattice data structure, the neighborhood is defined by the adjacency on the grid and the 
+distance between two spots is measured by the least number of spots that need to be visited 
+while moving from one spot to the other on the lattice. 
+ 
+Figure 3: Depending on the type of HST data, it can be considered as either point process data or areal 
+data. First, imaging-based HST data can be regarded as point process data. For example, cell locations 
+are analogous to the spatial coordinates of birds’ habitats in the US. Its spatial information is modeled 
+through the distance among habitats. Sequencing-based HST data, on the other hand, can be regarded as 
+areal data on a regular grid. Here the spot, which is a group of cells, can be compared to the states' 
+aggregated bird habitats. Its spatial information is modeled through the adjacency or neighborhood 
+structure. 
+ 
+
+Imaging-basedHST
+Sequencing-based HsT
+PointProcessData
+ArealDataAs shown in Figure 4, there are several key experimental factors that can affect the 
+generation of spatial features in HST data, including the choice of tissue area, size of the fields of 
+view (FoVs), the number of FoVs, and the number of cells or spots, where FoVs are defined as 
+the region on a tissue captured by an HST experiment. Note that such selection of FoVs and 
+tissue area is needed as it is often not possible to capture the whole tissue using the HST 
+experiment. These experimental factors can affect capturing transcripts at a specific location on 
+a tissue [59] or lead to a different context for capturing the region of interest, e.g., building a 
+neighborhood network [60]. Hence, the power analysis for HST data needs to take these 
+experimental factors into account to estimate the minimum number of samples to achieve a 
+specific analysis goal using HST data. First, the size of FoVs determines how large we measure 
+spatial features and gene expression locally (i.e., local capture efficiency). On the other hand, the 
+number of FoVs affects how many different regions on a tissue we check on a tissue (i.e., global 
+capture efficiency). Second, because these FoVs are not qualitatively and biologically identical, it 
+also matters where we capture on the tissue. For example, for the tissue architecture identification, 
+one might want to include the regions that contain interesting and/or rare cell sub-populations. 
+Likewise, for the cell-cell communication prediction, one might hope that the regions with active 
+cell-cell interactions are included in our HST data. Third, because the number of cells and spots 
+can affect signal-to-noise ratios of the generated HST data, one needs to make sure that sufficient 
+cells and spots are captured to avoid potential analytical and computational issues. In summary, 
+a rigorous experimental design that systematically considers these experimental factors will 
+facilitate the effective use of resources (e.g., experimental cost) by improving efficiency in 
+capturing the spatial features with gene expression data. 
+ 
+
+ 
+Figure 4: Key experimental factors in designing HST experiments include: (1) the choice of tissue area, (2) 
+the number and sizes of fields of view (FoVs), and (3) the number of cells and spots. These experimental 
+factors can affect the statistical power to achieve the research goals, e.g., those mentioned in Figure 2. 
+For example, the choice of tissue area, along with the number and sizes of FoVs, can determine the degree 
+that biological aspects of our interest (e.g., interesting cell sub-populations, or cell-cell communications) 
+are captured in the generated HST data. Likewise, the number of cells and spots can affect the signal-to-
+noise ratios (effect sizes) of the generated HST data. 
+ 
+4.2 Literature reviews of power analysis for HST data 
+ 
+Recently, Bost et al. [61] implemented several experiments to figure out how the number 
+of FoVs and their widths affect the coverage of the true clusters in a tissue. By changing the 
+number and the size of FoVs, they examined the ratio of the number of covered clusters to the 
+true number of clusters. It was the first attempt to investigate how the experimental design affects 
+the HST data analysis. For example, they calculated the required number of FoVs to discover the 
+true clusters in the cell phenotype and compared it between tumor samples and healthy samples. 
+The result showed that a larger number of FoVs are needed to capture the true clusters in tumor 
+samples compared to healthy samples, likely because of the complex and heterogeneous tissue 
+
+TissueArea
+ExperimentalFactor
+NumberofFoVsandSizeofFoVs
+NumberofCellsandSpotsstructure generated through tumorigenesis. They also applied this experiment to real data on 
+heart disease and breast cancer. They concluded that different types of data, such as human 
+body and animal tissue, have different required numbers and sizes of FoVs to recover the true 
+clusters. Moreover, the technologies of generating the HST data also affect the relationship 
+between the identification of cell clustering and the number and size of FoVs. However, the 
+investigation of Bost et al. [61] is limited in the sense that it was based on an empirical equation 
+that was not justified by any statistical model or machine learning model. Moreover, its ratio of 
+discovering the true cluster is not the power to discover the true clusters, whose computation 
+requires a large number of iterations. 
+In contrast to Bost et al. [61], which used an empirical equation to calculate the ratio of 
+covering true clusters, Baker et al. [62] employed a simulated HST approach to investigate the 
+design of HST experiments. Here, they performed a spatial power analysis experiment with their 
+devised HST data generation, called "in silico” approach. Using the in silico approach, they 
+generated various types of HST data as spatial profiling data such as cells in random states or 
+cells in self-preference states to proceed with an exploratory computational framework. They 
+pointed out three experimental factors to be considered in calculating the power: the number of 
+cells, the number of FoVs, and the size of FoVs. They applied their approach to two analytical 
+tasks, including cell type discovery (tissue architecture identification) and cell-cell communication. 
+Based on these simulation strategies, they used statistical models such as the Gamma-Poisson 
+model to predict how many FoVs are required to discover the cell types or cell interactions. 
+Through their simulation studies, they discovered that the size of FoVs and the number of FoVs 
+impacted the statistical power. First, in cell type discovery, they concluded that the nature of tissue 
+structure affects the required number of cells and FoVs to discover the true cell types. They 
+demonstrated this by applying the power analysis model to unstructured data of human breast 
+cancer, highly ordered and heterogeneous data from the mouse brain, and complex and 
+recurrently structured data from the mouse spleen. Second, for the cell-cell communication task, 
+they argued that the interactions among the cells might not be captured with the insufficient FoV 
+size. However, the investigation of Baker et al. [62] also has multiple limitations. First, it is hard to 
+directly apply their approach to point-referenced data (point process data). Specifically, the 
+simulation data generation model ("in silico”) is based on the blank tissue scaffold where the 
+random circle packing forms a planar graph, which requires strong prior knowledge for cluster 
+labels. This cannot capture all the variations in point reference data whose spatial locations are 
+randomly distributed, and the resulting pattern often exhibits non-trivial microscale variation. 
+Second, their investigation was limited to the number and sizes of FoVs while they ignored other 
+
+important experimental factors that can affect the statistical power, e.g., the choice of tissue area 
+and the number of cells/spots mentioned in Figure 4. In summary, at this point, the optimal 
+strategies for statistical power analysis for HST experiments remain to be explored. 
+ 
+5. Conclusions 
+ 
+The advancement of transcriptomic technology has allowed researchers to expand their 
+scope of questioning. In order to guarantee biologically meaningful findings, rigorous experimental 
+design is critical, including statistical power analysis that carefully considers research questions 
+and data characteristics. In this review paper, we investigated the power analysis for three distinct 
+types of transcriptomic technologies from a practical standpoint. First, in the case of the bulk RNA-
+seq experiment, the primary objective is to identify DEGs and we recommend the R package 
+‘ssizeRNA’ as a tool for power analysis. Second, in the case of the scRNA-seq experiment, two 
+main analytical goals are cell subpopulation identification and DEG detection. Specifically, 
+regarding cell subpopulation detection, we recommend ‘SCOPIT’ for detecting cell 
+subpopulations and ‘scPOST’ for inferring proportional differences across cell subpopulations. 
+Regarding DEG detection, we recommend ‘scPower’ for DEG detection across multiple cell sub-
+populations using multiple samples, and ‘POWSC’ for DEG detection across cell sub-populations 
+with a single sample and within a cell subpopulation under varying experimental conditions. Third, 
+in the case of the HST experiment, its power analysis framework is still under-developed and we 
+highlight key aspects that need to be considered for the power analysis framework of HST 
+experiments, including research questions (SVG, tissue architecture, cell-cell communications), 
+technological variations (imaging- and sequencing-based HST), and experimental factors (tissue 
+area, the number and size of FoVs, and the number of cells or spots). We believe that this review 
+paper can be a useful guideline for the future design and statistical power analysis of 
+transcriptomic experiments. 
+ 
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+a toolbox for integrative analysis and visualization of spatial expression data. Genome Biol 2021, 22, 78, 
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+ 
+ 
+ 
+

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+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
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+1 
+IAC-22-B3.7.5 
+ 
+Categorisation of future applications for Augmented Reality in human lunar exploration 
+ 
+Paul Topf Aguiar de Medeirosa, Paul Njayoub, Flavie A. A. S. D. T. Rometschc, Dr. Tommy Nilssond, Leonie 
+Beckere, Dr. Aidan Cowleyf 
+  
+a European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+b European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+c European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+d European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+e European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+f European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany 
+[email protected] 
+ 
+Abstract 
+The European Space Agency (ESA) has a clear mission to go forward to the Moon in preparation of human 
+presence on Mars. One of the technologies looked at to increase safety and efficiency of astronauts in this context 
+is Augmented Reality (AR). This technology allows digital visual information to be overlaid onto the user's 
+environment through some type of display or projector. In recent years separate studies have been conducted to 
+test the potential value of AR for astronauts by implementing a few functionalities on an AR display followed by 
+testing in terrestrial analogue environments. One of the groups contributing to these investigations is Spaceship 
+EAC (SSEAC). SSEAC is a group of interns and trainees at the European Astronaut Centre (EAC) focusing on 
+emerging technologies for human space exploration. 
+This paper presents an outcome of SSEAC's activities related to AR for lunar extravehicular activities (EVAs), in 
+which an approach similar to design thinking was used to explore, identify, and structure the opportunities offered 
+by this technology. The resulting categorization of AR use cases can be used to identify new functionalities to test 
+through prototyping and usability tests and can also be used to relate individual studies to each other to gain insight 
+into the overall potential value AR has to offer to human lunar exploration. 
+The approach adopted in this paper is based on the Fuzzy Front End (FFE) model from the innovation management 
+domain. Utilising a user-driven instead of technology-driven method resulted in findings that are relevant 
+irrespective of the hardware and software implementation. Instead, the outcome is an overview of use cases in 
+which some type of AR system could provide value by contributing to increased astronaut safety, efficiency and/or 
+efficacy. 
+ An initial overview of AR functions for lunar EVAs was created based on existing literature. These were 
+expanded on through a multidisciplinary brainstorm within SSEAC. A subsequent clustering activity resulted in 
+a categorisation of potential AR applications.  
+The following categories were defined: EVA navigation, Scientific measurements and observations, Sample 
+Collection, Maintenance, Repair, Overhaul (MRO) and Construction, Logistics and Inventory Management, 
+Medical Procedures, Biomedical and System Status Monitoring, Collaboration and Support. 
+ 
+Keywords:  Augmented Reality, use case classification, user centred design, Fuzzy Front End, lunar exploration, 
+astronaut systems
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+2 
+Acronyms / abbreviations 
+AR 
+ 
+Augmented Reality 
+COTS  
+Commercial Off The Shelf 
+ESA 
+ 
+European Space Agency 
+EVA 
+ 
+Extravehicular Activity 
+FFE 
+ 
+Fuzzy Front end 
+SLS 
+ 
+Space Launch System 
+HUD 
+ 
+Heads Up Display 
+ISS 
+ 
+International Space Station 
+MRO 
+ 
+Maintenance, Repair, Overhaul 
+NASA 
+National Aeronautics and Space 
+Administration 
+xEMU 
+eXploration Extravehicular 
+Mobility Unit, 
+ 
+1. Introduction 
+The international aerospace community is once again 
+preparing for the exploration of the lunar surface by 
+astronauts. Leading up to the anticipated crewed 
+Artemis missions, scientists and engineers are 
+working to define what lunar exploration will look like 
+in the 21st century. Humanity has come a long way 
+since the Apollo era, and one should expect higher 
+standards of safety, increased science return and 
+hopefully missions with a longer duration leading to 
+the establishment of a sustainable human presence on 
+the Moon. The new technological paradigm affects 
+every single aspect of future missions, from the suits 
+used during Extravehicular Activities (EVAs) to the 
+communication infrastructure and the tools used for 
+in-situ science and sample return. 
+This paper presents the results of a project 
+which aimed to create an overview and classification 
+of potential use cases of Augmented Reality (AR) in 
+the context of Lunar EVAs. Through a review of 
+literature, a list of applications which have been 
+investigated was made. Subsequently a guided 
+brainstorm served to generate new ideas and concepts 
+for novel use cases. Through a clustering activity, all 
+the use cases were grouped together, and a 
+classification 
+was 
+made 
+to 
+describe 
+distinct 
+application areas. 
+The 
+aim 
+was 
+not 
+to 
+make 
+a 
+fully 
+comprehensive categorization, but rather to lay the 
+groundwork for a user-centred design approach which 
+can take these and other application areas into account 
+in the design and development of the entire AR 
+system. Secondarily, the overview made in this project 
+can be helpful to others wishing to evaluate the 
+potential benefits of AR for lunar EVAs across use 
+cases. This more complete view of the benefits which 
+could be derived from such a technology development 
+could aid in decision-making regarding the allocation 
+of funds for a lunar EVA AR interface. 
+This paper is the result of an investigation into 
+the potential of AR applications for Lunar exploration 
+which was performed by interns at the European 
+Astronaut Center (EAC) and more specifically within 
+the Spaceship EAC group. This group consists of 
+interns and trainees and aims to investigate low 
+Technology Readiness Level technologies for space 
+exploration. 
+ 
+1.1 Lunar exploration context 
+Although there have been fluctuations in the level of 
+interest in and funding for human space exploration 
+since the end of the Apollo program, there are 
+indications that the current upwards trend will 
+continue. There is international support for a strategy 
+in which human exploration of the Moon will be used 
+as a steppingstone towards human exploration of Mars 
+[1]. This year NASA’s Space Launch System (SLS) 
+and the Orion spacecraft, a collaborative achievement 
+between NASA and ESA, are scheduled to launch as 
+part of the Artemis I mission. This inaugural uncrewed 
+mission will prove the system’s capability to bring 
+humans into Lunar orbit. Meanwhile, an international 
+collaboration of space agencies has started working on 
+the next long-term human orbital outpost called 
+‘Lunar Gateway’, for the first time in history to be 
+built in Lunar orbit. NASA’s next-generation EVA 
+spacesuit is also being developed with Lunar surface 
+operations in mind [2]. The Human Landing System 
+is the last piece of the puzzle which will allow 
+astronauts to access the Lunar surface, and its 
+development is being funded by NASA [3]. 
+Later phases of the Artemis program aim to 
+establish longer-duration crewed lunar missions. ESA 
+has also envisioned the establishment of an 
+international lunar village, an outpost for long 
+duration manned planetary missions. This would be an 
+ideal platform not only for detailed science, but also 
+to prepare for the first manned Mars missions [4]. 
+With the prospect of increasing human deep 
+space exploration, the return of planetary EVAs and 
+all the challenges related to long-term astronaut 
+presence on lunar and planetary surfaces, we must 
+consolidate efforts to develop optimized state-of-the-
+art technologies and tools to enable astronauts to work 
+safely and efficiently.  
+ 
+1.2 Augmented reality 
+One such technology which has gained some interest 
+in the context of EVAs is AR. Augmented reality 
+involves the overlay of digital information onto the 
+user’s physical environment. There are three main 
+types of AR technologies currently on the market [5]: 
+Optical See-through AR consists of a transparent 
+display which allows the user to see their physical 
+environment behind digital projections. Video See-
+through AR, commonly used in Mobile Augmented 
+Reality found on smartphones, consists of a display 
+which shows a real-time video feed from a camera 
+with overlayed digital information. Finally, Spatial 
+AR does not make use of a display, but rather projects 
+digital information directly onto the physical 
+environment.  
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+3 
+Augmented reality emerged several decades 
+ago and has since then been developed in a multitude 
+of technologies for various applications. Some of the 
+earliest examples were found in military cockpits to 
+aid pilots. Other use cases have been found in 
+education, training, industry and more. The use of 
+Augmented Reality for astronauts is also not a new 
+concept. As far back as the 1980s and 90s, concepts 
+were made for Heads up displays (HUDs) to be 
+integrated in EVA suit helmets [6][7]. Practical 
+experience has since been gained in microgravity 
+through experimentation with both bespoke and 
+Commercial Off the Shelf (COTS) AR interfaces on 
+board the International Space Station (ISS)[8][9]. 
+ 
+1.3 AR for lunar exploration 
+The integration of a HUD system has been 
+documented as being one of the design goals for 
+NASA’s next generation EVA suit, called xEMU [10]. 
+Although there have been some published tests with 
+AR in the xEMU helmet [11], based on the lack of 
+publicly available information it appears that this 
+functionality is currently not on the critical design path 
+for the system. 
+Numerous studies have been performed in 
+which some specific functionality was implemented 
+as a prototype on either bespoke or COTS hardware, 
+to enable testing of AR functionalities in use cases 
+analogous to astronaut operations in space [7] [12] 
+[13] [11] [14] [15] [16] [17] [18] [19] [20] [21] [22] 
+[23] [24] [25]. With some exceptions, the studies do 
+not tend to adopt user-centred design processes, 
+instead opting to work with available technology to 
+demonstrate the benefits of AR in a specific use case.  
+In the setup of these studies, it is rarely 
+mentioned why the hardware used for the study was 
+chosen. If it is mentioned, it tends to be in the form of 
+an evaluation of a few available options, comparing 
+the suitability of these technologies to the specific use 
+case intended for the study. There seems to be a 
+knowledge gap concerning the wider context of 
+potential applications for AR. This makes it difficult 
+to 
+select 
+optimal 
+technologies 
+and 
+system 
+architectures for development, since one cannot 
+predict the suitability of any given technology for all 
+use cases if no overview of use cases exists. 
+The practical studies listed above choose a few 
+highly specific use cases or applications, but do not 
+tend to elaborate on how the choice for a specific use 
+case was made, beyond establishing that they are 
+relevant to the human space exploration context. This 
+presents a limitation in the state of the art, since one 
+must assume that a complex and presumably 
+expensive system such as an AR interface rated for use 
+inside an EVA suit, should be used for as broad a 
+range of applications as is possible and useful.  
+Although individual studies have contributed 
+significantly 
+to 
+showing 
+applications 
+of 
+AR 
+technology for human space exploration and the 
+benefits which can be derived from them, there seems 
+to be a need for a more comprehensive study of 
+potential applications of this technology [26]. Such an 
+overview would allow for a better understanding of 
+the full benefits which can be derived from an AR 
+system across use cases, which could form a stronger 
+basis for the allocation of the necessary funding to 
+develop such a system. Additionally, understanding 
+potential use cases of AR irrespective of the 
+technology used for implementation allows for a user-
+centred instead of a technology-driven design 
+approach.  
+ 
+2. Approach 
+The aim of this project was to create an overview and 
+classification of potential use cases of AR in the 
+context of Lunar EVAs. The adopted approach finds 
+similarities in the ‘Fuzzy Front End’ (FFE) phase of 
+the product development process from the innovation 
+management domain.  
+Defined as “the period between when an 
+opportunity is first considered and when an idea is 
+judged ready for development” [27], the FFE 
+approach assumes that significant value can be 
+derived from properly understanding the contexts, 
+stakeholder needs and problem definitions of a new 
+product before investing heavily into its development. 
+This is reflected in the first half of the British Design 
+Council’s Double Diamond model for a structured 
+design approach (Figure 1) [28], a widely utilized 
+model in the Industrial Design Engineering industry. 
+FFE also shares common attributes with the widely 
+known 
+‘Design 
+Thinking’ 
+approach 
+which 
+emphasizes a human-centred, iterative approach 
+including analysis and synthesis phases which 
+employ, amongst other things, brainstorms, and 
+clustering activities [29].  
+FFE aims to develop more optimized products 
+by spending time to properly understand what is being 
+developed and why. This should result in a higher 
+return on investment and can prevent costly late-stage 
+design changes which might incur significant delays 
+in the delivery of a product or system [30]. 
+Additionally, integrating relevant data in new ways 
+during a well-structured FFE phase can lead to novel 
+and innovative solutions. [31] 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+4 
+ 
+Figure 1, the Double Design approach as described by the British Design Council [28] 
+ 
+Characteristics of a well-structured FFE phase 
+tend to be multi-disciplinary, collaborative, and 
+iterative. The process should consist of multiple 
+rounds of convergent and divergent activities and can 
+include guided brainstorm sessions with experts, users 
+and/or stakeholders. This allows a learning process to 
+take place in which the problem is further defined, the 
+user is better understood, and the context is further 
+mapped out. Breuer et al. describe a classic FFE 
+approach in which certain inputs are given to an expert 
+brainstorm, which triggers a wide range of ideas 
+(divergent) 
+which 
+are 
+subsequently 
+clustered 
+(convergent) to form search areas. These search areas 
+can then form the basis for further investigation, 
+definition, ideation, and design (Figure 2) [32] 
+The classification generated in this project can 
+be seen as analogous to the search areas in FFE, in that 
+they do not specify a design or technology but rather 
+represent insights into user needs and context factors 
+such as science goals, and form demarcated areas 
+which aid further ideation and concept development, 
+breaking 
+free 
+from 
+convention 
+and 
+existing 
+assumptions about the applications of AR to develop 
+user-centred solutions. 
+The approach to forming the classification also 
+reflects processes commonly employed in FFE. 
+Starting with contextual research, existing literature 
+was studied to create an overview of applications 
+which have previously been described and/or  
+ 
+Figure 2. The iterative divergent and convergent 
+process as described by Breuer et al. [32] 
+ 
+ 
+investigated.  Subsequently, a guided brainstorm with 
+a multi-disciplinary team of SSEAC interns and staff 
+served to generate a large quantity of ideas for 
+potential use cases. These were then clustered to 
+create a categorization of AR use cases for lunar 
+surface exploration. Finally, the categorization was 
+tested against the applications described in literature 
+to ensure they were representative of the existing body 
+of work. 
+During the project it was decided to limit the 
+scope to applications and use cases of AR during lunar 
+EVAs. Although an even wider evaluation of 
+applications for all elements of a human lunar 
+exploration mission could be valuable, the more 
+limited scope helped to gather useful insights within 
+the limited timeframe of the project. 
+Publications related to among others NASA’s 
+IDEAS system, Holo-SEXTANT, SUITS program 
+
+★
+Stuetions-
+Twodustering
+Specificalionaf
+impulsesbrainstoming
+contentafeach
+&oorcept-
+searchfelds
+searchfieid.
+brainstomingENGAGEMENT
+DESIGN
+PRINCIPLES
+OUTCOME
+METHODS
+BANK
+LEADERSHIP 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+5 
+were included in the review of existing literature. Due 
+to the scope of the project, publications related to real-
+world experiments with AR in terrestrial industry and 
+on the ISS such as ESA’s MobiPV4Hololens were 
+purposefully omitted. Inclusion of a wider selection of 
+studies could be beneficial to find more potential 
+applications, however the limitation of the scope was 
+instrumental to complete the project within its limited 
+timeframe. 
+The guided brainstorm was organized on 
+August 3, 2020. Due to restrictions related to the 
+COVID-19 pandemic, the brainstorm was organized 
+remotely, and an online whiteboard tool was used in 
+conjunction with video conference software. This 
+allowed a group of interns, trainees, and staff from 
+SSEAC with a wide variety of backgrounds from 
+computer science to aerospace engineering and 
+industrial design to join the session and contribute to 
+the ideation of potential use cases of AR for human 
+lunar exploration. 
+The first step in the brainstorm was not to 
+directly talk about AR applications for lunar 
+exploration. Instead, the ‘principle of detour’ [32] was 
+applied and participants were asked to write down 
+abstracted 
+potential 
+values 
+offered 
+by 
+AR 
+technologies regardless of their application area. 
+Additionally, participants were asked to write down as 
+many activities as they could think of that could 
+possibly be a part of future human lunar exploration, 
+without thinking about AR at all.  
+Subsequently, participants were asked to 
+combine these two inputs and generate a large number 
+of use cases. They were also instructed that not all use 
+cases had to be linked to inputs which were defined in 
+the previous step. To the contrary, the synthesis of use 
+cases from insights should ideally trigger new ideas 
+and insights, thereby leading to the identification of 
+more use cases. The brainstorm lasted 2.5 hours, and 
+the resulting use cases are described in section 3. 
+After the divergent phase, the seemingly 
+random and chaotic collection of ideas needs to be 
+ordered in some way. More than just an organization 
+of ideas, the process of clustering also adds value to 
+the creative process. By linking ideas together and 
+choosing specific words to describe idea-spaces, new 
+search areas are created which can form the basis for 
+whole new concepts to be developed [33], indirectly 
+triggered by the earlier discovery and definition steps. 
+The clustering activity was performed by two 
+authors, in an iterative process that included feedback 
+from other co-authors. The resulting classification can 
+be found in the section 3. 
+Finally, the classification was tested against the 
+applications found in existing literature. Through this 
+process, it was realized that there was no category 
+accurately representing the display of telemetry from 
+various external sensors and that science operations 
+outside of geological sampling had not been discussed 
+during the brainstorm. To address this, a category was 
+added to represent these use cases. 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+6 
+3. Results
+16 publications were included in the review of 
+applications mentioned and/or investigated by 
+existing literature. Table 1 shows an overview of the 
+applications per publication, worded as they are in the 
+original text. 
+ 
+ 
+ 
+ 
+ 
+ 
+Reference 
+AR applications which are investigated or suggested 
+ 
+Griffin, B. (1990)[7] 
+Map-type graphics for navigation, pre-recorded video instructions, remote 
+live-streamed video from cameras, gauge readings for consumables 
+Hogson, E. et al. (2003) [12] 
+Life support and comfort control, communications, mission and task 
+planning, localization and situational awareness, navigation, task execution 
+Di Capua, M. (2008) [13] 
+Life support and comfort control, mission and task planning, localization 
+and situational awareness, navigation, task execution and human-robot 
+interfaces 
+Stolen, M. et al. (2008) [11] 
+Monitor the status of their own and other’s biometrics, monitor the status 
+of their and other’s spacesuit systems, monitor the status of robotic systems 
+Jacobs, S. et al. (2009) [14] 
+Navigation package, remaining consumables, crewmember health, suit 
+status 
+Villorin, A. (2016) [15] 
+Procedure lists and task instructions, consumables status, camera tools, 
+video communications, sensor telemetry views 
+Morrison, M. et al. (2017) [16] 
+Procedure checklists, navigational aids, display of biomedical data 
+Anandapadmanaban, E. et al. 
+(2018) [17] 
+Traverse plans 
+Gibson, A. et al. (2018) [18] 
+Obstacle avoidance and wayfinding 
+Mitra, P. (2018) [19] 
+Cuff checklist, suit data display, camera control, communications, caution 
+and warning system 
+Valencio D’souza, G. (2019) 
+[20] 
+Maintenance task, navigation and rocks sample collection task 
+Fox, K. (2020) [21] 
+Task instructions 
+McHenry, N. et al. (2020) [22] 
+Visual display of suit vitals, telemetry, waypoints and checklist items 
+Radway, S. et al. (2020) [23] 
+Task instruction, sampling assistance, note taking, telemetry monitoring 
+and display 
+Rometsch, F. (2020) [24] 
+Geological site inspection, data logging, photo documentation, taking site 
+coordinates, verbal field notebook, waypoints, display of suit diagnostics 
+Miller, L. et al. (2021) [25] 
+Livestream of biometric values, procedure overview, reference resources 
+to support activities with detailed information 
+Table 1: Applications described and investigated in existing literature. 
+ 
+To generate a list which is more workable than the 
+information in table 1, the list in table 2 was made, 
+somewhat generalizing, and grouping specific 
+applications together. 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
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+ 
+ 
+ 
+ 
+7 
+Application 
+References 
+Navigation 
+[7][12][13][14], [16][17][18][20][22][24] 
+Procedure information 
+[7][12][13][15][16][19][20][21][22][23] 
+Camera live feed 
+[7][15][19] 
+Consumables monitoring 
+[7][11][14][15] 
+Life support control 
+[12][13] 
+Communications 
+[12][13][15][19] 
+Procedure planning 
+[12][13] 
+Situational awareness 
+[12][13] 
+Human-robot 
+and 
+Human-machine 
+interfaces 
+[13] [11][15] 
+Biometrics monitoring 
+[11] [14][16] 
+Suit system status monitoring 
+[11][14][19][22][23] 
+Note taking and data logging 
+[23][24] 
+Table 2: Generalized overview of applications described and investigated in existing literature. 
+ 
+ 
+As described in the approach section, a brainstorm 
+was organized in which participants were asked to 
+document ideas for potential values derived from AR 
+irrespective of application type, and to document 
+potential activities which might be a part of future 
+human lunar exploration missions. 
+ 
+The following types of value which could be derived 
+from a lunar AR system were identified: 
+ 
+For astronauts 
+- 
+Reduce cognitive load 
+- 
+More agency in accessing data 
+- 
+Increase amount of information crew can 
+access 
+- 
+Enhance capabilities to control vessel in 
+flight 
+- 
+Easier crew to crew communication 
+- 
+Free hands 
+- 
+Increased situational awareness 
+- 
+Ground can send information directly to 
+crew’s feed 
+- 
+Enhanced 
+communication 
+between 
+astronauts and ground 
+- 
+Faster assembly / maintenance 
+- 
+Decrease time needed to perform a task 
+- 
+Live adaptable instructions 
+- 
+Visual text-based communication messages 
+- 
+Sharing target of attention 
+- 
+Extend visual senses 
+- 
+Ability to reconfigure the multipurpose 
+interface 
+- 
+Adaptable setting 
+- 
+Integration into existing hardware 
+ 
+Programmatic value 
+- 
+Enhanced PR content 
+- 
+Lower risk for accidents 
+- 
+More collaborative possibilities 
+- 
+Increased general well-being of astronauts 
+- 
+Avoid distractions for astronauts 
+- 
+Increase astronauts’ focus 
+- 
+Less need for training 
+- 
+Improved emergency response 
+ 
+The following lunar activities were described: 
+Gateway 
+- 
+Communication, planning and preparation of 
+day-to-day tasks 
+- 
+Crop cultivation 
+- 
+Hardware troubleshooting 
+- 
+Tele-medicine 
+- 
+Payload deployment 
+- 
+Retrieving regolith samples 
+- 
+Hardware status observations 
+- 
+Construction of infrastructure 
+- 
+Post- and pre- EVA activities 
+- 
+Performing experiments 
+- 
+Leak detection 
+- 
+In-Situ medical care 
+- 
+Post- and pre- flight activities 
+- 
+Spare part manufacture 
+- 
+Payload maintenance 
+- 
+Resting/ sleeping 
+- 
+Cargo and stowage logistics 
+- 
+Payload upgrading 
+ 
+ 
+ 
+Human Landing System 
+- 
+Dust mitigation in habitat 
+- 
+Collaboration between Gateway and lunar 
+surface 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+8 
+- 
+Terrain awareness 
+- 
+Live flight data 
+- 
+Hardware troubleshooting 
+- 
+Retrieving/handling regolith samples 
+- 
+Tele-medicine 
+- 
+Preparing samples for return to Earth 
+- 
+System integrity checks 
+- 
+Leak detection 
+- 
+Communication, planning and preparation of 
+day-to-day tasks 
+- 
+Resting/sleeping 
+- 
+Post- and pre- EVA activities 
+- 
+Proof of concepts for fuel and oxygen storage 
+and transportation 
+- 
+In-situ medical care 
+- 
+Synthetic landing site markers 
+ 
+Lunar surface 
+- 
+Harvesting lunar volatiles 
+- 
+Terrain awareness 
+- 
+In-situ analysis of geological samples 
+- 
+Crop cultivation 
+- 
+Tele-geology 
+- 
+Exploration of Permanently Shadowed 
+Regions 
+- 
+Retrieving regolith samples 
+- 
+Hardware troubleshooting 
+- 
+Dust mitigation on equipment 
+- 
+Tele-medicine 
+- 
+Traverse over rough terrain 
+- 
+proof-of-concepts for fuel and oxygen 
+storage and transportation 
+- 
+Co-bot operations 
+- 
+Performing experiments 
+- 
+Mapping and characterization of macro 
+geological features 
+- 
+Construction of infrastructure 
+- 
+Leak detection 
+- 
+Construction of roads or landing pads 
+- 
+Live checklists 
+- 
+Communication, planning and preparation of 
+day-to-day tasks 
+- 
+Spare part manufacture 
+- 
+Construction of infrastructure 
+- 
+In-situ medical care 
+ 
+Subsequently, participants were asked to write down 
+as many use cases of AR for lunar exploration as they 
+could come up with. Each use case should have a title, 
+and one or two sentences detailing the function and 
+added value of AR in this use case (Table 3). 
+ 
+ 
+ 
+Use Case Title 
+Function 
+Value 
+1 
+Rover / 
+instrument 
+maintenance 
+Display procedures, schematics to do 
+maintenance work on an instrument  
+Less training required as procedures are 
+automatic and updated accordingly, easy 
+to follow and highlights and displays 
+overlays on the  
+2 
+Construction of 
+roads / landing 
+pads 
+Helps astronauts in selecting areas to 
+construct basic infrastructure and helps 
+them in finding level ground to build on. 
+Support for construction tasks that would 
+require additional hardware, integrated 
+into a HUD. 
+3 
+Instructions 
+Overlay 
+overlay visual assembly or maintenance 
+cues (highlight next screw holes, 
+insertion path/orientation of parts etc.) 
+Faster assembly / maintenance, less 
+training required, fewer errors. 
+4 
+Sample 
+selection HUD 
+HUD provides overlay of information 
+from an IR camera to provide more 
+information about potential sample 
+composition 
+Increased science return from samples 
+more efficient use of astronaut time 
+5 
+Communication 
+between 
+astronauts 
+during EVA 
+HUD allows astronauts to communicate 
+by highlighting physical objects, and by 
+transferring data from one to another 
+(e.g., location, health monitoring). 
+Reduces the likelihood of 
+misunderstandings, increases the ability 
+of astronauts to assist each other (e.g., 
+rescue), makes communication more 
+effective, decreases the amount of verbal 
+communication needed. 
+6 
+Sample 
+retrieval 
+Display the location of a sample and 
+protocols to follow for retrieval 
+Minimize sample retrieval time 
+7 
+Classic flight / 
+landing HUD 
+Will display flight data, landing data and 
+environmental data on a classic HUD 
+allowing astronauts to observe the Lunar 
+environment during critical phases. 
+ less accidents, better situational 
+awareness 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+9 
+8 
+Non-vocal  
+one-way 
+communication 
+Messages by ground control or Gateway 
+can be sent to the astronaut’s HUD and 
+displayed there. 
+no need for vocal communication 
+9 
+Medical 
+information in 
+HUD 
+Displaying personal vitals and vitals of 
+crew members. Basic vitals (e.g., blood 
+pressure, heart rate, O2sat). Can also 
+display energy expenditure and give 
+warnings if overexerting oneself. 
+Reduces the need to request medical 
+information. Can increase safety, increase 
+emergency response 
+10 
+Checklists in 
+HUD 
+Checklists of items (i.e., deployment of 
+stuff, or procedures). Collaborative 
+checklists could possibly be 
+synchronized in real time. 
+No need for an additional device for 
+checklists 
+11 
+Construction 
+enhancer 
+Simulate beams and loads and payloads 
+to calculate the optimal structure or 
+deployment 
+ 
+12 
+ Mission 
+markers 
+Visual representation of items to be 
+interacted with 
+Good overview of where to go for the 
+next objective 
+13 
+Remote support 
+during medical 
+operations 
+Enables an expert on the ground (i.e., 
+medical doctor) to provide relevant 
+visual information to an astronaut 
+performing a minor surgery. This 
+information can be: checklists in text 
+format, pre-recorded visual instructions, 
+virtual pointer/highlighting to guide 
+astronaut, live video feed from 
+instructor. 
+Reduces the amount of training needed, 
+increases the odds of success of surgery, 
+increases the flexibility in terms of 
+performable operations (instructor can 
+adapt to exact situation) 
+14 
+Telepresence of 
+expert / 
+instructor 
+Overlay of video-feed of expert or 
+instructor enabling additional 
+communication channels (gestures, 
+demonstration of movements etc.) 
+Higher quality communication, easier 
+interaction with instructor or expert 
+15 
+EVA mini map 
+Display current position around ISS, or 
+on lunar and/or planetary surface relative 
+to base camp (including surface features 
+etc. from satellite imagery) as well as 
+teammate [Gä1] ’s positions. 
+Increased situational or locational 
+awareness of self and crew. This is good 
+for safety, efficiency, and cooperation. 
+Table 3: Use cases resulting from the brainstorm 
+ 
+After the brainstorm, the resulting use cases were 
+clustered in a collaborative and iterative process 
+amongst the co-authors of this publication. The 
+following classification (Table 4.) was deemed to be 
+representative of all use cases, while maintaining 
+sufficient differentiation between each class. It should 
+be noted that each class of use cases can contain 
+multiple specific use cases and each use case can 
+involve a combination of AR applications (e.g., 
+waypoints, procedure list) and UI elements (e.g., 
+video feed, overlaid data on the physical terrain). 
+Table 4, ‘related use cases from literature’ only refers 
+to use cases found in literature listed in Table 1
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+10 
+Use case 
+classification 
+Description 
+Related use cases from literature 
+EVA navigation 
+Navigation on the surface with or without 
+vehicle. Positioning, situational awareness and 
+interpretation of terrain features. 
+Navigation, Procedure planning, 
+Situational 
+awareness, 
+Human-
+Robot 
+and 
+human-machine 
+interfaces 
+Scientific 
+measurements and 
+observations 
+Observation and interpretation of data from 
+science 
+instruments, 
+control 
+of 
+science 
+instruments, annotation and tagging of data. 
+Camera live feed 
+Sample collection 
+Sample collection process, sample and site 
+documentation and data logging. 
+Procedure information, procedure 
+planning, Camera live feed 
+MRO and 
+construction 
+Maintenance, Repair and Overhaul (MRO) and 
+construction 
+procedures, 
+instructions, 
+annotation, simulation, compliance testing and 
+data logging. 
+Procedure information, procedure 
+planning, Human-Robot and human-
+machine interfaces 
+Logistics and 
+inventory 
+management 
+Inventory 
+tracking, 
+equipment 
+and 
+consumables management, process and storage 
+optimization. 
+ 
+Medical procedures 
+Diagnostic, 
+emergency, 
+and 
+scientific 
+procedures. 
+Procedure information, procedure 
+planning, Huma-Robot and human-
+machine interfaces 
+Biomedical and 
+system status 
+monitoring 
+Monitoring of crew member’s vitals and 
+critical system telemetry. 
+Consumables 
+monitoring, 
+Life 
+support control, Human-machine 
+interfaces, biometric monitoring, 
+suit system status monitoring. 
+Collaboration and 
+support 
+Collaboration between crew members, crew 
+and ground, EVA crew and crew inside a 
+habitat, lunar surface crew and Gateway crew 
+or crew and (semi)-autonomous robotic 
+systems. 
+Camera live feed, Communications, 
+Human-robot and Human-machine 
+interfaces 
+Table 4, classification of use cases of a lunar EVA AR
+4. Discussion 
+The results of this project encompass a wide variety of 
+applications, and the classification should be useful in 
+the generation of new concepts and the development 
+of a user-centred system design. 
+Although efforts were made to include a wide 
+variety of activities and use cases, the overview of use 
+cases cannot be seen as comprehensive, even within 
+the limited scope of lunar EVAs. This is evidenced by 
+the fact that a significant group of activities was not 
+found during the brainstorm and was instead added 
+later, which indicates that there are likely to be other 
+use cases which have not been found during this 
+project. Ostensibly, making a complete overview of 
+activities might not be possible until the actual mission 
+profiles have been decided on. Until that time, one can 
+however assume a certain value to be inherent in 
+insights which aim to be diverse if not complete. 
+A certain transition is evident between the 
+‘applications’ 
+of 
+technology-driven 
+design 
+developments and evaluations - which constitute most 
+of the existing literature - and the ‘use cases’ which 
+are more relevant for the user-centred approach. The 
+difference 
+can 
+be 
+described 
+as 
+applications 
+representing 
+technical 
+functions 
+(i.e., 
+placing 
+waypoints, displaying a list of procedures, controlling 
+the Life Support System, see ‘Table 1’) whereas use 
+cases 
+represent 
+activities 
+with 
+more 
+clear 
+stakeholders, contexts and goals (i.e., ‘guiding non-
+geologists during geological inspection tasks’ [34]). 
+The latter feeds directly into user-centred concept 
+development and could allow designs to let go of 
+conventions informed by the paradigm of outdated 
+technologies. Any realistic system should however 
+keep in mind the proven processes and designs which 
+have been in use for decades. Future designs should 
+incorporate these to benefit from their reliability and 
+compatibility with existing systems. 
+Although a user-centred approach can lead to 
+novel and optimized designs, one could argue that 
+technical limitations should be given as much 
+importance as design considerations as user needs. 
+Especially for a technology which should work inside 
+an EVA suit in use, extreme technical challenges need 
+to be overcome to create a functioning system. For 
+example, the electronics must be safe to use in the 
+oxygen-rich environment inside a suit, integration of 
+multiple systems such as GPS and IoT networks can 
+rapidly increase complexity and cost, and redundancy 
+must be built into systems which are critical for 
+mission success and astronaut safety. All this 
+considered, the technology-driven approach does not 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+11 
+guarantee that these limitations are considered, since 
+many studies are based on terrestrial COTS systems 
+and would not fulfil these requirements. And a user-
+centred approach would include considerations for 
+technical limitations in the design embodiment and 
+detailing phases, as represented for example in the 
+iterative ‘develop and deliver’ diamond shown in 
+Figure 1. 
+This project has proven that there are relevant 
+methodologies from the innovation management 
+domain that could be applied to the development of 
+complex systems for human space exploration. Future 
+studies could potentially identify more opportunities 
+for the development of user-centred systems for 
+astronauts when applying methodologies from the 
+innovation management and design engineering 
+domains, as also evidenced by Rometsch et al. [35]. 
+ 
+The main subject of this project was the 
+classification of potential AR use cases for human 
+lunar exploration. Although the outcome should be 
+useful in its current form, one can imagine an even 
+more comprehensive classification process which 
+would not limit the scope to EVAs but to all activities 
+related to human lunar and planetary exploration.  
+Furthermore, the approach which was used to 
+create the classification could be formalized further, 
+ensuring 
+that 
+the 
+resulting 
+categorization 
+is 
+comprehensive and individual classes are sufficiently 
+differentiated from each other. An example of an 
+excellent formalized classification of AR use cases 
+was performed by Röltgen and Dumitrescu and could 
+serve as an inspiration for further work in the subject 
+area of this publication [36]. 
+By focusing specifically on visual AR systems, 
+the potential value of multi-modal AR systems might 
+have been overlooked. Multi-modal AR systems use a 
+mix of stimuli to provide data to the user instead of 
+solely using visual displays. For example, Gibson et 
+al. studied the use of haptic feedback in astronaut 
+boots for obstacle avoidance 
+[18]. Although 
+challenging, it is likely worthwhile to include multi-
+modal interfaces as a consideration in the further 
+development and evaluation of AR systems for lunar 
+exploration. 
+ 
+5. Conclusion 
+This project has fulfilled its aim of generating a 
+classification of potential use cases of AR for human 
+lunar surface exploration. Although the scope had to 
+be narrowed down to AR for EVAs, the hope is that 
+future work can identify use cases for every potential 
+context of use for an astronaut AR system . A more 
+formalized process for classification might yield 
+results which are more comprehensive with more 
+precisely defined categories. However, it is expected 
+that the results from this project already in their 
+current form can help to evaluate potential AR 
+technologies, support concept development of novel 
+AR functions and provide a framework to bring 
+together results from individual studies and start to 
+form a picture of the full potential value which might 
+be gained from the development of an AR system for 
+human space exploration. 
+The following categories were defined: EVA 
+navigation, 
+Scientific 
+measurements 
+and 
+observations, 
+Sample 
+Collection, 
+Maintenance, 
+Repair, Overhaul (MRO) and Construction, Logistics 
+and Inventory Management, Medical Procedures, 
+Biomedical 
+and 
+System 
+Status 
+Monitoring, 
+Collaboration and Support. 
+ 
+
+ 
+ 
+ 
+ 
+IAC-22- B3.7.5 
+ 
+ 
+ 
+ 
+ 
+12 
+References 
+ 
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+  
+

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@@ -0,0 +1,1305 @@
+Highlights
+Gaussian process regression and conditional Karhunen-Lo´eve mod-
+els for data assimilation in inverse problems⋆
+Yu-Hong Yeung, David A. Barajas-Solano, Alexandre M. Tartakovsky
+• We propose CKLEMAP as an efficient alternative to the maximum
+a posteriori probability (MAP) method of parameter estimation for
+partial differential equations.
+• The efficiency is due to the use of a conditional Karhunen-Lo´eve repre-
+sentation of the parameter field and an acceleration scheme for Jacobian
+computations.
+• CKLEMAP and MAP scale as N 1.3 and N 3, where N is the number
+of nodes of degrees of freedom in the discretization of the governing
+partial differential equation.
+• CKLEMAP is as accurate as MAP but significantly faster for large-
+scale parameter estimation problems.
+arXiv:2301.11279v1  [cs.LG]  26 Jan 2023
+
+Gaussian process regression and conditional
+Karhunen-Lo´eve models for data assimilation in inverse
+problems
+Yu-Hong Yeunga, David A. Barajas-Solanoa, Alexandre M. Tartakovskya,b,∗
+aPhysical and Computational Sciences Directorate, Pacific Northwest National
+Laboratory, Richland, 99354, WA, USA
+bDepartment of Civil and Environmental Engineering, University of Illinois
+Urbana-Champaign, Urbana, 61801, IL, USA
+Abstract
+We present a model inversion algorithm, CKLEMAP, for data assimilation
+and parameter estimation in partial differential equation models of physi-
+cal systems with spatially heterogeneous parameter fields. These fields are
+approximated using low-dimensional conditional Karhunen-Lo´eve expansions
+(CKLEs), which are constructed using Gaussian process regression (GPR)
+models of these fields trained on the parameters’ measurements. We then
+assimilate measurements of the state of the system and compute the max-
+imum a posteriori (MAP) estimate of the CKLE coefficients by solving a
+nonlinear least-squares problem. When solving this optimization problem,
+we efficiently compute the Jacobian of the vector objective by exploiting
+the sparsity structure of the linear system of equations associated with the
+forward solution of the physics problem.
+The CKLEMAP method provides better scalability compared to the stan-
+dard MAP method. In the MAP method, the number of unknowns to be
+estimated is equal to the number of elements in the numerical forward model.
+⋆This research was partially supported by the U.S. Department of Energy (DOE) Ad-
+vanced Scientific Computing program. Pacific Northwest National Laboratory is operated
+by Battelle for the DOE under Contract DE-AC05-76RL01830.
+∗Corresponding author
+Email addresses: [email protected] (Yu-Hong Yeung),
+[email protected] (David A. Barajas-Solano), [email protected]
+(Alexandre M. Tartakovsky)
+Preprint submitted to Journal of Computational Physics
+January 27, 2023
+
+On the other hand, in CKLEMAP, the number of unknowns (CKLE coeffi-
+cients) is controlled by the smoothness of the parameter field and the num-
+ber of measurements, and is in general much smaller than the number of
+discretization nodes, which leads to a significant reduction of computational
+cost with respect to the standard MAP method. To show this advantage
+in scalability, we apply CKLEMAP to estimate the transmissivity field in a
+two-dimensional steady-state subsurface flow model of the Hanford Site by
+assimilating synthetic measurements of transmissivity and hydraulic head.
+We find that the execution time of CKLEMAP scales nearly linearly as N 1.33,
+where N is the number of discretization nodes, while the execution time of
+standard MAP scales as N 2.91. The CKLEMAP method improved execu-
+tion time without sacrificing accuracy when compared to the standard MAP
+method.
+Keywords:
+Model inversion, Gaussian process regression, conditional
+Karhunen-Lo´eve expansion, maximum a posteriori (MAP)
+1. Introduction
+Parameter estimation is a critical part of developing partial differential
+equation (PDE) models of natural or engineered systems. In heterogeneous
+systems, parameters vary in space (and, possibly, time), and the destructive
+nature and high cost of collecting measurements limit the number of direct
+parameter measurements that can be gathered. As a consequence, modelers
+are tasked with solving the inverse problem, i.e., estimating parameters from
+a limited number of direct measurements and, usually, a larger number of
+indirect measurements, e.g., measurements of the states in the PDE model.
+In the context of subsurface flow and transport, such observables include
+hydraulic head and tracer breakthrough measurements at observation wells,
+among others.
+The heterogeneity of parameters gives rise to two challenges: (1) spa-
+tial heterogeneity must be parameterized, either naively, using the grid dis-
+cretization of the PDE’s domain, or through some other scheme; and (2)
+sparse-in-space measurements are often not enough to fully characterize spa-
+tial heterogeneity, thus it is necessary to introduce assumptions about spatial
+heterogeneity that regularize the inverse problem.
+Once parameterization and regularization schemes have been selected,
+one can compute the maximum a posteriori (MAP) estimate of the model
+2
+
+parameters. The MAP estimate is computed by solving a PDE-constrained
+optimization problem consisting of minimizing a certain norm of the differ-
+ence between predicted and measured observables (data misfit term) plus a
+regularizing penalty. Assuming that the solution is obtained at a global min-
+imum, the MAP estimate is equivalent to the largest mode of the Bayesian
+posterior with the data misfit term corresponding to the (negative) Bayesian
+log-likelihood and the regularizing penalty corresponding to the (negative)
+Bayesian log-prior [1, 2, 3]. One can drop the PDE constraint by modeling
+the predicted observables via a “surrogate” model, at the cost of constructing
+said model either on the fly (e.g., [4]) or ahead of tackling the inverse prob-
+lem (e.g., [5, 6, 7]). Alternatives to MAP estimation for nonlinear problems
+include iterative linear filtering and smoothing [8, 9]. In this work, by “MAP
+method” we will refer to MAP estimation via nonlinear least-squares using
+the parameterization in terms of the degrees of freedom of the spatial grid
+discretization of the forward solver scheme.
+The pilot point method (PPM) [10, 11, 12] provides parameterization
+and regularization by modeling parameter fields as a regressor computed
+from a set of spatially discrete values (“pilot points”) of the parameter fields.
+These pilot points then become the parameters to be estimated via PDE-
+constrained optimization. The choice of the number and locations of pilot
+points is not trivial and significantly affects the quality and time-to-solution
+of the inverse problems. To address these challenges, [12] proposed to use
+the singular value decomposition of the sensitivities of observables with re-
+spect to the pilot points to reduce the effective dimension of the pilot point
+parameterization. Beyond PPM, other parameterizations and regularization
+schemes have been proposed. For example, [13] represented the parameter
+field with a deep neural network and [14, 5] used the latent space represen-
+tation of the parameter fields defined by a variational autoencoder and a
+convolutional adversarial autoencoder, respectively.
+Scientific machine learning (SciML) algorithms provide both an alter-
+native and a complement to the PDE-constrained optimization and linear
+filtering-based approaches to inverse problems described above. SciML ap-
+proaches for inverse problems can be roughly classified into two families:
+physics-informed deep learning (DL) and DL for constructing surrogate mod-
+els. In physics-informed DL methods [15, 16, 17], the parameters and states
+of PDE models are represented by DL models such as feed-forward or con-
+volutional neural networks; then, the parameters of these DL models are
+estimated by minimizing an objective consisting of the data misfit term plus
+3
+
+a weighted penalty on the PDE model residuals evaluated at certain points in
+the simulation domain. This objective corresponds to the so-called “penalty”
+approximation of the corresponding constrained minimization problem with
+a fixed penalty weight [18]. The physics-informed DL approaches rely on
+the expressive capacity of DL models to accurately represent parameters and
+states. On the other hand, DL surrogate modeling approaches use DL models
+to approximate the map from parameters to observables [5, 6, 7, 19]. These
+approaches rely on the capacity of DL models to approximate functions of
+high-dimensional inputs. Other recent developments include “neural opera-
+tor” methods, which aim to learn the PDE solution as an explicit function
+of the model parameters [20].
+Karhunen-Lo`eve expansions (KLEs) are extensively employed to param-
+eterize spatially heterogeneous fields for both uncertainty quantification and
+model inversion tasks. In [21], the conditional KLE of the parameter field
+was conditioned on the direct field’s measurements, leading to conditional
+KL expansions (CKLEs). It was demonstrated that using CKLEs instead of
+KLEs reduces the variance of the stochastic model of the parameter field and
+reduces uncertainty in the forward models. In [22, 23], CKLEs were used to
+represent both parameter and state fields for solving inverse problems. The
+CKLE parameters were estimated by minimizing the residuals of the govern-
+ing equations. The resulting “physics-informed CKLE” algorithm (PICKLE)
+was shown to provide approximate solutions to the inverse problem of accu-
+racy comparable to PDE-constrained optimization-based methods but at a
+significantly lower computational cost.
+Here, we propose solving inverse problems in PDE models by representing
+the parameter fields using CKLEs conditioned on available direct measure-
+ments of these fields and then estimating the CKLE coefficients via nonlinear
+least-squares. We refer to this combination of MAP estimation and CKLEs
+as “CKLEMAP.” Compared to PICKLE, CKLEMAP is free of the errors
+introduced by the approximation of the state with the CKLE expansions
+and the penalty approximation of the PDE constraint, which leads to more
+accurate solutions to the inverse problem at the cost of having to solve the
+forward problem during the nonlinear least-squares minimization procedure.
+Nevertheless, we significantly reduce the execution time of model inversion
+with respect to the MAP method by drastically reducing the number of pa-
+rameters to be estimated. We note that while KLEs, and more generally
+the spectrum of Gaussian process covariance models, have been extensively
+used to parameterize heterogeneous fields in Bayesian parameter estimation
+4
+
+(e.g., [24, 25, 26, 6]), the application of KLE in deterministic inverse meth-
+ods has not been explored and is the subject of our work. Furthermore, we
+demonstrate the advantage of using the CKLE representation as opposed to
+the one based on KLE.
+We apply CKLEMAP to a high-dimensional (approximately 1000 param-
+eters in the CKLE are needed to accurately represent the transmissivity field)
+stationary groundwater flow model of the Hanford Site, a former nuclear pro-
+duction complex on the west shore of the Columbia River in the Columbia
+Basin in the southeast part of the state of Washington in the United States
+and currently operated by the United States Department of Energy.
+We
+use CKLEMAP to estimate the transmissivity field from synthetic measure-
+ments of the transmissivity and hydraulic head fields. These measurements
+are generated using the hydraulic conductivity measurements and boundary
+conditions obtained in the Hanford Site calibration study [27].
+We compare the CKLEMAP and MAP methods and find that both meth-
+ods are very close in accuracy with respect to the reference field. On the other
+hand, we find that the computational cost of MAP increases with the prob-
+lem size (the number N of finite volume cells) as N 2.91, while the cost of
+CKLEMAP increases as N 1.33. We also observe that for N = 5900, the ex-
+ecution time of CKLEMAP is one order of magnitude smaller than that of
+MAP, and for NFV = 23600, we estimate that CKLEMAP would be more
+than two orders of magnitude faster than MAP (the execution time of CK-
+LEMAP is found to be ≈ 8 × 102 s, and the execution time of MAP of
+approximately 2 × 105 s is estimated from the scaling relationship). The
+choice of synthetic (as opposed to the field) measurements of the hydraulic
+head allows us to have a reference transmissivity field for comparing the accu-
+racy of the MAP and CKLEMAP methods while preserving the complexity
+of boundary conditions and the transmissivity field of the Hanford Site.
+2. Groundwater flow model
+We consider two-dimensional flow in a heterogeneous porous medium in
+the domain D ⊂ R2. Given some sparse measurements of the transmissivity
+T(x): D → R+ and the hydraulic head u(x): D → R, our goal is to estimate
+the spatial distribution of transmissivity. Flow in porous media is described
+5
+
+by the boundary value problem (BVP)
+∇ · [T(x)∇u(x)] = 0,
+x ∈ D,
+(1)
+T(x)∇u(x) · ⃗n(x) = −qN(x),
+x ∈ ΓN,
+(2)
+u(x) = uD(x),
+x ∈ ΓD,
+(3)
+where ΓN and ΓD are the disjoint subsets of the boundary of the domain D,
+where the Neumann and Dirichlet boundary conditions (BCs) are prescribed,
+respectively.
+The flux qN ∈ R at the Neumann boundary ΓN is in the
+direction of the outward-pointing unit vector ⃗n ∈ R2 normal to ΓN. The
+prescribed hydraulic head at ΓD is denoted as uD ∈ R.
+In groundwater models, Dirichlet BCs describe water levels in the lakes
+and rivers connected to the aquifer.
+Since it is possible to measure the
+water levels relatively accurately, we treat the Dirichlet boundary conditions
+as deterministic. Furthermore, we assume that the homogeneous Neumann
+boundary condition (qN = 0) is imposed over the subset of ΓN formed by
+the impermeable boundaries of the aquifer. The rest of ΓN is assumed to be
+formed by recharge areas where the values of qN > 0. The boundary fluxes
+from recharge areas are difficult to measure; therefore, we treat the non-zero
+fluxes as random variables and estimate them along with the transmissivity
+field T as part of the inverse solution.
+The MAP method (described in detail in Section 3) requires solving the
+governing equation for different BCs and realizations of T, which in general
+must be done numerically. In this study, we solve the governing equation
+using a cell-centered finite volume (FV) scheme with N quadrilateral cells,
+and the fluxes across cell faces are approximated using the two-point flux ap-
+proximation (TPFA). For simplicity, we assume that ΓN and ΓD are entirely
+composed of cell faces. Let ˆxi denote the ith cell center, with i ∈ [1, N].
+We denote by ui ≡ u(ˆxi) and yi ≡ y(ˆxi) the discrete values of the hydraulic
+head field u and log-transmissivity y ≡ ln T field evaluated at the ith FV
+cell centers.
+These discrete values are organized into the column vectors
+u �� [u1, . . . , uN]⊤ ∈ RN and y ≡ [y1, . . . , yN]⊤ ∈ RN, respectively.
+Then, the FV-TPFA discretization of the BVP (1)–(3) yields the system
+of equations linear in u,
+l(u, y) ≡ A(y)u − b(y) = 0,
+(4)
+with stiffness matrix A: RN → RN×N and right-hand vector side b: RN →
+RN. Here, l: RN ×RN → RN denotes the vector of discretized BVP residuals
+6
+
+whose entries correspond to the mass balance for each FV cell. The set of FV
+cells C can be partitioned into three subsets: N, the set NN of cells adjacent
+to ΓN, D, the set ND of cells adjacent to ΓD, and the set of “interior”
+cells I = C \ (D ∪ N) (that is, the cells to which boundary conditions do
+not contribute directly to their mass balance). The set I has cardinality
+NI = N − NN − ND.
+3. MAP formulation
+We assume that Nus and Nys measurements of u and y, denoted by us
+and ys, respectively, are collected at the cell centers indicated by the vectors
+of observation indices Iu and Iy, respectively. That is,
+[us]i ≡ u(ˆx[Iu]i),
+[ys]i ≡ y(ˆx[Iu]j),
+i ∈ [1, Nus], j ∈ [1, Nys].
+Using these measurements, we aim to estimate y.
+The MAP estimator [1] of y is computed by minimizing the sum of the
+ℓ2-norm of the discrepancy between measurements and model predictions,
+plus a regularization penalty on y, that is, by solving the PDE-constrained
+minimization problem
+min
+u,y
+1
+2∥us − Huu∥2
+2 + 1
+2∥ys − Hyy∥2
+2 + γR(y),
+s.t.
+l(u, y) = 0,
+(5)
+where R(y) is the regularization penalty, γ > 0 is a regularization weight, and
+Hu : RNus×N and Hy : RNys×N are observation matrices, which downsample
+u and y using the observation indices Iu and Iy, respectively. Specifically,
+Hu ≡ IN[Iu], and Hy ≡ IN[Iy] are submatrices of the N ×N identity matrix
+IN corresponding to the rows of indices Iu and Iy, respectively.
+For y, we employ the so-called “H1 regularization,” which penalizes the
+H1 seminorm of y (the ℓ2-norm of the gradient of y). In the discrete case,
+the H1 seminorm penalty is of the form ∥Dy∥2
+2, where D is the TPFA dis-
+cretization of the gradient operator such that Dy is equal to the gradients
+of y across the interior faces of the FV discretization. The resulting PDE-
+constrained minimization reads
+min
+u,y
+1
+2∥us − Huu∥2
+2 + 1
+2∥ys − Hyy∥2
+2 + γ
+2∥Dy∥2
+2,
+s.t.
+l(u, y) = 0,
+(6)
+7
+
+The MAP estimates ˆu and ˆy obtained from Eq. (6) are equivalent to the
+largest mode (ˆu, ˆy) of the joint posterior distribution of (u, y) in a Bayesian
+interpretation of the inverse problem, in which the data misfit terms corre-
+spond to a Gaussian negative log-likelihood and the regularization penalty
+to a Gaussian negative log-prior.
+4. CKLEMAP method for inverse problems
+4.1. Parameterizing y(x) via conditional Karhunen-Lo´eve expansions
+As in the PICKLE method [22, 23], we represent the unknown parameter
+field y(x) using the truncated CKLE
+yc(x, ξ) ≡ ¯yc(x) +
+Ny
+�
+i=1
+φy
+i (x)
+�
+λy
+i ξi,
+(7)
+where ξ ≡ (ξ1, ξ2, . . . , ξNy)⊤ is the vector of CKLE coefficients and the eigen-
+pairs {φy
+i (x), λy
+i }Ny
+i=1 are the solutions of the eigenvalue problem
+�
+D
+Cc
+y(x, x′)φy(x′) dx′ = λyφy(x).
+(8)
+Here, ¯yc(x) and Cc
+y(x, x′) denote the mean and covariance of y(x) conditioned
+on the measurements yc.
+The CKLE is truncated (i.e., Ny is selected) such as to achieve a desired
+relative tolerance
+rtoly ≡
+N
+�
+i=Ny+1
+λy
+i /
+N
+�
+i=1
+λy
+i ,
+(9)
+where N is the number of FV cells.
+The GPR (or Kriging) equations are used to compute yc(x) and Cc
+y(x, y):
+¯yc(x) = C(x)C−1
+s ys,
+(10)
+Cc
+y(x, x′) = Cy(x, x′) − C(x)C−1
+s C(x′),
+(11)
+where Cs is the Nys × Nys observation covariance matrix with elements
+[Cs]ij = Cy(ˆx[Iy]i, ˆx[Iy]j) and C(x) is the Nys-dimensional vector function
+with components [C(x)]i = Cy(x, ˆx[Iy]i).
+8
+
+The prior covariance kernel Cy(x, y) is estimated as in the GPR method
+by choosing a parameterized covariance model and computing its hyperpa-
+rameters by minimizing the marginal log-likelihood of the data ys [28]. In
+this work, we employ the 5/2-Mat´ern kernel as the prior covariance model,
+Cy(x, y) = σ2
+�
+1 +
+√
+5|x − y|
+l
++ 5
+3
+|x − y|2
+l2
+�
+exp
+�
+−
+√
+5|x − y|
+l
+�
+,
+with hyperparameters σ and λ, which correspond to the standard deviation
+and the correlation length, respectively.
+By representing y(x) via the CKLE (7), we replace the discrete vector
+y as the unknown of the inverse problem with the CKLE coefficients ξ.
+Specifically, we propose parameterizing y in the MAP problem (6) via the
+discrete CKLE
+yc(ξ) ≡ ¯yc + Ψyξ,
+(12)
+where
+[¯yc]i ≡ ¯yc(ˆxi),
+[Ψy]ij ≡
+�
+λy
+jφy
+j(ˆxi).
+We refer to this approach as the “CKLEMAP” method.
+Given that, for
+sufficiently smooth log-transmissivity fields, the number of CKLE coefficients
+required to accurately represent yc is much smaller than the number of FV
+cells, i.e., Ny ≪ N, the CKLEMAP method is less computationally expensive
+than the MAP method.
+4.2. CKLEMAP minimization problem formulation
+By solving Eq. (4) with y = yc(ξ), it can be seen that u can be expressed
+as a function of ξ; specifically,
+u(ξ) = [A(ξ)]−1 b(ξ),
+(13)
+where A (ξ) = A (yc(ξ)) and b (ξ) = b (yc(ξ)). By expressing u as a func-
+tion of ξ, we can remove the PDE constraint from Eq. (6), leading to the
+CKLEMAP unconstrained minimization problem
+min
+ξ
+1
+2∥us − Huu(ξ)∥2
+2 + 1
+2∥ys − Hyyc(ξ)∥2
+2 + γ
+2∥Dyc(ξ)∥2
+2.
+(14)
+To solve the CKLEMAP problem Eq. (14), we recast it as the nonlinear
+least-squares minimization problem
+min
+ξ
+1
+2 ∥f(ξ)∥2
+2 ,
+f(ξ) =
+�
+�
+us − Huu(ξ)
+ys − Hyyc(ξ)
+√γ Dyc(ξ)
+�
+� ,
+9
+
+which we solve using the Trust Region Reflective algorithm [29]. The least-
+square minimization algorithm requires the evaluation of the Jacobian Jξ
+of the objective vector of the least-squares problem, f, which is also the
+most computationally demanding part of the least-square minimization. This
+Jacobian evaluation is done in two steps. First, we evaluate the Jacobian of
+the objective vector with respect to yc, which reads
+Jξ = Jyc
+� ∂yc
+∂ξ
+I
+�
+=
+�
+�
+−Hu
+∂u(yc)
+∂yc
+−Hy
+√γ D
+�
+�
+�Ψy
+I
+�
+.
+(15)
+The partial derivative ∂u/∂yc is evaluated via the chain rule [3, 23] as de-
+scribed in Section 4.3. We note that most elements of Jyc are constant over
+iterations except the partial derivatives in the first block row. These con-
+stant values are computed once before the least-square minimization and
+reused in each iteration. With Jyc computed, Jξ can then be evaluated by
+postmultiplying the first block column by Ψy.
+4.3. Computations of partial derivatives in the evaluation of Jacobian
+In this section we describe how the partial derivative ∂u/∂yc, required to
+evaluate the Jacobian of Eq. (15), are evalauted. Let p denote yc
+i. Differen-
+tiating Eq. (4) with respect to p yields
+dl
+dp = ∂l
+∂u
+∂u
+∂p + ∂l
+∂p = A∂u
+∂p +
+�∂A
+∂p u − ∂b
+∂p
+�
+= 0,
+(16)
+which can be readily solved for ∂u/∂p, leading to the expression
+∂u
+∂p = −A−1
+�∂A
+∂p u − ∂b
+∂p
+�
+= −A−1 ∂l
+∂p
+����
+u
+.
+(17)
+It can be seen that evaluating ∂u/∂yc requires evaluating the sensitivities of
+the TPFA stiffness matrix A and right-hand side vector b with respect to
+yc. Substituting Eq. (17) into the first row block of Eq. (15) and taking the
+transpose yields
+� ∂l
+∂yc
+����
+u
+�⊤
+A−1H⊤
+u,
+(18)
+by the fact that A is symmetric.
+10
+
+Note that in the MAP method, the Jacobian is given as
+Jy =
+�
+�
+−Hu
+∂u(y)
+∂y
+−Hy
+√γ D
+�
+� ,
+(19)
+and the partial derivatives are computed as in the CKLEMAP method, with
+y being treated the same way as yc.
+4.4. Accelerated CKLEMAP method
+In the “accelerated” CKLEMAP method, we compute A−1H⊤
+u efficiently
+by exploiting the sparsity structure of the Cholesky factor of A. Recall that
+each column of H⊤
+u = (IN[Iu])⊤ has only one non-zero entry. Therefore, if
+the sparsity structure of the Cholesky factor L of A is known, the sparsity
+structure of each column of Z = L−1H⊤
+u is {closureL(i) | i ∈ Iu}, that is,
+the subset of vertices in the graph G(L) that have a path from each vertex
+i ∈ Iu [30].
+Figure 1 shows an example of a closure.
+Furthermore, the
+graph of a Cholesky factor L is a directed tree, and any closure induced by a
+vertex i is all the vertices along the path from i to the root of the tree [31].
+This enables a simple algorithm to find the sparsity structure of the solution
+of LZ = H⊤
+u. Figure 2 illustrates this algorithm together with a graphical
+example. Once we have the sparsity structure Zi of zi, the column i of Z, we
+only need the submatrix L[Zi, Zi] instead of the whole matrix L to solve for
+zi. Such submatrix is highlighted in blue dots in the lower triangular matrix
+L in Figure 2b. This eliminates the unnecessary computations involving the
+part of L that does not contribute to the final solutions, thus accelerating
+the computations. Furthermore, since the topology of the FV discretization
+is static, the sparsity structure of the Cholesky factor L is fixed throughout
+the entire least-square minimization procedure. Given this, together with
+the fact that Hu is constant, it follows that Zi is also fixed and only needs to
+be computed once. Figure 3 shows the closures of two observation locations
+in Iu on the Hanford Site experiment to be discussed in detail in Section 5.
+The gray lines indicate the cells that do not contribute to the columns of the
+Jacobian corresponding to either of these two locations.
+We note that, although the computations of the Jacobian can be acceler-
+ated by 3–4 times using the procedure described above, the overall execution
+time reduction in solving the minimization problems exhibited by the nu-
+merical experiments of Section 5 is 10–20%. This is because the nonlinear
+11
+
+1
+2
+3
+4
+5
+6
+7
+8
+Figure 1: Closure of a unit column vector e3 ≡ [0, 0, 1, 0, . . .]⊤ in a graph G(A). The
+nonzero entries of A−1e3 are those nodes in the closure, i.e., {3, 4, 6, 7, 8}.
+1: procedure FindSparsity(L, x)
+2:
+j ← x
+3:
+S ← {j}
+4:
+while j ̸= N do
+5:
+j ← argmini>jL[i, j] ̸= 0
+6:
+S ← S ∪ {j}
+7:
+end while
+8:
+return S
+9: end procedure
+(a) Algorithm
+L
+× z = ex
+(b) Graphical Example
+Figure 2: Algorithm for finding the sparsity structure S of z = L−1ex.
+least-squares minimization algorithm, the Trust Region algorithm, dominates
+most of the execution time. The execution times can be further reduced by
+optimizing the implementation of the Trust Region algorithm.
+5. Numerical experiments
+5.1. Case study
+We evaluate the performance of the proposed CKLEMAP formulation
+against MAP with a case study of parameter estimation in a steady-state
+two-dimensional groundwater model of the Hanford Site. The reference log-
+transmissivity field ˜y and boundary conditions uD and qN are based on the
+data obtained from a three-dimensional Hanford Site calibration study [27]
+and are shown in Figure 4. The details of the reference transmissivity field
+generation are given in [23].
+To study the scalability of the CKLEMAP
+and MAP methods with the problem size (i.e., the number of cells in the FV
+model), we generate the reference field at two additional resolutions with four
+times and 16 times the number of cells in the base FV model, respectively.
+12
+
+directed tree G(L) and its root
+closure 1
+closure 2
+common closure
+Figure 3: The directed tree G(L) structure on with two closures from different cells.
+13
+
+The numbers of cells in the low, medium, and high-resolution models
+are 1475, 5900, and 23600, respectively. For a higher resolution mesh, we
+divide each cell in a lower resolution model into four equiareal subcells and
+interpolate ˜y at the centers of each subcell, as well as uD and qN at the
+midpoints of each boundary edge of the boundary subcells.
+There are 558 wells at the Hanford Site where u can be potentially mea-
+sured [27]. Some of these wells are located in the same coarse or fine cells.
+Figure 4 shows the locations of the cells in the low-resolution FV model that
+contain at least one well.
+Since our model uses exclusively cells but not
+points to specify spatial locations, multiple wells are treated as a single well
+if they are located in the same cell. As a result, there are 323 wells in the
+low-resolution FV model, while the medium-resolution model has 408 wells.
+The aforementioned Hanford Site calibration study defined the Dirichlet
+and Neumann boundaries ΓD and ΓN as shown in Figure 4, and provides the
+estimates of the heads uD and the fluxes qN at these boundaries. In setting
+boundary conditions for our comparison study, we assume that uD and qN
+are both known and are given by the estimate.
+For each reference log-transmissivity field ˜y, we generate the hydraulic
+head field ˜u by solving the Darcy flow equation on the corresponding FV mesh
+with the Dirichlet and (deterministic) Neumann boundary conditions that are
+set as described above. The values of the reference y and u fields at all cell
+locations ˆxi are organized into the vectors ˜y and ˜u, respectively. Then, we
+randomly pick Nys well locations and treat the values of ˜y at these locations
+as y measurements to form ys.
+Similarly, we draw Nus measurements of
+the hydraulic head u from ˜u to form us. These measurements are treated
+as synthetic data sets and used in the CKLEMAP and MAP methods to
+estimate the entire y and u fields.
+We note that the aquifer at the Hanford Site is unconfined, and the
+use of Eq. (1) to describe flow at the Hanford Site relies on a conceptual
+simplification. A more accurate linear conceptual model for flow in an un-
+confined aquifer with a horizontal confining layer can be obtained based on
+the Dupuit–Forchheimer approximation in the form [32]
+∇ · [K(x)∇v(x)] = 0, x ∈ D,
+(20)
+where v(x) = u2(x) and K(x) is the depth-averaged conductivity. Mathe-
+matically, Eqs. (1) and (20) are identical, although the field u(x) computed
+using these two equations will be different. Therefore, solving the inverse
+14
+
+Umtanum Ridge
+Cold Creek Valley
+Dry Creek Valley
+Rattlesnake Hills
+Rattlesnake Springs
+Recharge Area
+Gable Butte
+Gable Mountain
+Columbia River
+Yakima River
+well locations
+Dirichlet boundary conditions
+Neumann boundary conditions
+no-flow boundary condition
+Figure 4: The coarse-resolution mesh of (NF V = 1475) cells with well locations marked,
+and the parts of boundaries colored for different types of prescribed boundary conditions.
+15
+
+problem for Eq. (1) is equivalent in complexity to solving the inverse prob-
+lem for Eq. (20). We also note that applying the Dupuit–Forchheimer ap-
+proximation to the Hanford Site aquifer will produce additional linear terms
+in Eq. (20) due to the variations in the elevation of the bottom confining
+layer of the aquifer.
+The implementation of CKLEMAP and MAP are written in Python using
+the NumPy and SciPy packages. All CKLEMAP and MAP simulations are
+performed using a 3.2 GHz 8-core Intel Xeon W CPU and 32 GB of 2666 MHz
+DDR4 RAM.
+The weight γ in the CKLEMAP and MAP minimization problems is
+empirically found to minimize the error with respect to the reference y fields
+as γ = 10−6. When a reference field is not known, these weights can be found
+using cross-validation [33].
+5.2. Performance of CKLEMAP as a function of the number of KL terms
+Table 1: Performance of CKLEMAP in estimating the coarse-resolution (NF V = 1475)
+mesh with Nys = 100 as functions of number of KL terms Ny.
+Ny
+200
+400
+600
+800
+1000
+least square
+iterations
+99–218
+44–335
+25–69
+28–177
+20–65
+execution
+time (s)
+17.55–
+42.14
+12.37–
+86.31
+9.76–
+24.73
+14.60–
+94.86
+14.25–
+36.29
+relative
+ℓ2 error
+0.265–
+0.568
+0.137–
+0.239
+0.081–
+0.098
+0.072–
+0.082
+0.072–
+0.083
+absolute
+ℓ∞ error
+13.08–
+42.69
+6.56–
+16.32
+3.71–5.63
+3.68–5.22
+3.46–5.31
+First, we study the relative ℓ2 and absolute ℓ∞ errors in the CKLEMAP
+solution for y as well as the time-to-solution and the number of iterations
+of the minimization algorithm as functions of Ny, the number of terms in
+the CKLE of y for Nys = 100. The relative ℓ2 and absolute ℓ∞ errors are
+16
+
+200
+400
+600
+800
+1000
+10−1
+10−0.5
+Number of KL terms
+ℓ2 errors
+Figure 5: Relative ℓ2 errors versus the number of KL terms.
+computed on the FV mesh, respectively, as
+ε2(y) ≡ ∥ˆy − ˜y∥2
+∥˜y∥2
+.
+(21)
+and
+ε∞(y) ≡ ∥ˆy − ˜y∥∞.
+(22)
+We find that for the considered inverse problem, all these quantities
+strongly depend on the locations of y measurements. Therefore, we compute
+these quantities for 10 different distributions of the measurement locations.
+The ranges of the ℓ2 and ℓ∞ errors, execution times, and the numbers of
+iterations are reported in Table 1. The ℓ2 error and its bounds as functions
+of Ny are also plotted in Figure 5. We find that the ℓ2 errors decrease with
+increasing Ny and converge to asymptotic values for Ny ≈ 800. The lower
+bound of ℓ∞ continues to decrease even for Ny greater than 800, while the
+upper bound increases from 5.22 to 5.31 as Ny increases from 800 to 1000.
+However, the relative changes of ℓ∞ are insignificant for Ny > 800. What
+is surprising is that the execution time does not significantly change with
+increasing Ny. While the time per iteration increases with Ny, the number of
+iterations tends to decrease. Therefore, in the rest of the numerical examples,
+we set Ny = 1000, which corresponds to rtoly on the order of 10−8.
+5.3. CKLEMAP and MAP errors versus the number of y measurements
+Next, we study the accuracy of the CKLEMAP and MAP methods in
+estimating y as the function of the number of y measurements. We assume
+17
+
+that u measurements are available at all wells.
+We start with the low-resolution model. Figure 6 shows the locations of y
+measurements, the y fields estimated by the MAP and CKLEMAP methods
+for Nys = 25, 50, 100, and 200, and the distributions of point errors in the
+MAP and CKLEMAP estimates of y relative to the reference field ˜y. For the
+considered measurement locations, we observe that the MAP and CKLEMAP
+methods have comparable accuracy for all Nys.
+Table 2 shows the ranges of relative ℓ2 and absolute ℓ∞ errors in the
+MAP and CKLEMAP y estimates as well as the number of iterations in the
+minimization algorithm and the execution times (in seconds) for Nys ranging
+from 25 to 200. Also included in this table are the execution times of the
+accelerated CKLEMAP method. We note that the accuracy (including the
+ℓ2 and absolute ℓ∞ errors) and the number of iterations in the accelerated
+CKLEMAP and CKLEMAP methods are the same.
+As expected, the accuracy of the MAP and CKLEMAP methods increases
+with Nys. The MAP and CKLEMAP methods are almost equally accurate,
+with ℓ2 and ℓ∞ errors in the CKLEMAP method being slightly smaller. How-
+ever, we observe that CKLEMAP is faster than MAP for all considered values
+of Nys except for Nys = 25, where the MAP’s lower bound of the execution
+time is less than that of the CKLEMAP. Accelerated CKLEMAP is about
+20% faster than CKLEMAP and for all considered values of Nys. Accelerated
+CKLEMAP is also faster than MAP for all considered cases; however, the
+speedup depends on Nys.
+In all examples reported in Table 2, the number of unknowns in the
+CKLEMAP method is 1000 (the number of terms in the CKLE expansion),
+while in the MAP method, this number is 1475 (the number of cells in the FV
+model). The reason for CKLEMAP being slower than MAP for Nys = 25 and
+certain y measurement locations is that for such locations MAP converges
+much faster. For example, the lower execution time bands in MAP and CK-
+LEMAP correspond to 29 and 50 iterations, respectively. However, because
+there are fewer unknowns in the CKLEMAP method, the CKLEMAP com-
+putational time per iteration is smaller than that in MAP. As a result, the
+computational time in the CKLEMAP is only 20% larger than that of MAP
+for these limiting cases. The time per iteration is further reduced in the
+accelerated CKLEMAP method, resulting in the execution time of acceler-
+ated CKLEMAP being less than that of MAP by 20%. We also note that
+for Nys > 25, MAP requires more iterations than CKLEMAP, making the
+computational advantages of CKLEMAP even more significant.
+18
+
+reference
+0
+2
+4
+6
+8
+10
+12
+Nys
+25
+50
+100
+200
+observation
+locations
+CKLEMAP
+estimates
+0
+2
+4
+6
+8
+10
+12
+CKLEMAP
+point errors
+0
+1
+2
+3
+4
+5
+6
+MAP
+estimates
+0
+2
+4
+6
+8
+10
+12
+MAP
+point errors
+0
+1
+2
+3
+4
+5
+6
+Figure 6: The fine-resolution (NF V = 5900) reference y fields, the CKLEMAP and MAP
+estimates of the y field and their point errors as functions of Nys.
+19
+
+Next, we perform a similar study for the medium-resolution model with
+N = 5900 cells. Table 3 provides a comparative summary of the models con-
+sidered for this case. Here, we find that CKLEMAP is slightly more accurate
+than MAP for all considered values of Nys and one to two orders of magni-
+tude faster than MAP. Accelerated CKLEMAP is approximately 10% faster
+than CKLEMAP. The computational advantage of CKLEMAP significantly
+increases with the problem size as the number of unknown parameters in the
+MAP linearly increases with the problem size while the number of parameters
+in the CKLEMAP is independent of the problem size.
+5.4. Scaling of the execution time with the problem size
+Table 2:
+Performance of MAP and CKLEMAP in estimating the coarse-resolution
+(NF V = 1475) mesh as functions of Nys.
+Nys
+solver
+25
+50
+100
+200
+least square
+iterations
+MAP
+29–95
+29–106
+41–60
+28–80
+CKLEMAP
+50–96
+26–70
+20–65
+33–62
+execution
+time (s)
+MAP
+31.36–
+91.50
+57.98–
+199.61
+76.39–
+123.08
+32.88–
+80.32
+CKLEMAP
+37.01–
+71.04
+21.71–
+51.86
+14.25–
+36.29
+17.73–
+40.57
+accelerated
+CKELMAP
+25.45–
+47.97
+17.00–
+40.00
+12.07–
+30.41
+12.78–
+29.09
+relative
+ℓ2 error
+MAP
+0.092–
+0.111
+0.084–
+0.101
+0.073–
+0.084
+0.068–
+0.073
+CKLEMAP
+0.091–
+0.109
+0.082–
+0.101
+0.072–
+0.083
+0.064–
+0.071
+absolute
+ℓ∞ error
+MAP
+5.38–6.61
+4.95–6.55
+4.06–6.35
+3.88–6.74
+CKLEMAP
+4.96–6.25
+4.73–6.11
+3.46–5.31
+5.63–5.71
+The comparison of Tables 2 and 3 shows that the execution times of
+the MAP, CKLEMAP, and accelerated CKLEMAP increase with the mesh
+resolution; however, the execution times of CKLEMAP and accelerated CK-
+LEMAP increase slower than that of MAP. To study the scalability of these
+20
+
+Table 3:
+Performance of MAP and CKLEMAP in estimating the fine-resolution
+(NF V = 5900) mesh as functions of Nys.
+Nys
+solver
+25
+50
+100
+200
+least square
+iterations
+MAP
+78–99
+71–97
+69–83
+23–76
+CKLEMAP
+53–114
+20–142
+36–60
+15–83
+execution
+time (s)
+MAP
+3907.00–
+4868.21
+3528.90–
+4580.40
+3533.06–
+4190.20
+1247.37–
+3733.05
+CKLEMAP
+88.76–
+181.67
+48.45–
+200.08
+62.59–
+100.04
+42.86–
+148.03
+accelerated
+CKELMAP
+77.05–
+141.90
+39.50–
+156.63
+52.14–
+81.19
+38.28–
+120.18
+relative
+ℓ2 error
+MAP
+0.0954–
+0.112
+0.081–
+0.105
+0.074–
+0.088
+0.065–
+0.073
+CKLEMAP
+0.0906–
+0.111
+0.081–
+0.105
+0.068–
+0.079
+0.061–
+0.069
+absolute
+ℓ∞ error
+MAP
+4.96–7.21
+5.45–7.28
+4.00–6.48
+4.37–5.20
+CKLEMAP
+4.21–6.66
+4.94–6.74
+3.79–5.71
+3.82–5.28
+21
+
+1475
+5900
+23600
+101
+102
+103
+104
+105
+3.7 · 10−8x2.91
+1.11 · 10−3x1.33
+8.11 · 10−4x1.35
+Number of FV cells
+Execution time (s)
+MAP
+CKLEMAP
+accelerated CKLEMAP
+Figure 7: Execution times of MAP, CKLEMAP, and accelerated CKLEMAP methods
+versus the number of FV cells. The execution times of MAP for the mesh with 23600 FV
+cells are estimated by extrapolation.
+methods with the problem size, we use these methods to estimate y in the
+high-resolution FV model with N = 23600 and, in Figure 7, we plot the
+execution times of these methods as functions of N. The number of y mea-
+surements in all simulations reported in this figure is set to Nys = 100.
+We also show the power-law models fitted to the scalability curves com-
+puted using MAP, CKLEMAP, and accelerated CKLEMAP. We note that
+for N = 23600, the MAP method did not converge after running for two
+days. Therefore, the power law relationship for the MAP method is obtained
+based on the execution times for N = 1475 and 5900 and used to estimate
+the MAP’s execution time for the highest resolution by extrapolation. We
+find that the MAP, CKLEMAP, and accelerated CKLEMAP execution times
+scale as N 2.91, N 1.33, and N 1.35, respectively.
+Therefore, the CKLEMAP
+methods have a computational advantage over the MAP method for large
+problems. The CKLEMAP and accelerated CKLEMAP methods have ap-
+proximately the same scalability, but for the same problem size, the acceler-
+ated CKLEMAP method is 10–20% faster than the CKLEMAP method.
+22
+
+6. Discussion and Conclusions
+We proposed the CKLEMAP method as an alternative to the MAP meth-
+ods for solving inverse PDE problems and used it for estimating the trans-
+missivity and hydraulic head in a two-dimensional steady-state groundwater
+model of the Hanford Site. The CKLEMAP method is based on the ap-
+proximation of unknown parameters (log-transmissivity in this case) with
+CKLEs. The advantage of using a CKLE over other representations (like
+DNNs in [13]) is that it enforces (i.e., exactly matches) the field measure-
+ments and the covariance structure, that is, it models the field as a realization
+of the conditional Gaussian field with a prescribed covariance function. As a
+general conclusion, we found that the accuracy of the MAP and CKLEMAP
+methods is essentially the same (with CKLEMAP being a few percents more
+accurate under most tested conditions), but CKLEMAP is faster than MAP.
+Specifically, we demonstrated that the CKLEMAP and MAP execution
+times scale with the problem size as N 1.33 and N 2.91, respectively, where N is
+the number of FV cells. The close-to-linear scaling of CKLEMAP’s execution
+time with problem size gives CKLEMAP a computational advantage over
+the MAP method for large-scale problems. We consider this to be the main
+advantage of the CKLEMAP method.
+For the same number of measurements, the accuracy of MAP and CK-
+LEMAP can depend on the measurement locations.
+Both the MAP and
+the CKLEMAP methods are, on average, equally accurate in terms of abso-
+lute ℓ∞ errors. The CKLEMAP method is slightly more accurate than the
+MAP method in terms of relative ℓ2 errors. The execution times of MAP
+and CKLEMAP increase, and their accuracy decreases, as the number of y
+measurements decreases.
+In the CKLEMAP method, execution time and accuracy increase with
+the increasing number of CKL terms. In this work, as a baseline, we used
+Ny = 1000, which corresponds to rtol < 10−8. We stipulate that this criterion
+is sufficient to obtain a convergent estimate of y with respect to the number
+of CKL terms.
+To further reduce the computational time, we proposed the accelerated
+CKLEMAP method, which takes advantage of the sparse structure of the
+stiffness matrix in the FV discretization of the residual term. We demon-
+strated that the scalability of the accelerated CKLEMAP and CKLEMAP
+methods is approximately the same; however, for the same problem size,
+accelerated CKLEMAP is 10–20% faster than the CKLEMAP method.
+23
+
+7. Acknowledgments
+This research was partially supported by the U.S. Department of Energy
+(DOE) Advanced Scientific Computing program and the United States Geo-
+logical Survey. Pacific Northwest National Laboratory is operated by Battelle
+for the DOE under Contract DE-AC05-76RL01830. The data and codes used
+in this paper are available at https://github.com/yeungyh/cklemap.git.
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+CSRCZ: A Dataset About Corporate Social Responsibility in Czech
+Republic
+Xhesilda Vogli
+Department of Management
+Faculty of Economics and Management
+Czech University of Life Sciences
+[email protected]
+Erion Çano
+Digital Philology
+Data Mining and Machine Learning
+University of Vienna, Austria
+[email protected]
+Abstract
+As stakeholders’ pressure on corporates for
+disclosing their corporate social responsibility
+operations grows, it is crucial to understand
+how efficient corporate disclosure systems are
+in bridging the gap between corporate social
+responsibility reports and their actual practice.
+Meanwhile, research on corporate social re-
+sponsibility is still not aligned with the recent
+data-driven strategies, and little public data are
+available. This paper aims to describe CSRCZ,
+a newly created dataset based on disclosure re-
+ports from the websites of 1 000 companies
+that operate in Czech Republic.
+Each com-
+pany was analyzed based on three main param-
+eters: company size, company industry, and
+company initiatives. We describe the content
+of the dataset as well as its potential use for
+future research. We believe that CSRCZ has
+implications for further research, since it is the
+first publicly available dataset of its kind.
+1
+Introduction
+Corporate Social Responsibility (CSR) has evolved
+from a “why” in the early 1950s (Carroll and
+Brown, 2018) to a “must” in recent years. Gen-
+erally, CSR is considered a self-regulating business
+model which helps companies to contribute to so-
+cietal goals and be socially accountable to them-
+selves and the public. It is highly influenced by the
+legal context (LIANG and RENNEBOOG, 2017)
+and the socio-political context (Tilt, 2016) of the
+countries where the companies operate. Globally,
+more and more companies are engaging in CSR
+initiatives. They are therefore providing more so-
+cial information to the public. As a result, CSR
+disclosure has grown to be one of the main study
+directions for researchers of this field (Goyal et al.,
+2015; Halkosa and Skouloudis, 2016).
+While reaching adequate standards of sustain-
+ability disclosure or reporting is desirable, there are
+several obstacles to overcome. Sustainability re-
+porting is optional, in contrast, to strictly regulated
+financial reporting, and it is consequently charac-
+terized by a lack of uniformity (Braam and Peeters,
+2018; Bhattacharyya and Cummings, 2015). Prior
+studies have been generally focused on the fac-
+tors that drive the disclosure of these initiatives, the
+given information, the mode of communication and
+their impact on the company’s performance and im-
+age (Gonçalves and Gaio, 2023; Benoit-Moreau
+and Parguel, 2011). These factors that may in-
+fluence CSR disclosure reports of a company are
+usually classified as: (i) internal, such as company
+size, industry sector, financial performance, and
+corporate governance; (ii) external, such as country
+of origin, stakeholders, media, or social and politi-
+cal environment (Fifka, 2013; Morhardt, 2009).
+Considering the limited research that is avail-
+able, a few studies also try to investigate the pos-
+sibility that “country” can influence CSR initia-
+tives and disclosure levels (Kansal et al., 2014;
+Fufa and Roba, 2021; Khan et al., 2021). On one
+hand, deeper correlations between other factors and
+the CSR initiatives of companies are mostly miss-
+ing. On the other hand, most of the studies (e.g.,
+those cited above) are methodologically “conserva-
+tive” and do not exploit data-driven approaches that
+have surged in the last decade (Pugna et al., 2022;
+Abuimara et al., 2022; Çano, 2018). This trend
+towards data-driven research is mostly conducted
+using English language resources (e.g., datasets)
+which are the most numerous on the internet. There
+are still several studies and resources in Czech or
+other languages becoming common and available
+(Çano and Bojar, 2019; Sestino and Mauro, 2022).
+In this paper, we try to foster data-driven re-
+search about CSR by creating and describing
+CSRCZ, a freely available dataset containing pub-
+lic information of 1 000 companies operating in
+the Czech Republic.1 In the following sections,
+we present the information retrieval process steps
+that were followed. We also describe the available
+1https://zenodo.org/record/7495802
+arXiv:2301.03404v1  [econ.GN]  5 Jan 2023
+
+Attribute
+Content Type
+Company Name
+String
+Number of employees
+Integer
+Has a CSR page
+Binary
+Industry Sector
+String
+Size of company
+Categorical
+Initiatives
+String
+Website
+URL
+Table 1: Data attributes and their respective types.
+data fields (especially those related to CSR), their
+characteristic values, and some relevant statistics.
+Finally, we discuss potential utilization of CSRCZ
+content in the context of future CSR research.
+2
+Dataset Content
+The sources for constructing the CSRCZ dataset
+were collected from the public websites of 1 000
+companies currently operating in the Czech Repub-
+lic. Initially, the websites of those companies were
+retrieved by jobs.cz. Each website was analyzed
+and only the information relating to CSR was col-
+lected. The relevant attributes that were considered
+are presented in Table 1.
+Company Name represents the official name of
+the company as it is registered in the Czech Re-
+public. It is saved as a text string. Number of
+employees is an integer that includes the total num-
+ber of full-time employees, part-time employees,
+seasonal workers, and partners. Has a CSR page is
+a binary value with ‘1’ indicating that this company
+includes in its website some page with information
+regarding CSR policies or practices, and ‘0’ indi-
+cating that it does not. Industry Sector is a string
+describing the market segment of the company or
+the type of activity it mostly performs.
+Size of company is a categorical variable that de-
+scribes the size of the company. Any company with
+fewer than 10 employees is considered as ‘Micro’.
+Those with up to 50 employees are ‘Small’ compa-
+nies. The companies are considered ‘Medium’ if
+they have 51 up to 250 employees. Any company
+with 251 or more employees is ‘Large’. Initia-
+tives is probably the most important attribute with
+respect to the CSR analysis. It is a long string
+describing any CSR-related policies, practices or
+initiatives that the company outlines. Finally, Web-
+site is the URL from which the information was
+retrieved.
+Size
+Number
+Percent
+Unknown
+1
+0.1
+Micro
+125
+12.5
+Small
+214
+21.4
+Medium
+330
+33
+Large
+330
+33
+Table 2: Size statistics of the selected companies
+3
+Dataset Statistics
+In the following sections, CSRCZ content is dis-
+cussed in detail. The characteristics values of the
+respective fields are analyzed and presented in a
+tabular format. The codes for deriving the statistics
+are available online.2
+3.1
+Size and Employees
+The size of a company is an important factor that
+is usually related to the capacities that a company
+has to implement goals and practices in fulfilment
+of its CSR strategy. One way to determine the size
+of a company is by using the number of its employ-
+ees, same as we described in Section 2. This is
+obviously a simplistic approach, since other factors
+like different types of assets the company owns (un-
+fortunately, this type of information is not always
+public) do also indicate how big it is.
+We inspected the collected data and found that
+most of the companies are large or medium, with
+each category representing 33 % of the instances.
+There are also 214 small companies which make
+up 21.4 % of the total. There are also 125 compa-
+nies (representing 12.5 % of the total) which are
+considered to be very small or “Micro”. For one
+of the sampled companies, it was not possible to
+determine its size. The full statistics are presented
+in Table 2 and depicted in Figure 1.
+We also checked the number of employees for
+each size category. Specifically, we found the min-
+imum, maximum and average number of employ-
+ees in the ‘Micro’, ‘Small’, ‘Medium’, and the
+‘Large’ companies in CSRCZ data. In the case
+of ‘Micro’ companies, there are at least 5 and at
+most 9 employees, with an average of 6.54. The
+same statistics for the case of ‘Small’ companies
+are 10, 49 and 31.97 respectively. Companies of
+a ‘Medium’ size have an average of 169.62 em-
+ployees. Finally, the ‘Large’ companies do have
+a maximum of 10000 employees (the biggest in
+2https://github.com/erionc/csrcz-stats
+
+Figure 1: Size distribution of the selected companies.
+Company
+Min
+Max
+Avg
+Micro
+5
+9
+6.54
+Small
+10
+49
+31.97
+Medium
+50
+249
+169.62
+Large
+299
+10000
+1635.58
+Table 3: Minimum, maximum and average number of
+employees for each company category.
+CSRCZ) with an average of 1635.58. The statistics
+are summarized in Table 3 and depicted in Figure 3.
+Figure 2: Average number of employees in each com-
+pany size category.
+3.2
+Industry Sector
+The industry sector is an interesting attribute since
+it could shed light on important trends that relate
+to the CSR initiatives and the different sectors the
+companies operate. According to GICS (Global
+Industry Classification Standard), eleven industry
+sectors represent the majority of industry types
+nowadays.3
+3https://www.msci.com/our-solutions/
+indexes/gics
+Sector
+Number
+Percent
+Unknown
+607
+60.7
+Communication Services
+17
+1.7
+Consumer Discretionary
+91
+9.1
+Consumer Staples
+31
+3.1
+Energy
+15
+1.5
+Financials
+28
+2.8
+Health Care
+16
+1.6
+Industrials
+111
+11.1
+Information Technology
+56
+5.6
+Materials
+25
+2.5
+Real estate
+3
+0.3
+Utilities
+0
+0
+Table 4: Sector statistics of the selected companies
+Communication Services is an industry that in-
+cludes media and entertainment or any of the
+telecommunication services.
+Consumer Discretionary involves the retail in-
+dustry, hotels, restaurants, leisure, and house-
+hold durables.
+Consumer Staples is an industry category that
+groups all food products, beverages, and to-
+bacco.
+Energy includes oil, gas, consumable fuels, and
+energy services.
+Financials is a category grouping all banking ser-
+vices, capital markets, and insurance services.
+Health Care involves health care providers and
+pharmaceuticals.
+Industrials includes transportation services such
+as airlines, marine, road & rail and all services
+related to it.
+Information Technology involves IT services,
+software, technology hardware, storage, and
+peripherals.
+Materials includes all industry sectors that pro-
+duce chemicals, construction materials, pack-
+aging, metals, and mining.
+Real estate includes real estate investment trusts
+and real estate services.
+Utilities includes electric, gas, and water utilities
+services.
+
+33.0%
+33.0%
+30
+25
+21.4%
+20
+15
+12.5%
+10
+5 -
+0
+0.1%
+Unknown
+Micro
+Small
+Medium
+Large1635.58
+1600
+1400
+1200
+1000
+800
+600
+400
+200
+169.62
+6.54
+31.97
+0
+Micro
+Small
+Medium
+LargeCSR Initiatives
+Min
+Max
+Avg
+Characters
+0
+32023
+1218.01
+Tokens
+0
+4870
+191.73
+Table 5: Minimum, maximum and average number of
+characters and tokens for each CSR initiative.
+We explored the data and identified the num-
+ber and percentage of the companies belonging
+to each of the above listed industry sectors. The
+gathered statistics are summarized in Table 4. Un-
+fortunately, this indicator is not available for many
+of the data records. Among the available sectors
+we found, ‘Industrials’ is the most popular, with
+111 companies or 11.1 % of the total. The sec-
+tor ‘Consumer Dicretionary’ comes next with 91
+companies. ‘Information Technology’, ‘Cosumer
+Staples’ and ‘Financials’ are also common, with
+56, 31 and 28 records each. The most unpopular
+sectors are ‘Real estate’ and ‘Utilities’, with 3 and
+0 companies.
+3.3
+CSR Initiatives
+The most important record attribute of the CSRCZ
+dataset is probably ‘Initiatives’, where the CSR
+mission, goals and practices of the companies are
+summarized. This information usually comes as
+a sequence of sentences, or sometimes as a few
+paragraphs. A trivial statistical evaluation here is
+to check its length in characters or tokens, despite
+the fact that a short or long ‘Initiatives’ text in the
+website does not necessarily mean that the CSR
+commitment of a company is low or high.
+We used NLTK word tokenizer to tokenize the
+texts.4 Unfortunately, a high number of the sam-
+pled companies (more specifically 610 which have
+0 length of characters and tokens) have not pro-
+vided such a description in their websites. The
+longest CSR initiatives texts have 32023 charac-
+ters and 4870 tokens. The average length of this
+attribute is about 1218 characters and 191 tokens.
+These statistics are summarized in Table 5.
+4
+Discussion
+Despite the fact that information is broadly avail-
+able for a lot of organizations, many companies
+regularly fail to present the CSR data in a consis-
+tent way and assorted according to a framework.
+As the attention towards CSR is raising and the
+community becoming more watchful, the need for
+4https://www.nltk.org/
+a standardized definition and CSR framework has
+been rising. The need for applying data-driven
+methodologies and providing structured datasets is
+also in rise.
+The purpose of this work is to foster data-driven
+CSR research by providing and describing CSRCZ,
+a recently created dataset. We believe that using
+CSRCZ can provide a better view of the current
+understanding of CSR in companies that operate
+in the Czech Republic and in a global context as
+well. Various correlations between internal and
+external company factors and its CSR initiatives
+can be found. Those findings could be used to de-
+velop further frameworks and management strate-
+gies in order to better communicate CSR initiatives
+to stakeholders being those external or internal.
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+

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+page_content='CSRCZ: A Dataset About Corporate Social Responsibility in Czech  Republic  Xhesilda Vogli  Department of Management  Faculty of Economics and Management  Czech University of Life Sciences  vogli@pef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
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+page_content='cz  Erion Çano  Digital Philology  Data Mining and Machine Learning  University of Vienna, Austria  erion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
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+page_content='at  Abstract  As stakeholders’ pressure on corporates for  disclosing their corporate social responsibility  operations grows, it is crucial to understand  how efficient corporate disclosure systems are  in bridging the gap between corporate social  responsibility reports and their actual practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Meanwhile, research on corporate social re-  sponsibility is still not aligned with the recent  data-driven strategies, and little public data are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' This paper aims to describe CSRCZ,  a newly created dataset based on disclosure re-  ports from the websites of 1 000 companies  that operate in Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Each com-  pany was analyzed based on three main param-  eters: company size, company industry, and  company initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' We describe the content  of the dataset as well as its potential use for  future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' We believe that CSRCZ has  implications for further research, since it is the  first publicly available dataset of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  1  Introduction  Corporate Social Responsibility (CSR) has evolved  from a “why” in the early 1950s (Carroll and  Brown, 2018) to a “must” in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Gen-  erally, CSR is considered a self-regulating business  model which helps companies to contribute to so-  cietal goals and be socially accountable to them-  selves and the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' It is highly influenced by the  legal context (LIANG and RENNEBOOG, 2017)  and the socio-political context (Tilt, 2016) of the  countries where the companies operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
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+page_content=' They are therefore providing more so-  cial information to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' As a result, CSR  disclosure has grown to be one of the main study  directions for researchers of this field (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
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+page_content=' Halkosa and Skouloudis, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  While reaching adequate standards of sustain-  ability disclosure or reporting is desirable, there are  several obstacles to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Sustainability re-  porting is optional, in contrast, to strictly regulated  financial reporting, and it is consequently charac-  terized by a lack of uniformity (Braam and Peeters,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Bhattacharyya and Cummings, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Prior  studies have been generally focused on the fac-  tors that drive the disclosure of these initiatives, the  given information, the mode of communication and  their impact on the company’s performance and im-  age (Gonçalves and Gaio, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Benoit-Moreau  and Parguel, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' These factors that may in-  fluence CSR disclosure reports of a company are  usually classified as: (i) internal, such as company  size, industry sector, financial performance, and  corporate governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' (ii) external, such as country  of origin, stakeholders, media, or social and politi-  cal environment (Fifka, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Morhardt, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Considering the limited research that is avail-  able, a few studies also try to investigate the pos-  sibility that “country” can influence CSR initia-  tives and disclosure levels (Kansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Fufa and Roba, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' On one  hand, deeper correlations between other factors and  the CSR initiatives of companies are mostly miss-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' On the other hand, most of the studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=',  those cited above) are methodologically “conserva-  tive” and do not exploit data-driven approaches that  have surged in the last decade (Pugna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Abuimara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Çano, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' This trend  towards data-driven research is mostly conducted  using English language resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=', datasets)  which are the most numerous on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' There  are still several studies and resources in Czech or  other languages becoming common and available  (Çano and Bojar, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Sestino and Mauro, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  In this paper, we try to foster data-driven re-  search about CSR by creating and describing  CSRCZ, a freely available dataset containing pub-  lic information of 1 000 companies operating in  the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1 In the following sections,  we present the information retrieval process steps  that were followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' We also describe the available  1https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='org/record/7495802  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='03404v1  [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='GN]  5 Jan 2023  Attribute  Content Type  Company Name  String  Number of employees  Integer  Has a CSR page  Binary  Industry Sector  String  Size of company  Categorical  Initiatives  String  Website  URL  Table 1: Data attributes and their respective types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  data fields (especially those related to CSR), their  characteristic values, and some relevant statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Finally, we discuss potential utilization of CSRCZ  content in the context of future CSR research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  2  Dataset Content  The sources for constructing the CSRCZ dataset  were collected from the public websites of 1 000  companies currently operating in the Czech Repub-  lic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Initially, the websites of those companies were  retrieved by jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Each website was analyzed  and only the information relating to CSR was col-  lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The relevant attributes that were considered  are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Company Name represents the official name of  the company as it is registered in the Czech Re-  public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' It is saved as a text string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Number of  employees is an integer that includes the total num-  ber of full-time employees, part-time employees,  seasonal workers, and partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Has a CSR page is  a binary value with ‘1’ indicating that this company  includes in its website some page with information  regarding CSR policies or practices, and ‘0’ indi-  cating that it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Industry Sector is a string  describing the market segment of the company or  the type of activity it mostly performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Size of company is a categorical variable that de-  scribes the size of the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Any company with  fewer than 10 employees is considered as ‘Micro’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Those with up to 50 employees are ‘Small’ compa-  nies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The companies are considered ‘Medium’ if  they have 51 up to 250 employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Any company  with 251 or more employees is ‘Large’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Initia-  tives is probably the most important attribute with  respect to the CSR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' It is a long string  describing any CSR-related policies, practices or  initiatives that the company outlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Finally, Web-  site is the URL from which the information was  retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Size  Number  Percent  Unknown  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1  Micro  125  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='5  Small  214  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='4  Medium  330  33  Large  330  33  Table 2: Size statistics of the selected companies  3  Dataset Statistics  In the following sections, CSRCZ content is dis-  cussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The characteristics values of the  respective fields are analyzed and presented in a  tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The codes for deriving the statistics  are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1  Size and Employees  The size of a company is an important factor that  is usually related to the capacities that a company  has to implement goals and practices in fulfilment  of its CSR strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' One way to determine the size  of a company is by using the number of its employ-  ees, same as we described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' This is  obviously a simplistic approach, since other factors  like different types of assets the company owns (un-  fortunately, this type of information is not always  public) do also indicate how big it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  We inspected the collected data and found that  most of the companies are large or medium, with  each category representing 33 % of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  There are also 214 small companies which make  up 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='4 % of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' There are also 125 compa-  nies (representing 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='5 % of the total) which are  considered to be very small or “Micro”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' For one  of the sampled companies, it was not possible to  determine its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The full statistics are presented  in Table 2 and depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  We also checked the number of employees for  each size category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Specifically, we found the min-  imum, maximum and average number of employ-  ees in the ‘Micro’, ‘Small’, ‘Medium’, and the  ‘Large’ companies in CSRCZ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' In the case  of ‘Micro’ companies, there are at least 5 and at  most 9 employees, with an average of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The  same statistics for the case of ‘Small’ companies  are 10, 49 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='97 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Companies of  a ‘Medium’ size have an average of 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='62 em-  ployees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Finally, the ‘Large’ companies do have  a maximum of 10000 employees (the biggest in  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='com/erionc/csrcz-stats  Figure 1: Size distribution of the selected companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Company  Min  Max  Avg  Micro  5  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='54  Small  10  49  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='97  Medium  50  249  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='62  Large  299  10000  1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='58  Table 3: Minimum, maximum and average number of  employees for each company category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  CSRCZ) with an average of 1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The statistics  are summarized in Table 3 and depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Figure 2: Average number of employees in each com-  pany size category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='2  Industry Sector  The industry sector is an interesting attribute since  it could shed light on important trends that relate  to the CSR initiatives and the different sectors the  companies operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' According to GICS (Global  Industry Classification Standard), eleven industry  sectors represent the majority of industry types  nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='3  3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='msci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='com/our-solutions/  indexes/gics  Sector  Number  Percent  Unknown  607  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='7  Communication Services  17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='7  Consumer Discretionary  91  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1  Consumer Staples  31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1  Energy  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='5  Financials  28  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='8  Health Care  16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='6  Industrials  111  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1  Information Technology  56  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='6  Materials  25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='5  Real estate  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='3  Utilities  0  0  Table 4: Sector statistics of the selected companies  Communication Services is an industry that in-  cludes media and entertainment or any of the  telecommunication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Consumer Discretionary involves the retail in-  dustry, hotels, restaurants, leisure, and house-  hold durables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Consumer Staples is an industry category that  groups all food products, beverages, and to-  bacco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Energy includes oil, gas, consumable fuels, and  energy services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Financials is a category grouping all banking ser-  vices, capital markets, and insurance services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Health Care involves health care providers and  pharmaceuticals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Industrials includes transportation services such  as airlines, marine, road & rail and all services  related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Information Technology involves IT services,  software, technology hardware, storage, and  peripherals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Materials includes all industry sectors that pro-  duce chemicals, construction materials, pack-  aging, metals, and mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Real estate includes real estate investment trusts  and real estate services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Utilities includes electric, gas, and water utilities  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='0%  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='0%  30  25  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='4%  20  15  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='5%  10  5 -  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1%  Unknown  Micro  Small  Medium  Large1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='58  1600  1400  1200  1000  800  600  400  200  169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='62  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='54  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='97  0  Micro  Small  Medium  LargeCSR Initiatives  Min  Max  Avg  Characters  0  32023  1218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='01  Tokens  0  4870  191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='73  Table 5: Minimum, maximum and average number of  characters and tokens for each CSR initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  We explored the data and identified the num-  ber and percentage of the companies belonging  to each of the above listed industry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The  gathered statistics are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Un-  fortunately, this indicator is not available for many  of the data records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Among the available sectors  we found, ‘Industrials’ is the most popular, with  111 companies or 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='1 % of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The sec-  tor ‘Consumer Dicretionary’ comes next with 91  companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' ‘Information Technology’, ‘Cosumer  Staples’ and ‘Financials’ are also common, with  56, 31 and 28 records each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The most unpopular  sectors are ‘Real estate’ and ‘Utilities’, with 3 and  0 companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='3  CSR Initiatives  The most important record attribute of the CSRCZ  dataset is probably ‘Initiatives’, where the CSR  mission, goals and practices of the companies are  summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' This information usually comes as  a sequence of sentences, or sometimes as a few  paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' A trivial statistical evaluation here is  to check its length in characters or tokens, despite  the fact that a short or long ‘Initiatives’ text in the  website does not necessarily mean that the CSR  commitment of a company is low or high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  We used NLTK word tokenizer to tokenize the  texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='4 Unfortunately, a high number of the sam-  pled companies (more specifically 610 which have  0 length of characters and tokens) have not pro-  vided such a description in their websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The  longest CSR initiatives texts have 32023 charac-  ters and 4870 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The average length of this  attribute is about 1218 characters and 191 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  These statistics are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  4  Discussion  Despite the fact that information is broadly avail-  able for a lot of organizations, many companies  regularly fail to present the CSR data in a consis-  tent way and assorted according to a framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  As the attention towards CSR is raising and the  community becoming more watchful, the need for  4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='nltk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='org/  a standardized definition and CSR framework has  been rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The need for applying data-driven  methodologies and providing structured datasets is  also in rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  The purpose of this work is to foster data-driven  CSR research by providing and describing CSRCZ,  a recently created dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' We believe that using  CSRCZ can provide a better view of the current  understanding of CSR in companies that operate  in the Czech Republic and in a global context as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Various correlations between internal and  external company factors and its CSR initiatives  can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Those findings could be used to de-  velop further frameworks and management strate-  gies in order to better communicate CSR initiatives  to stakeholders being those external or internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  References  Tareq Abuimara, Brodie W Hobson, Burak Gunay, and  William O’Brien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
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+page_content=' Corporate responsibility re-  porting and its determinants in comparative perspec-  tive – a review of the empirical literature and a meta-  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Business Strategy and The Environment,  22:1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Tolossa Fufa and Yadessa Roba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Internal and ex-  ternal determinants of corporate social responsibility  practices in multinational enterprise subsidiaries in  developing countries: evidence from ethiopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Fu-  ture Business Journal, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Tiago Cruz Gonçalves and Cristina Gaio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Cor-  porate sustainability disclosure and media visibil-  ity: Mixed method evidence from the tourism sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Journal of Business Research, 155:113447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Praveen Goyal, Zillur Rahman, and Absar Ahmad  Kazmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Identification and prioritization of cor-  porate sustainability practices using analytical hier-  archical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Journal of Modelling in Manage-  ment, 10(1):23–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  George Halkosa and Antonis Skouloudis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Ex-  ploring the current status and key determinants of  corporate disclosure on climate change: Evidence  from the greek business sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Environmental Sci-  ence & Policy, 56(1):22–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Monika Kansal, Mahesh Joshi, and Gurdip Singh Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Determinants of corporate social responsibil-  ity disclosures: Evidence from india.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Advances in  Accounting, 30(1):217–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Majid Khan, James Lockhart, and Ralph Bathurst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The institutional analysis of csr: Learnings  from an emerging country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Emerging Markets Re-  view, 46:100752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Corporate Social Responsibility  in Emerging Markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  HAO LIANG and LUC RENNEBOOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' On the  foundations of corporate social responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' The  Journal of Finance, 72(2):853–910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Emil Morhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Corporate social responsibil-  ity and sustainability reporting on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Busi-  ness Strategy and The Environment, 19:436–452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Irina Bogdana Pugna, Dana Maria Boldeanu, Mirela  Gheorghe, Gabriel Cozgarea, and Adrian Nicolae  Cozgarea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Management perspectives towards  the data-driven organization in the energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Energies, 15(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Andrea Sestino and Andrea De Mauro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Leverag-  ing artificial intelligence in business: Implications,  applications and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Technology Analysis &  Strategic Management, 34(1):16–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  Carol A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content=' Corporate social responsibility re-  search: The importance of context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}
+page_content='  International  Journal of Corporate Social Responsibility (JCSR),  1(2):1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'}

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+                                   TITLE  
+ 
+Using Gamma Functions in the Mathematical Formulation of the 
+Impact Crater Size-Age Frequency Distribution on Earth and Mars. 
+                                  Author: William F Bruckman 
+                                                   Abstract 
+A review of a mathematical formulation that describes the number of impact craters as 
+function of diameter and time of formation is presented, where the use of Gamma 
+functions is emphasized. The application of this formalism for the description of the 
+impact crater data of Planets Earth and Mars is also discussed. 
+ 
+1. Introduction  
+      When solving differential or integral equations an ideal outcome is to express the 
+solution in terms of elementary or special functions. In that case the mathematical and 
+physical interpretation of the solutions is clarified. Moreover, with the use of algebraic 
+computing, the comparison of the prediction of theoretical models with the observational 
+data is greatly facilitated. 
+      This paper will consider work in reference1 (Earth and Mars Crater Size Frequency 
+Distribution and Impact Rates: Theoretical and Observational Analysis; William 
+Bruckman, Abraham Ruiz, and Elio Ramos; Arxiv: 1212.3273), which presented a 
+theoretical formulation describing impact crater data on Earth and Mars, giving the 
+number of craters as functions of diameter, and time of formation, successfully 
+reproducing the observations. The revision will emphasize the presentation of the 
+solutions of the models in terms of Gamma functions. 
+2. General Considerations  
+       Impact craters, of a given diameter 𝐷, are formed at a certain rate 𝛷, and are also 
+depleted, as they get older, by a variety of processes, at a rate proportional to their 
+already existing number of craters, 𝑁. Hence, the number of craters eliminated in the 
+time interval dt can be express as 𝐶𝑁𝑑𝑡, where 𝐶 is a parameter representing the rate of 
+elimination per crater. On the other hand, in this time interval we also have that the 
+number of craters produced by impacts is 𝛷𝑑𝑡, and thus the net change in the number 
+of crater numbers, 𝑑𝑁, is given by 
+
+                        𝑑𝑁 = 𝛷𝑑𝑡 − 𝐶𝑁𝑑𝑡 =   ( 
+𝛷
+𝐶 −  𝑁 )𝐶𝑑𝑡 .                                                       (1)   
+This equation is expected to represent well the observational data if the number of 
+craters is large enough to justify the assumptions that analytical mathematical continuity 
+is a good approximation to the discrete and probabilistic nature of the problem.  
+       We see from Eq. (1) that 𝑁= constant implies that 
+                         𝑁 =  
+𝛷
+𝐶 =  𝛷𝜏𝑚,                                                                                            (2)                    
+                   𝜏𝑚 ≡ 1/𝐶.                                                                                                      (3) 
+In this situation (saturation) the number of craters produced by impacts is equal to the 
+number of craters eliminated. The dimension of  𝜏𝑚 is time, and we will see later in this 
+section that this time is related to the concept of “craters mean life.” 
+       Equation (1) was integrated in reference (1) to obtain 
+                        𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`)
+𝜏
+0
+𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`.                                                        (4) 
+                        𝐶̅ ≡
+∫ 𝐶𝑑𝜏`
+𝜏
+0
+𝜏
+ ,                                                                                                    (5) 
+where 𝐶̅ is the time average of 𝐶, and 𝑁(𝐷, 0, 𝜏) defined in Eq. (4) denotes the number 
+of craters of diameter 𝐷, per bin size, observed at the present time (𝜏` = 0), with age 
+younger than 𝜏 . Accordingly, defining the term “per bin”, we have that the integral  
+                    𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫
+𝑁(𝐷, 0, 𝜏)𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+,                                                                (6) 
+                 
+gives the total number of craters with diameters in the interval between 𝐷𝑖 and 𝐷𝑓, 
+observed at the present time, with age of formation younger than 𝜏. Also, 𝛷(𝐷, 𝜏) is the 
+rate of meteorite impacts, per bin, forming craters of diameter 𝐷 at time 𝜏, so that 𝛷𝐶: 
+                            𝛷𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫
+{𝛷(𝐷, 𝜏)
+𝐷𝑓
+𝐷𝑖
+}𝑑𝐷,                                                                (7) 
+Is the cumulative impact rate of formation of craters with diameters in the interval 
+between 𝐷𝑖 and 𝐷𝑓. For instance, if 𝐷𝑓 → ∞, which is of common use, the above integral 
+is the total cumulative impact rate of formation of craters with diameters larger than 𝐷𝑖. 
+        Equations (6) and (4) can be generalized so that the lower 𝜏 limit is different from 
+zero: 
+                   𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫
+𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+  ,                                                         (8) 
+
+where 
+                  𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) = ∫ {𝛷(𝐷, 𝜏`)
+𝜏𝑓
+𝜏𝑖
+𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`.                                                    (9) 
+Thus, Eqs. (8) and (9) refer to craters with ages between 𝜏𝑖 and  𝜏𝑓.  
+       Further discussion and applications of Eq. (8) to the Earth’s crater record will be 
+continued in Section 4. For the planet Mars, however, we will be applying Eq. (4) in the 
+next section, but now continue its interpretation below. 
+        Since the quantity 𝛷(𝐷, 𝜏`)𝑑𝜏`. Is the number of craters formed at time 𝜏`, during 
+the interval 𝑑𝜏`, and the integrand in Eq. (4): 𝛷(𝐷, 𝜏`)𝑑𝜏`𝐸𝑥𝑝[−𝐶̅𝜏`],  is the number of 
+these craters, of age 𝜏` , that remain at the present time, then the expression 𝐸𝑥𝑝[−𝐶̅𝜏`] 
+represents the fraction of these formed craters that survive after the time 𝜏`. It is then 
+usual to call the inverse of 𝐶̅  “the mean life”: 𝜏𝑚𝑒𝑎𝑛,   
+                     𝜏𝑚𝑒𝑎𝑛 ≡
+1
+𝐶̅ ; 1/𝜏𝑚𝑒𝑎𝑛 = 𝐶̅ ≡
+∫ 𝐶𝑑𝜏`
+𝜏
+0
+𝜏
+= 
+∫ (1
+𝜏𝑚)𝑑𝜏`
+𝜏
+0
+𝜏
+.                                         (10)                     
+Thus, in this context 𝜏𝑚𝑒𝑎𝑛 can be viewed as the mean life of craters of diameter 𝐷. 
+Also, this interpretation suggests thinking of 𝛷 as a probability of impact, rather than an 
+impact flux, thus emphasizing the statistical nature of the impacts of asteroids and 
+comets. Conversely, if we start with the definition of 𝐸𝑥𝑝[−𝐶̅𝜏`] as the fraction of craters 
+surviving after the interval 𝜏` from their formation, then we can construct Eq. (4) to 
+represent the sum of all the contributions, to the present number, for all times 𝜏` 
+younger than 𝜏, and then find that 𝑁 satisfies the differential equation implied in Eq. (1). 
+        Consider the following definition: 
+                             𝑇(𝐷, 𝜏, ) ≡ ∫ 𝐶𝑑𝜏`
+𝜏
+0
+ = 𝐶̅ 𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛   .                                                   (11) 
+Hence  𝑇 is a dimensionless time that measures the numbers of mean-life in an interval 
+𝜏. From Eq. (11) it follows that 
+                      𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏)    ,                                                                               (12) 
+where 𝐷 is considered here as a constant parameter. Since crater elimination is a 
+decay process, where 𝐶 is strictly positive, we have 
+                     𝑑𝑇/𝑑𝜏 > 0 .                                                                                            (13) 
+Consequently, the function  𝑇(𝐷, 𝜏, ) can be inverted to express 𝜏 as a function of 𝑇 and 
+𝐷:  𝜏(𝐷, 𝑇). Likewise, 𝐶 and 𝛷 are each expressible as functions of 𝑇 and 𝐷. We can 
+then rewrite Eq. (4), using Eqs. (11), (12) and (3), in the form 
+
+                   𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`)
+𝜏
+0
+𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏` = ∫ {(
+𝛷
+𝐶)
+𝑇
+0
+𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` =
+                          ∫ {(𝛷𝜏𝑚)
+𝑇
+0
+𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`.                                                                           (14)                                   
+where, in the right-hand side of Eq. (14), 
+𝛷
+𝐶 = 𝛷𝜏𝑚 is considered now a function of 𝑇, 
+and the parameter 𝐷. For instance, if 𝛷𝜏𝑚 is a sum like 
+                   𝛷𝜏𝑚 = 𝛴𝑎𝑠𝑇𝑠 , 𝑎𝑠 and 𝑠 are independent of 𝑇,                                         (15)                     
+then we have, from Eq. (14), 
+                      𝑁(𝐷, 0, 𝑇) = 𝛴𝑎𝑠 ∫ {𝑇`𝑠
+𝑇
+0
+𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` = 𝛴𝑎𝑠 𝛾(𝑠 + 1, 𝑇) ,                        (16) 
+where the lower incomplete gamma function notation was used above: 
+                       𝛾(𝑠 + 1, 𝑇) = ∫ {𝑇`𝑠
+𝑇
+0
+𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`.                                                         (17) 
+If 𝑠 is a whole number, as in a Taylor-Maclaurin series, we can also write 
+                    𝛾(𝑠 + 1, 𝑇) = 𝑠! (1 - 𝑒−𝑇 ∑
+𝑇𝑘
+𝑠
+𝑘=0
+/𝑘! ).                                                     (18) 
+This is our first encounter with the use of gamma functions expressing the number of 
+craters as a function of diameter and age. We will see further use of gamma functions 
+when considering applications to Earth’s impact crater data in Section (4). We will now 
+focus our attention on applications of Eq. (14) to the planet Mars. 
+3. Applications to the Crater-Size Frequency Distribution of Mars 
+        It was discussed in Section 2 that the product  
+𝛷
+𝐶 =  𝛷𝜏𝑚 represents the value of 𝑁 
+when the production and the elimination of craters are equal and  𝑑𝑁 = 0. Then in a 
+steady state situation we will have 𝑁 = 𝛷𝜏𝑚 = constant. However, in general, 𝛷𝜏𝑚 
+could depend on time, since both 𝛷 and 𝜏𝑚 could depend on time. On the other hand, 
+since 𝐶 is by definition the rate of crater elimination per number of craters, we have 
+then that 𝜏𝑚 ≡ 1/𝐶 is strongly influenced by the elimination of old craters due to impacts 
+forming new craters. Therefore, an increase or decrease of 𝛷 would be correlated with 
+an increase or decrease of 𝐶. Consequently, if obliterations by impacts are important, 
+the changes in time of  
+𝛷
+𝐶 =  𝛷𝜏𝑚 are smoothed out relative to the individual changes in 
+time of 𝛷, 𝐶,or 𝜏𝑚. In such a heuristic and realistic situation, a model in which it is 
+assumed that  
+𝛷
+𝐶 =  𝛷𝜏𝑚 is constant should be a good representation of the 
+observations. In this case, Eq. (14) becomes 
+                       𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇)                                                                                       (19) 
+
+        With the above simple model we were able to represent (reference 1) remarkably 
+well the pioneering Mars crater database catalog of Barlow (1988), as illustrated in Fig. 
+(1). Also, Fig. 2 compares the model with the more recent Mars data catalog of Robbins 
+and Hynek (2012), and also the model is in very good agreement with observations 
+(Bruckman 2019). The values of  𝛷𝜏𝑚 and 𝑇 for Barlow’s model are  
+                  𝛷𝜏𝑚 = 
+1.43𝑥105
+𝐷1.8
+,                                                                                                                 (20)       
+                    𝑇 = 𝐶̅𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 =
+2.48𝑥104
+𝐷2.5
+,                                                                        (21) 
+and then Eq. (19) becomes 
+                      𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇) =  
+1.43𝑥105
+𝐷1.8
+(1 − 𝐸𝑥𝑝[− 
+2.48𝑥104
+𝐷2.5
+]),                                      (22)  
+where the unit of 𝐷 is kilometers.  It can also be shown (Appendix A), using the 
+assumption that 𝛷𝜏𝑚 is independent of 𝑇, that  
+                  𝛷̅𝜏 = ∫ 𝛷𝑑𝜏 
+𝜏
+0
+= 𝛷𝜏𝑚𝑇 =  
+3.55𝑥109
+𝐷4.3
+,                                                                 (23)  
+where 𝛷̅ is the time average of 𝛷, and  𝛷̅𝜏 is the total number of craters, of diameter 𝐷, 
+per bin, created over the total time of production 𝜏 . The corresponding expression for 
+the number of craters created with diameters in the interval between 𝐷𝑖 and 𝐷𝑓 is then : 
+                           𝜏𝛷̅𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫
+𝛷̅𝜏
+𝐷𝑓
+𝐷𝑖
+𝑑𝐷. = (3.55/3.3)109(
+1
+𝐷𝑖3.3 - 
+1
+𝐷𝑓3.3)  .                       (24)  
+It is common to take the upper limit 𝐷𝑓 to be infinite to obtain the total number of craters 
+produced larger than 𝐷𝑖 : 
+                                𝜏𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = 1.076𝑥109(
+1
+𝐷𝑖3.3) ,                                                        (25)          
+or 
+                        𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = (1.076𝑥109/𝜏)(
+1
+𝐷𝑖3.3) ,                                                     (26)                                                 
+where 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) is the time average of the cumulative impact rate for the formation of 
+craters larger than 𝐷𝑖. For instance, it is interesting to note that for 𝐷𝑖 = 1 km, 
+approximately 109 such impacts were produced. Therefore, assuming that the total time 
+of crater production  𝜏 was 3000 to 4000 million years, we get an average of one 
+impact, making craters larger than 1 km, approximately every three to four years. Since 
+the energy associated to impacts with a diameter of 1 km is close to one megaton, this 
+
+result is of concern for explorations of Mars, assuming that the present impact flux 
+average is comparable to that given by Eq. (26). 
+        It is expected that also the corresponding impact rate for Earth has, similar to 
+Mars, a crater diameter dependency of the form 
+1
+𝐷𝑖3.3, and indeed, we found that for our 
+planet such a relation is consistent with the observations (Appendix B). 
+       Let us continue our analysis of the implications of the above model, by looking at 
+Eq. (21), rewritten in the form 
+                   
+𝜏𝑚𝑒𝑎𝑛
+𝜏
+= 𝐷2.5/2.48𝑥104 .                                                                             (27) 
+An interesting interpretation of the above equation (Reference 1) is that it represents a 
+proportionality relation between the mean life, of a crater of diameter ��, and the initial 
+volume of this crater. This conclusion is based on observations on Mars that 
+established that the initial depths of pristine craters are proportional to 𝐷𝑘/2, with 𝑘 ≈ 1, 
+and, consequently, the expected initial volumes for these craters are proportional to 
+𝐷2𝐷𝑘/2 ≈ 𝐷2.5. For instance, Garvin (2002) gives 𝑘 ≈ 0.98, while Boyce et al. (2007) 
+give 𝑘 ≈ 1.04. Furthermore, from the application to Earth of the above formalism, to be 
+discussed in the next section, it was concluded that craters in our planet also have their 
+mean-life proportional to ≈ 𝐷2.5. Thus, we have that the relation 𝜏𝑚𝑒𝑎𝑛 proportional to 
+the crater initial volume is not only intuitively appealing, but also helps us understand 
+why we have similar 𝐷 exponents in the 𝜏𝑚𝑒𝑎𝑛 for Earth and Mars, notwithstanding 
+these planets contrasting geological evolutions.  
+ 
+ 
+ 
+ 
+
+ 
+FIGURE (1): Log-Log plot of number of craters per bin, 𝑁(𝐷) 𝑣𝑠 𝐷 based on Barlow’s Mars catalog 
+(1988). The number 𝑁(𝐷) is calculated by counting the number of craters in a bin ∆𝐷 = 𝐷𝑅 − 𝐷𝐿, and 
+then dividing this number by the bin size. The point is placed at the mathematical average of 𝐷 in the 
+bin: (𝐷𝑅 + 𝐷𝐿)/2. The bin size is ∆𝐷 = (√2 − 1)𝐷𝐿, so that 
+𝐷𝑅
+𝐷𝐿 = √2. ). The curve is from the model 
+implied by Eq. (22). We see that the theoretical curve shown differs significantly from the observed data 
+for 𝐷 less than about 8𝑘𝑚. However, according to Barlow, the empirical data undercounts the actual 
+crater population for 𝐷 less than 8𝑘𝑚. However, more recent Mars crater data by Robbins et al. (2012) 
+was used to update the observations, yielding similar results to the model in Figure 1, but extending the 
+range to craters with diameters down to 1 km (see Fig. 2). 
+ 
+ 
+FIGURE (2): Log-Log plot of 𝑵(𝐷), 𝑣𝑠 𝐷(km), based on the Mars catalog of Robbins et al (2012), 
+(Bruckman (2019)). Bin size is  ∆𝐷 = (  21/6 − 1)𝐷𝐿. Note that for  𝐷 > ~300 𝑘𝑚,  the data points are 
+above the curve of the analytic model. However, we expect that the analytical model will be less 
+reliable when the number of craters in a given bin is so small that statistical continuous models break 
+down. Moreover, another source of discrepancy could be that these very large craters were being 
+formed at high proportions at older times 𝜏, thus perhaps belonging to the so-called heavy 
+bombardment era, characterized by a much higher impact flux.     
+          
+10
+20
+50
+100
+200
+500
+D
+0.1
+1
+10
+100
+1000
+N
+
+Log[N(D)]
+3 E
+2 E
+1上
+1.0
+1.5
+2.0
+Log[P] 4. Applications to Planet Earth  
+        The number of identified impact craters on Earth is close to 190 (Planetary and 
+Space Science Center: PASSC.com), while, in contrast, the number of craters used for 
+Mars in Fig. (1) was 42,284. Therefore, in the analysis of Earth’s crater data it is 
+convenient to use the cumulative number of impacts of craters, 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏), defined in 
+Eq. (6), instead of  𝑁(𝐷, 0, 𝜏), defined in Eq. (4). Furthermore, for  𝑁(𝐷, 0, 𝜏), the 
+simplified expression in Eq. (19) will be used, since it reproduced the Martian impact 
+data very well. Thus we have 
+                   𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫
+𝑁(𝐷, 0, 𝜏)𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+= ∫
+𝛷𝜏𝑚(1 – 𝑒−𝑇) 𝑑𝐷 =
+𝐷𝑓
+𝐷𝑖
+ 
+                        ∫
+𝛷𝜏𝑚𝑑𝐷 − ∫
+𝛷𝜏𝑚𝑒−𝑇𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+𝐷𝑓
+𝐷𝑖
+.                                                                    (28)   
+In addition, let us assume that    
+                    𝛷𝜏𝑚 = 
+𝐻
+𝐷𝑚`  ,                                                                                                                    (29)                                
+                        𝑇 = 𝐶̅𝜏 =
+𝐵𝜏
+𝐷𝑝 ,                                                                                            (30)   
+                    𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` 
+𝜏
+0
+= 𝛷𝜏𝑚𝑇 =  
+𝐴𝜏
+𝐷𝑚 .                                                                    (31) 
+where 𝐻, 𝑚`, 𝐵, 𝑝, 𝐴 𝑎𝑛𝑑 𝑚 are independent of 𝐷, and, from Eqs. (29), (30) and (31), 
+                   𝑚 =  𝑚` + 𝑝 ,                                                                                                       (32) 
+                     𝐴 = 𝐻𝐵  .                                                                                                             (33) 
+Equations (29), (30), and (31) are a generalization for Earth of the corresponding equations, 
+(20), (21) and (23), describing the crater distribution for Mars. For Mars, we have 𝐻 =
+1.43𝑥105, 𝐵𝜏 = 2.48𝑥104 , and 𝐴𝜏 = 3.55𝑥109. However, for our planet these values will 
+have to be redetermined. Also, the exponents 𝑚 and 𝑝 should come out from the fitting to 
+Earth data. As was discussed in previous section, a value of 𝑚 = 4.3, in the exponent of 𝐷 of 
+the impact flux 𝛷̅ is also consistent with the Earth observational impact rate data 
+(Appendix B). The value 𝑝 = 2.5 is also consistent with the Earth observations, to be 
+discussed in this section. 
+        After the substitution of the expressions in Eqs. (29), (30), and (31) in Eq. (28) the first 
+integral in the right-hand side is elementary, hence, we will turn our attention to the second 
+integral: 
+                      − ∫
+𝛷𝜏𝑚𝑒−𝑇𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+= − ∫
+𝐻
+𝐷𝑚` {𝐸𝑥𝑝 [
+−𝐵𝜏
+𝐷𝑝  ]} 𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+.                                         (34) 
+
+To emphasize that the variable of integration is now 𝐷, while 𝜏 is a fixed parameter, we 
+rename 𝑇  as 𝑈: 
+                      𝑇 = 𝑈 = 
+𝐵𝜏
+𝐷𝑝  ,                                                                                          (35) 
+or 
+                      𝐷 = [ 
+𝐵𝜏
+𝑈  ]1/𝑝 ,                                                                                         (36) 
+from which, differentiating with respect to 𝐷, holding 𝜏 fixed, 
+                          𝑑𝐷 = −[𝐵𝜏 ]
+1
+𝑝[ 𝑈 ]
+−1
+𝑝 −1𝑑𝑈/𝑝 .                                                                   (37) 
+Substituting Eqs. (35), (36), and (37) in Eq. (34) we get      
+ − ∫
+𝛷𝜏𝑚𝑒−𝑇𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+= {𝐻/(𝑝[𝐵𝜏 ]𝑛)} ∫
+𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈
+𝑈𝑓
+𝑈𝑖
+= {𝐻/(𝑝[𝐵𝜏 ]𝑛)}𝛤[𝑛, 𝑈𝑖,𝑈𝑓],(38)                              
+ where   
+                    𝑛 ≡ ( 𝑚` − 1)/𝑝  ,                                                                                      (39)  
+                    𝑈𝑖 = 
+𝐵𝜏
+𝐷𝑖𝑝       ,                                                                                           (40) 
+                    𝑈𝑓 = 
+𝐵𝜏
+𝐷𝑓𝑝    ,                                                                                              (41) 
+and 
+                    𝛤[𝑛, 𝑈𝑖,𝑈𝑓] = ∫
+𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈
+𝑈𝑓
+𝑈𝑖
+                                                         (42) 
+is the generalized incomplete gamma function. Consequently, we can rewrite Eq. (28) 
+in the form 
+                     𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫
+ 
+𝐻
+𝐷𝑚` 𝑑𝐷 + {
+𝐷𝑓
+𝐷𝑖
+𝐻/(𝑝[𝐵𝜏 ]𝑛)} 𝛤[𝑛, 𝑈𝑖,𝑈𝑓]                           (43) 
+        The above integral represents the number of craters with diameters in the interval 
+between 𝐷𝑖 and 𝐷𝑓, that are younger than 𝜏. Hence, the number of craters formed with 
+ages between 𝜏𝑖 and  𝜏𝑓 is 
+                      𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫
+𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+= 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑓) - 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑖) = 
+                      {𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛)} 𝛤 [𝑛,
+𝐵𝜏𝑓
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑓
+𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛)} 𝛤 [𝑛,
+𝐵𝜏𝑖
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑖
+𝐷𝑓𝑝]  .                (44) 
+
+Here, if 𝐵 is a function of time, it should be evaluated at the corresponding 𝜏𝑖 or 𝜏𝑓. 
+        Another useful concept is the statistical mean of a function of 𝐷: 𝑓(𝐷), which is 
+defined using 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓), as follows 
+ 𝑓̅ = ∫
+𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+/{∫
+𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 
+𝐷𝑓
+𝐷𝑖
+}=∫
+𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷
+𝐷𝑓
+𝐷𝑖
+/{𝑁
+̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓)}. (45)      
+For instance, if 𝑓 = 𝐷 we get, from definition (45), the average diameters of craters with 
+diameters and ages in the intervals 𝐷𝑖 ≤ 𝐷 ≤ 𝐷𝑓, and  𝜏𝑖 ≤ 𝜏 ≤  𝜏𝑓 , respectively. In this 
+case it follows that the numerator of Eq. (45) is  
+ ∫
+𝐷𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 = 
+𝐷𝑓
+𝐷𝑖
+{𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`,
+𝐵𝜏𝑓
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑓
+𝐷𝑓𝑝] − {𝐻(𝑝[𝐵𝜏𝑖 ]𝑛`)}𝛤 [𝑛`,
+𝐵𝜏𝑖
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑖
+𝐷𝑓𝑝],  (46)                  
+where 
+                      𝑛` ≡
+ 𝑚`−2
+𝑝
+= 𝑛 − 1/𝑝  .                                                                                (47)  
+Hence  
+ 𝐷̅ = [
+1
+𝑁̃(𝐷𝑖,𝐷𝑓,𝜏𝑖,𝜏𝑓)][{𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`,
+𝐵𝜏𝑓
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑓
+𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛`)} 𝛤 [����`,
+𝐵𝜏𝑖
+𝐷𝑖𝑝 ,
+𝐵𝜏𝑖
+𝐷𝑓𝑝]]         (48)           
+The above expression was adapted and applied to the Earth crater data in reference 
+(1). The value of 𝑝 was determined by the best fitting of the data to the model given in Eq. 
+(48), and yielded a value similar to that for Mars. As stated, this is interpreted to be the result of 
+the proportionality of 𝜏𝑚𝑒𝑎𝑛 to the initial volume of craters, and that this volume is in turn 
+proportional to 𝐷𝑝. From this fitting to observation also came an approximate value for 
+Earth’s parameter 𝐵. 
+      The expression 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) in Eq. (44), was also used in reference (1) to 
+describe the number of Earth’s craters as a function of diameter and age, as illustrated 
+in figures C1 and C2 in Appendix C. The values 𝑝 = 2.5, 𝑚 = 4.3,  and 𝑚` = 𝑚 − 𝑝 = 1.8 
+were assumed since they were observationally justified. The value of 𝐻 = 𝐴/𝐵 was also 
+needed, and, since 𝐵 was estimated from observations of  𝐷̅, then the value of 𝐴 remained to 
+be estimated, as described in Appendix C. A remarkable agreement of the model with 
+observations was obtained.  
+ 
+ 
+ 
+
+  Appendix A  
+     The number of impacts, during the time 𝜏, producing craters of diameter 𝐷, per bin, can be 
+expressed as 
+                         𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 
+𝜏
+0
+ 
+𝜏
+0
+.                                                              A1 
+ Using Eqs. (3) and (12), we get 
+                        𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏) =
+1
+𝜏𝑚.                                                                             A2 
+We can then rewrite the right hand side of Eq. (A1) in the form 
+                         ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = 
+𝜏
+0
+∫ 𝛷𝜏𝑚𝑑𝑇` .
+𝑇
+0
+                                                                 A3  
+If furthermore 𝛷𝜏𝑚 is independent of 𝑇 we have  
+                        ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = 
+𝜏
+0
+∫ 𝛷𝜏𝑚𝑑𝑇` =  
+𝑇
+0
+𝛷𝜏𝑚 ∫ 𝑑𝑇` =   𝛷𝜏𝑚𝑇
+𝑇
+0
+ .                           A4  
+Therefore, from Eqs. (A1) and (A4),  
+                        𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 
+𝜏
+0
+ 
+𝜏
+0
+=  𝛷𝜏𝑚𝑇.                                                A5 
+Note also that, since 
+                         𝑇 = 𝐶̅𝜏 =  𝜏/𝜏𝑚𝑒𝑎𝑛 ,                                                                               A6 
+from (A5) we get 
+                          𝛷̅𝜏𝑚𝑒𝑎𝑛 =  𝛷𝜏𝑚.                                                                                    A7 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+   Appendix B 
+       Let us investigate the observational implications of the assumption of an average impact 
+flux for Earth given by 
+                  𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` 
+𝜏
+0
+= 
+𝐴𝜏
+𝐷4.3,                                                                                      B1 
+which implies the following cumulative impact flux 
+                    𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = ∫
+𝛷̅
+,∞
+𝐷𝑖
+𝑑𝐷 =
+𝐴/3.3
+𝐷3.3 ,                                                                  B2 
+where we drop the 𝑖 sub index from 𝐷, in the right-hand side of Eq. (B2). The value of 𝐴 
+can be estimated for Earth from the result of Grieve and Shoemaker (1994) for  𝐷 =
+20𝑘𝑚: 
+                      𝛷̅𝐶(20𝑘𝑚, ∞, 𝜏) =
+(5.5∓2.7)10−9
+(𝑚𝑦)𝑘𝑚2 4𝜋𝑅2 ≈ 2.8[
+1∓0.50
+𝑚𝑦 ],                                       B3 
+where 𝑅 is the Earth’s radius, and 𝑚𝑦 is million years. Comparing Eq. (B2), evaluated at  
+𝐷 = 20𝑘𝑚, with Eq. (B3) we obtain 
+                       𝐴 = 9.24[1∓0.50]
+(20)3.3
+𝑚𝑦 ,                                                                            B4 
+and thus 
+                      𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) ≡ 𝛷̅𝐶 = 2.8[
+1∓0.50
+𝑚𝑦 ](20 𝐷
+⁄ )3.3                                                   B5 
+This equation is a generalization of the result of Grieve and Shoemaker (1994), which 
+gives the Earth’s impact rate for the formation of craters with diameters larger than 𝐷. It 
+incorporates the 3.3 exponent on 𝐷 that we deduced from the model and observations 
+from Mars. 
+       The diameter of a crater corresponds to an energy, 𝐸, associated to the impact, 
+and hence Eq. (B5) can be re-expressed as (reference 1) 
+                      𝛷̅𝐶(𝐸) = 
+[1∓0.5]
+14.5𝑦 𝐸0.86,                                                                                    B6 
+where 𝐸 is in megatons. Equation (B6) gives similar predictions to those of Poveda et 
+al. (1999). The predictions of Eq. (B6) are also in agreement with Silber et al. (2009), 
+that, for impacts with energies larger than a megaton, gives one Earth impact about 
+every 15 years. It is interesting to note that, according to Eq. (B6), events like the 2013 
+Chelyabinsk meteorite of energy of about 0.5 megatons are predicted to happen with a 
+
+periodicity near one every 8/(1 ∓ 0.5) years, so that this type of event is expected to be 
+repeated in the near future.                                                  
+      Observations in the last few decades of lunar meteorites, called Lunar Flashes, 
+provide a direct determination of the impact rate, at these low range of energies (see for 
+example Oberst et al. (2012), and Suggs et al. (2014)). For instance, Oberst et al. 
+(2012) interpreted data of lunar flashes, and concluded a rate of 10−3 impacts per 𝑘𝑚2 
+per year, for energies ≥ ~8𝑥10−6 kilotons. This result, translated to the total Earth`s 
+surface area, becomes approximately 5.1𝑥105 impacts per year for these energies, 
+while from Eq. (B6) we get about  6.3[1 ∓ 0.5]𝑥105 impacts per year, which is consistent 
+with the above result for lunar flashes.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Appendix C 
+       To reduce the uncertainties due to undercounting in the Earth crater data we 
+selected the following regions for the study in reference 1: 
+(a) Continental United States 
+(b) Canada up to the Arctic Circle 
+(c) Europe 
+(d) Australia 
+The crater data is taken from The Planetary and Space Science Centre 
+(www.passc.net). Then, in Eq. (44), instead of using for the total Earth’s impact flux  
+                      𝛷̅ = (1/𝜏) ∫ 𝛷𝑑𝜏` 
+𝜏
+0
+= 
+𝐴
+𝐷4.3,                                                                            C1 
+we used for our study the more accurate impact flux corresponding to the area under 
+consideration above in a,b,c,d, which is given by  
+                     𝛷̅𝑎𝑐𝑐 =
+𝐴𝑎𝑐𝑐
+𝐷4.3,                                                                                                C2 
+where  
+𝐴𝑎𝑐𝑐  ≡ 𝐴
+𝐴𝑟𝑒𝑎 𝑈𝑛𝑑𝑒𝑟 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑡𝑖𝑜𝑛
+𝐸𝑎𝑟𝑡ℎ`𝑠 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎
+,                                                                C3                   
+where 𝐴 is given, from Eq. (B4), by 
+𝐴 =  9.24[1 ∓ 0.50]
+(20)3.3
+𝑚𝑦
+= (1.82)105[1 ∓ 0.5]/𝑚𝑦. 
+            C4                  
+Accordingly, 𝐻 = 𝐴/𝐵 becomes  𝐻𝑎𝑐𝑐 =
+𝐴𝑎𝑐𝑐
+𝐵 , with 𝐵 estimated from the curve   𝐷̅ vs. 
+crater age , given by Eq. (48), fitting to the Earth’s data. Therefore, we can write the 
+theoretical 𝑁̃ with no free parameters, and compare it with the observations, as 
+described below. We do this first in table (I) and Figure (C1), for craters with 𝐷 ≥ 20𝑘𝑚 
+and cumulative age starting with 𝜏 = 1𝑚𝑦  up to 𝜏 = 2,000𝑚𝑦. Furthermore, we put 𝜏𝑓 =
+2,500𝑚𝑦   and   𝐷𝑓 = 300𝑘𝑚, since all craters in the field of study are within this bin size. 
+This theoretical curve, 𝑁̃(𝜏), is then compared with the corresponding observational 
+data, and the very good agreement between theory and observation is noteworthy. On 
+the other hand, we also compare theory and observation in Table II and Figure (C2), 
+where now 𝑁̃ cumulative represents the number of craters of all ages, 1𝑚𝑦 ≤ 𝜏 ≤
+2,500𝑚𝑦, with diameters greater than or equal to 𝐷. Again, the theoretical 𝑁̃(𝐷) is in 
+very good agreement with the observations for 𝐷 ≥ ~20𝑘𝑚, although not so good for 
+𝐷 ≤ ~20𝑘𝑚, which is as expected due to the undercounting of craters of these sizes. 
+
+ 
+Table l          
+𝜏(𝑚𝑦)         
+𝑁̃[𝜏, 𝐷 ≥ 20𝑘𝑚 ]        
+Observation 
+1                    
+33.14                  
+33 
+10 
+32.00                  
+32 
+20 
+30.80                  
+31 
+40 
+28.62                  
+29 
+50 
+27.62                  
+28 
+100 
+23.40 
+24 
+150 
+20.24 
+20 
+200 
+17.80 
+17 
+300 
+14.20 
+13 
+400 
+11.70 
+10 
+600 
+8.50                    
+8 
+800 
+6.50 
+5                  
+1000 
+5.00 
+5                    
+1200 
+3.89                                  4                      
+1400 
+2.99 
+3                  
+1600 
+2.25 
+3                   
+1800 
+1.62 
+2                    
+2000 
+1.08   
+1                      
+   
+
+ 
+FIGURE (C1): 𝐿𝑜𝑔[𝑁] 
+̃ 𝑣𝑠 𝐿𝑜𝑔[𝜏 ≡ 𝐴𝑔𝑒], for all diameters 𝐷 ≥ 20𝑘𝑚. See Table l. 
+ 
+Table ll 
+D 
+𝑁̃[𝐷, 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 ] 
+Observation 
+1 
+166.00 
+121 
+2 
+165.00                             
+118 
+4 
+137.00 
+99 
+8 
+82.60 
+72 
+16 
+42.40 
+37 
+20 
+33.14 
+33 
+32 
+18.18 
+16 
+45 
+10.37 
+10 
+64 
+4.79 
+5 
+91 
+1.82 
+2 
+128 
+0.62 
+1 
+ 
+ 
+
+Log NaccAge,D>20km
+1.4 F
+1.2 E
+1.0 F
+180
+0.6 E
+0.4 E
+0.2 E
+0.5
+Log Age
+1.0
+1.5
+2.0
+2.5
+3.0          
+ 
+FIGURE (C2): [𝐿���𝑔[𝑁̃] vs. 𝐿𝑜𝑔[𝐷𝐴𝑐𝑐 ≡ 𝐷], for all ages between 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 (Table II).         
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Log NaccD
+2.0 E
+1.5 上
+1.0 F
+0.5
+0.5
+1.0
+1.5
+2.0References 
+1. Bruckman, W.F., Ruiz, A., Ramos, E. (2012). 
+Earth and Mars Crater Size Frequency Distribution 
+and Impact Rates: Theoretical and Observational 
+Analysis; arXiv:1212.3273(astro-ph) 
+2. Barlow, N.G. (1988). Icarus 75,  285. 
+3. Robbins, S.J., and Hynex, B.M. (2012). Global 
+Database of Mars Impact Craters ≥ 1𝑘𝑚.; Journal 
+of Geophysical Research: Planets 117(E5) 
+4. Bruckman, W.F. (2019). Researchgate preprint. 
+DOI: 10.13140/R.G.2.2.33363.43047 
+5. Garvin, J.B. (2002). Lunar and Planetary Science 33, 1255. 
+6. Boyce, J.M., Garbeil, H. (2007). Geophysical Research Letters 34(16). 
+7. Planetary and Space Science Centre (PASSC), Earth Impact Database 
+(http://www.passc.net/EarthImpactDatabase/ 
+8. Grieve and Shoemaker (1994). The Record of Past Impacts on Earth. In: Hazards 
+Due To Comets And Asteroids, T. Gehrels, ed., The University of Arizona Press. 
+9. Poveda, A., Herrera, M.A., Garcia, J.L., Curioca, K. (1999) Planetary and Space 
+Science 47, 679.  
+10. Silber, E.A., Revelle, D.O., Brown, P.G., Edwards, W.N. (2009). Journal of 
+Geophysical Research 114, E08006. 
+11. Oberst, J., A., Christou, A., Suggs, R., Moser, D., Daubar, I.J., McEwenf, A.S., 
+Burchell, M., Kawamura, T., Hiesinger, H., Wünnemann, K., Wagner, R., Robinson, 
+M.S. (2012); The Present day Flux of Large Meteoroids on the Lunar Surface. A 
+synthesis of Models and Observational Techniques. Planetary and Space Science 74, 
+179–193 
+12. Suggs, R.M., Muser, D.E., Cooke, W.J., Suggs, R.J. (2014). The Flux of Kilogram-
+Sized Meteoroids From Lunar Impact Monitoring. Icarus April 2014. 
+ 
+ 
+  
+