Title: Causal Discovery from Subsampled Time Series with Proxy Variables
Venue: NeurIPS 2023
Accepted: yes
Keywords: causal discovery

time series

subsampling

proxy variables
Abstract: Inferring causal structures from time series data is the central interest of many scientific inquiries. A major barrier to such inference is the problem of subsampling, *i.e.*, the frequency of measurement is much lower than that of causal influence. To overcome this problem, numerous methods have been proposed, yet either was limited to the linear case or failed to achieve identifiability. In this paper, we propose a constraint-based algorithm that can identify the entire causal structure from subsampled time series, without any parametric constraint. Our observation is that the challenge of subsampling arises mainly from hidden variables at the unobserved time steps. Meanwhile, every hidden variable has an observed proxy, which is essentially itself at some observable time in the future, benefiting from the temporal structure. Based on these, we can leverage the proxies to remove the bias induced by the hidden variables and hence achieve identifiability. Following this intuition, we propose a proxy-based causal discovery algorithm. Our algorithm is nonparametric and can achieve full causal identification. Theoretical advantages are reflected in synthetic and real-world experiments.

Causal Discovery from Subsampled Time Series with Proxy Variables

Abstract

Inferring causal structures from time series data is the central interest of many scientific inquiries. A major barrier to such inference is the problem of subsampling, i.e. , the frequency of measurement is much lower than that of causal influence. To overcome this problem, numerous methods have been proposed, yet either was limited to the linear case or failed to achieve identifiability. In this paper, we propose a constraint-based algorithm that can identify the entire causal structure from subsampled time series, without any parametric constraint. Our observation is that the challenge of subsampling arises mainly from hidden variables at the unobserved time steps. Meanwhile, every hidden variable has an observed proxy, which is essentially itself at some observable time in the future, benefiting from the temporal structure. Based on these, we can leverage the proxies to remove the bias induced by the hidden variables and hence achieve identifiability. Following this intuition, we propose a proxy-based causal discovery algorithm. Our algorithm is nonparametric and can achieve full causal identification. Theoretical advantages are reflected in synthetic and real-world experiments. Our code is available at https://github.com/lmz123321/proxy_causal_discovery .

1 Introduction

Temporal systems are the primary subject of causal modeling in many sciences, such as pathology, neuroscience, and economics. For these systems, a common issue is the difficulty of collecting sufficiently refined timescale data. That is, we can only observe a subsampled version of the true causal interaction. This can cause serious problems for causal identification, since full observation ( i.e. , causal sufficiency [1]) is believed to be an important condition for causal identification, and the violation of which ( e.g. , the existence of hidden variables) can induce strong bias [2].

Example 1.1 . Take the pathology study of Alzheimer's disease (AD) as an example. In AD, patients suffer from memory loss due to the atrophy of memory-related brain regions such as the Hippocampus [3]. To monitor such atrophy, the standard protocol is to perform a MRI examination on the brain every six months [4]. However, many studies have shown that the disease can progress much more rapidly than this [5, 6]. For this reason, when recovering the causal interactions between brain regions, those at the unobserved time steps constitute hidden variables and induce spurious edges in the causal graph, as shown in Figure 1 (b).

To resolve this problem, many attempts have been made to recover the correct causal relations from subsampled data. However, none of them achieved general identifiability results. For example, in

(a) consider subsampling

(b) ignore subsampling

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[10, 11, 12], identifiability was achieved only for linear data. As for nonlinear data, only a small part of the causal information ( i.e. , an equivalence class) could be identified [13, 14, 15, 16]. Indeed, encountering such difficulties is no coincidence, considering the fact that there are many hidden variables in the system. Particularly, the hidden mediators prevent us from distinguishing the direct causation from possible mediation and hence only allow us to recover some ancestral information.

In this paper, we propose a constraint-based algorithm that can identify the entire causal structure from subsampled data, without any parametric assumption. Our departure here is that in time series, every hidden variable has an observed descendent, which is essentially itself at some observable time in the future 1 . Therefore, we can use the observed descendent as its proxy variable [17] and adjust for the bias induced by it. For this purpose, we first represent the ancestral information, which is composed of both direct causation and hidden mediation, with the Maximal Ancestral Graph (MAG) [18]. Then, to distinguish causation from mediation, we use the observed descendants of the hidden mediators as their proxy variables and adjust for their induced mediation bias. We show that our strategy can achieve a complete identification of the causal relations entailed in the time series. It is essentially nonparametric and only relies on the smoothness and completeness of the structural equation to conduct valid proxy-based adjustment [19].

Back to the AD example, as shown in Figure 1, our method can recover causal pathways that align well with existing clinical studies [8, 9], while compared baselines fail to do so.

Our contributions are summarized below:

We establish nonparametric causal identification in subsampled time series.

We propose an algorithm to practically recover the causal graph.

We achieve more accurate causal identification than others on synthetic and real data.

The rest of the paper is organized as follows. First, in Section 2, we introduce the problem setting, basic assumptions, and related literature needed to understand our work. Second, in Section 3, we establish theories on structural identifiability. Then, based on these theories, in Section 4, we introduce the proposed causal discovery algorithm and verify its effectiveness in Section 5. Finally, we conclude the paper and discuss future works in Section 6.

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2 Preliminary

We start by introducing the causal framework and related literature needed to understand our work.

Time series model. Let X ( t ) ∶= [ X 1 ( t ) , ..., X d ( t )] be a multivariate time series with d variables defined at discrete time steps t = 1 , ..., T . We assume the data is generated by a first-order structural vector autoregression (SVAR) process [16]:

where f i is the structural function, PA i is the parent of X i , and N i is the exogenous noise.

Implicit in (1) is the assumptions that cause precedes effect, and that causation is invariant to time, i.e. , the structural function f i and the causal parents PA i keeps unchanged across time steps. These assumptions carry the fundamental beliefs of temporal precedence and stability on causality [1], therefore are widely adopted by existing works [10, 11, 12, 14, 16].

Subsampling. The subsampling problem means that model (1) can be only observed every k steps [10, 12, 16]. That is, we can only observe X ( 1 ) , X ( k + 1 ) , ..., X (⌊ T k ⌋ k + 1 ) .

Graph terminologies. Causal relations entailed in model (1) can be represented by causal graphs (Figure 2). We first introduce the full time Directed Acyclic Graph (DAG) [20], which provides a complete description of the dynamics in the system.

Definition 2.1 (Full time DAG) . Let G ∶= ( V , E ) be the associated full time DAG of model (1). The vertex set V ∶= { X ( t )} T t = 1 , the edge set E contains X i ( t -1 ) → X j ( t ) iff X i ∈ PA j .

We assume the full time DAG is Markovian and faithful to the joint distribution P ( X ( 1 ) , ..., X ( T )) .

Assumption 2.2 (Markovian and faithfulness) . For disjoint vertex sets A , B , Z ⊆ V , A ⊥ ⊥ B ∣ Z ⇔ A ⊥ ⊥ G B ∣ Z , where ⊥ ⊥ G denotes d -separation in G .

In practice, it is often sufficient to know the causal relations between time series as a whole, without knowing precisely the relations between time instants [21]. In this regard, we can summarize the causal relations with the summary graph [20].

Definition 2.3 (Summary graph) . Let G ∶= ( V , E ) be the associated summary graph of model (1). The vertex set V ∶= X , the edge set E contains X i → X j ( i ≠ j ) iff X i ∈ PA j .

In this paper, our goal is to recover the summary graph, given data observed at t = 1 , k + 1 , ..., ⌊ T k ⌋ k + 1 .

For this purpose, we need a structure to represent the (marginal) causal relations between observed variables and hence link the observation distribution to the summary graph. Here, we use the Maximal Ancestral Graph (MAG) [22], since it can represent casual relations when unobserved variables exist.

Specifically, for the variable set V ∶= { X ( t )} T t = 1 , let O ∶= { X ( 1 ) , ..., X (⌊ T k ⌋ k + 1 )} be the observed subset and L ∶= V / O be the unobserved subset. Let An ( A ) be the ancestor set of A . Then, given any full time DAG G over V , the corresponding MAG M G over O is defined as follows:

Definition 2.4 (MAG) . In the MAG M G , two vertices A,B ∈ O are adjacent iff there is an inducing path. 2 relative to L between them in G . Edge orientation is:

A → B in M G if A ∈ An ( B ) in G ;

A ← B in M G if B ∈ An ( A ) in G ;

A ↔ B in M G if A / ∈ An ( B ) and B / ∈ An ( A ) in G .

Remark 2.5 . In the MAG, the directed edge A → B means A is the ancestor of B . The bidirected edge A ↔ B means there is an unobserved confounder U between A and B .

Proximal causal discovery. Recent works [23, 19] found that one could use the descendant ( i.e. , proxy) of an unobserved variable M to differentiate direct causation A → B from hidden mediation A → M → B . Specifically, suppose that:

Assumption 2.6 (Smoothness) . For each causal pair A → B , the map a ↦ P ( B ∣ A = a ) is Lipschitz with respect to the Total Variation (TV) distance, that is, there exists a constant L such that:

Assumption 2.7 (Completeness) . For each causal pair A → B , the conditional distribution P ( B ∣ A ) is complete, that is, for any function g , we have:

E { g ( b )∣ a } = 0 almost surely if and only if g ( b ) = 0 almost surely .

Remark 2.8 . Roughly, Assumption 2.6 relies on the continuity of the structural function f i and Assumption 2.7 requires that the characterization function of the exogenous noise N i is non-zero. Please refer to [19] for a detailed discussion.

Under the above assumptions, [19] showed that:

Theorem 2.9 ([19]) . Suppose that A and B are mediated by an unobserved variable M , and that M has a proxy variable M ′ satisfying A ⊥ ⊥ G M ′ ∣ M . Then, we can identify whether A → B by testing whether pr ( b ∣ a ) ∼ pr ( m ′ ∣ a ) has a linear relation. If the linear relation exists, then A / → B , otherwise, we have A → B .

Remark 2.10 . The linear relation can be tested by checking the least square residual of regressing pr ( b ∣ a ) on pr ( m ′ ∣ a ) . Please refer to [19] for details.

To ensure the existence of the proxy variable, we require the following self causation assumption, which states that each variable X i ( t ) is influenced by its past X i ( t -1 ) .

Assumption 2.11 (Self causation) . In model (1), X i ∈ PA i for each i = 1 , ..., d .

Remark 2.12 . In time series statistics, this is also known as the autocorrelation assumption [24, 25].

3 Structural identifiability

In this section, we establish the identifiability of the summary graph. For this purpose, we connect the observational distribution to the summary graph with the bridge of the MAG.

Specifically, our analysis consists of three progressive results: Proposition 3.1, Proposition 3.3, and Theorem 3.5. First, in Proposition 3.1, we prove that the MAG can be identified from observational data. Then, in Proposition 3.3, we show that the identified MAG almost uniquely reflects the structure of the summary graph, in the sense that only edges connecting vertices and their ancestors cannot be determined due to hidden mediators. Finally, in Theorem 3.5, we identify these underdetermined edges with the proxies of the hidden mediators and therefore identify the whole summary graph.

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Proposition 3.1 (Identifiability of the MAG) . Assuming model (1) and Assumption 2.2, then the MAG over the observed variable set O is identifiable, i.e., its skeleton and edge orientations can be uniquely derived from the joint distribution P ( O ) .

Since the identifiability of the MAG's skeleton under the faithfulness assumption is a well-known result [18], we focus on explaining the identification of edge orientations. Specifically, we will divide the edges into two classes: the instantaneous edges and the lagged edges, and discuss their orientations respectively.

The instantaneous edge , e.g. , A ( t 3 ) ↔ M ( t 3 ) , is an edge that connects two vertices at the same time. Since we assume cause must precede effect, the instantaneous edge does not represent causation but latent confounding, therefore is bidirected according to Definition 2.4. For example, in Figure 2 (c), the instantaneous edge A ( t 3 ) ↔ M ( t 3 ) represents the latent confounding A ( t 3 ) ← A ( t 2 ) → M ( t 3 ) between A ( t 3 ) and M ( t 3 ) . On the other hand, the lagged edge , e.g. , A ( t 1 ) → M ( t 3 ) , is an edge that connects two vertices at different time. The lagged edge represents ancestral information and hence is directed ( → ) from the past to the future.

To connect the identified MAG to the summary graph, we first define the following graph structures, examples of which are shown in Figure 3:

Definition 3.2 (Graph structures) . In the summary graph,

A directed path p AB from A to B with length l is a sequence of distinct vertices A,V 1 , ..., V l , B where each vertex points to its successor. Two directed paths p AB 1 , p AB 2 are called disjoint if they do not share any non-startpoint vertex.

A confounding structure c AB between A and B with lengths ( r, q ) consists of a vertex U , a directed path p UA from U to A with length r , and a directed path p UB from U to B with length q , where p UA and p UB are disjoint.

In the following, we explain the relationship between the identified MAG and the summary graph. In particular, we will discuss what the directed and bidirected edges in the MAG ( resp. ) imply about the summary graph.

According to Definition 2.4, the directed edge A ( t ) → B ( t + k ) in the MAG means A is the ancestor of B . Therefore, in the summary graph, there is either A → B or a directed path from A to B . On the other hand, the bidirected edge A ( t + k ) ↔ B ( t + k ) in the MAG means there is a latent confounder between them, which can be either A ( t + 1 ) , i.e. , A ( t + k ) ←/uni22EF ← A ( t + 1 ) →/uni22EF → B ( t + k ) , or a third variable U ( t + 1 ) , i.e. , A ( t + k ) ←/uni22EF ← U ( t + 1 ) →/uni22EF → B ( t + k ) . Hence, in the summary graph, there is either a directed path from A to B or a confounding structure between them.

Summarizing the above observations, we have the following proposition:

Proposition 3.3 (MAG to summary graph) . If there are A ( t ) → B ( t + k ) and A ( t + k ) ↔ B ( t + k ) in the MAG, then, in the summary graph, there is either

A → B , 2. a directed path from A to B with length l ≤ k -2 , or

a directed path from A to B with length l = k -1 and a confounding structure between them with lengths ( r ≤ k -2 , q ≤ k -2 ) .

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Remark 3.4 . In Proposition 3.3, the maximum length of the directed path is k -1 , this is because any path longer than this can not be an inducing path and therefore can not induce A ( t ) → B ( t + k ) in the MAG. Please refer to the appendix for details.

Proposition 3.3 provides a necessary condition for having A → B in the summary graph, i.e. , there are both A ( t ) → B ( t + k ) and A ( t + k ) ↔ B ( t + k ) in the MAG. It also inspires us that, to make the condition sufficient, we need to distinguish the direct effect ( A → B ) from the indirect one ( A →/uni22EF → B ).

For this purpose, we propose to use the proximal causal discovery method. Before preceding any technical detail, we first brief our idea below. The indirect effect, unlike the direct one, relies on mediation and therefore can be d -separated by the (unobserved) mediator. To test the d -separation, we can use the observed proxy of the mediator, thanks to the self causation assumption in Assumption 2.11. For example, in Figure 4 (b), the indirect effect A ( t 1 ) → M ( t 2 ) → B ( t 3 ) from A to B can be d -separated by the mediator M ( t 2 ) , who has an observed proxy M ( t 3 ) .

For the general case, the separation set is a bit more complicated to ensure all paths except the one representing direct effect are d -separated. Specifically, for two vertices A ( t ) and B ( t + k ) , the separation set is the union of two sets M ( t + 1 )∪ S ( t ) . The set M ( t + 1 ) contains possible mediators, namely A ( t + 1 ) and any vertex M i ( t + 1 ) ( M i ≠ B ) such that A ( t ) → M i ( t + k ) in the MAG and M i is not B 's descendant 3 . The set S ( t ) is used to d -separate possible back-door paths between A ( t ) and B ( t + k ) , it contains any vertex S i ( t ) ( S i ≠ A ) such that S i ( t ) → B ( t + k ) or S i ( t ) → M j ( t + k ) for some M j ∈ M in the MAG.

Equipped with the separation set, we then have the following identifiability result:

Theorem 3.5 (Identifiability of the summary graph) . Assuming model (1), Assumption 2.2, and Assumptions 2.6, 2.7, 2.11, then the summary graph is identifiable. Specifically,

There is A → B in the summary graph iff there are A ( t ) → B ( t + k ) , A ( t + k ) ↔ B ( t + k ) in the MAG, and the set M ( t + 1 ) ∪ S ( t ) is not sufficient to d-separate A ( t ) , B ( t + k ) in the full time DAG.

The condition 'the set M ( t + 1 ) ∪ S ( t ) is not sufficient to d-separate A ( t ) , B ( t + k ) in the full time DAG' can be tested by the proxy variable M ( t + k ) of the unobserved set M ( t + 1 ) .

Theorem 3.5 provides a necessary and sufficient condition for having A → B in the summary graph. This condition is testable with the proxy M ( t + k ) of the unobserved set M ( t + 1 ) ( S ( t ) is observable). The validity of such proxying, i.e. , A ( t ) ⊥ ⊥ G M ( t + k )∣ M ( t + 1 ) , S ( t ) is proved in the appendix.

4 Discovery algorithm

Equipped with the identifiability results in Proposition 3.1 and Theorem 3.5, we can practically discover the summary graph with Algorithm 1. Specifically, given the observed data points, Algorithm 1 first recovers the MAG according to Proposition 3.1. Then, for each pair of vertices in the summary graph, it first uses the necessary condition of 'having A ( t ) → B ( t + k ) and A ( t + k ) ↔ B ( t + k ) in the MAG' to construct underdetermined edges. Finally, among these underdetermined edges, it uses the proxy variable approach to identify direct causation and remove the indirect one, according to Theorem 3.5.

Below, we first introduce the Partially Determined Directed Acyclic Graph (PD-DAG) , which will be used to represent the intermediate result of the algorithm.

Definition 4.1 (PD-DAG) . A PD-DAG is a DAG with two kinds of directed edges: solid ( → ) and dashed ( ⇢ ). A solid edge represents a determined causal relation, while a dashed one means the causal relation is still underdetermined. We use a meta-symbol, asterisk ( ∗→ ), to denote any of the two edges. In a PD-DAG G = ( V , E ) , the vertex A points to the vertex B if the edge A ∗→ B ∈ E . The definition of the directed path and confounding structure are the same as in the summary graph.

Remark 4.2 . Algorithm 1 can be generalized to cases where the subsampling factor k is unknown, by skipping the step (a) below line 3.

5 Experiment

In this section, we evaluate our method on synthetic data and a real-world application, i.e. , discovering causal pathways in Alzheimer's disease.

Compared baselines. Methods that account for subsampling : 1. SVAR-FCI [16] that extended the FCI algorithm to time series and recovered a MAG over observed variables; 2. NG-EM [10] that achieved identifiability with linear non-Gaussianity and used Expectation Maximization (EM) for estimation. Methods that neglect subsampling : 1. Dynotears [7] that extended the score-based Notears [27] algorithm to the time series; 2. PC-GCE [21] that modified the PC algorithm for temporal data with a new information-theoretic conditional independence (CI) test.

Subsampling factor

Sample size

Metrics. We use the F 1 -score, precision, and recall, where precision and recall ( resp. ) measure the accuracy and completeness of identified causal edges, and F 1 ∶= 2 ⋅ precision ⋅ recall precision + recall .

Implementation details. The significance level is set to 0 . 05 . For the MAG recovery, we use the FCI algorithm implemented in the causallearn package 4 . Temporal constraints, e.g. , causal precedence, time invariance, and self causation are added as the background knowledge.

5.1 Synthetic study

Data generation. We generate radnom summary graphs with the Erdos-Renyi model [28], where the vertex number is set to 5 , the probability of each edge is set to 0 . 3 . For each graph, we generate temporal data with the structural equation X i ( t ) = ∑ j ∈ PA i f ij ( X j ( t -1 )) + N i , where the function f ij is randomly chosen from { linear, sin, tanh, sqrt } , the exogenous noise N i is randomly sampled from { uniform,gauss.,exp.,gamma } . We consider different subsampling factors k = { 2 , 3 , 4 , 5 } and sample sizes n = { 600 , 800 , 1000 , 1200 } . For each setting 5 , we replicate over 100 random seeds.

Comparison with baselines. Figure 5 shows the performance of our method and baselines under different subsampling factors (upper row) and sample sizes (lower row). As we can see, our method significantly outperforms the others in all settings. Specifically, compared with methods that account for subsampling (SVAR-FCI and NG-EM), our method achieves both higher precision and recall, indicating less detection error and missing edges. This advantage can be attributed to the fact that our method enjoys an identifiability guarantee ( v.s. SVAR-FCI) and meanwhile requires no parametric assumption on the structural model ( v.s. NG-EM). Compared with methods that ignore subsampling (Dynotears and PC-GCE), our method achieves higher F 1 -score, precision, and is comparable in recall. This result shows that our method can effectively reduce the spurious detection induced by unobserved time steps. Besides, we can observe that Dynotears and PC-GCE slightly outperform SVAR-FCI and NG-EM, which again demonstrates the necessity of establishing nonparametric identification in subsampling problems.

Intermediate results. We evaluate the intermediate results of Algorithm 1 (MAG, edges identified by proxies, summary graph) and report the results in Figure 6. As shown, both the recovered MAG and edges identified by proxies have high accuracy under moderate subsampling factors and sample sizes, hence explaining the effectiveness of our algorithm in recovering the summary graph. Meanwhile, we can also observe that the performance slightly decreases when the subsampling factor k is large. This

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is explained as follows. When k is large, there are many unobserved steps between two observations, which weakens the correlation pattern in data and therefore breaks our faithfulness assumption.

5.2 Discovering causal pathways in Alzheimer's disease

Background. Alzheimer's disease (AD) is one of the most common neuro-degenerative diseases. In AD, patients suffer from memory loss due to atrophy of memory-related brain regions, such as the Hippocampus [3] and Temporal lobe [29]. A widely accepted explanation for such atrophy is: the disease releases toxic proteins, e.g. , Aβ [30] and tau [31]; along anatomical pathways, these proteins spread from one brain region to another, eventually leading to atrophy of the whole brain [32]. Recovering these anatomical pathways, i.e. , underlying causal mechanisms, will benefit the understanding of AD pathology and inspire potential treatment methods.

Dataset and preprocessing. We consider the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset [4], in which the imaging data is acquired from structural Magnetic Resonance Imaging (sMRI) scans. We apply the Dartel VBM [33] for preprocessing and the Statistical Parametric Mapping (SPM) [34] for segmenting brain regions. Then, we implement the Automatic Anatomical Labeling (AAL) atlas [35] to partition the whole brain into 90 regions. In total, we use n = 558 subjects with baseline and month-6 follow-up visits enrolled in ADNI-GO/1/2/3 periods.

Results. Figure 7 shows the recovered summary graph over 90 brain regions. For illustration, in Figure 1 (a), we further group the brain regions into 8 meta-regions according to their anatomical structures. We can observe that i) most causal pathways between meta-regions are unidirectional; ii) the identified sources of atrophy are the Hippocampus, Amygdala, and Temporal lobe, which are all early-degenerate regions [3, 29]; iii) the topology order induced from our result, i.e. , { Hip.,Amy.,Tem.,Cin.,Ins.,Fro.,Par.,Occ. } coincides with the temporal degeneration order found in [8, 9]. These results constitute an important finding that can be supported by existing studies: the atrophy (releasing of toxic proteins) is sourced from the Hippocampus and gradually propagates to other brain structures along certain anatomical pathways [32, 36].

In contrast, in Figure 1 (b), the causal relations recovered by the Dynotears baseline 6 are less clear. For example, most of the identified interactions are bidirectional, which may be due to the spurious edges induced by subsampling. Besides, the identified atrophy source is the Frontal lobe, with AD-related regions identified as outcomes, which is inconsistent with clinical studies. These results, from another perspective, show the importance of modeling subsampling in time series causal discovery.

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6 Conclusion

In this paper, we propose a causal discovery algorithm for subsampled time series. Our method leverages the recent progress of proximal causal discovery and can achieve complete identifiability without any parametric assumption. The proposed algorithm can outperform baselines on synthetic data and recover reasonable causal pathways in Alzheimer's disease.

Limitation and future work. Our method relies on an accurate test of the conditional independence (CI) and therefore may suffer from low recall when the CI patterns in data are weak. To solve this problem, we will investigate theories on high-efficiency CI tests and pursue a dedicated solution.

Acknowledgments

This work was supported by National Key R&D Program of China (2022ZD0114900).

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors suggest a method of causal discovery for multivariate time series under the regime when the data is being sampled at constant skips in the time dimension. Under mild assumptions, they prove that their method works asymptotically.\n\nSTRENGTHS:\nThe paper's contribution is clear, the methods are interesting and novel, and the subject (causal inference in multivariate time series) is of interest to many. Other than some places I will mention, the explanations are clear.\n\nWEAKNESSES:\nI would like to see a little more discussion on previous work. I see citations about work that inspired the current one, for example [17, 19, 25] on using descendants of an unobserved variable to differentiate direct causation from hidden mediation, but I do not see citations for competing methods for causality under the regime of sampling on the time dimension. It would be good to be explicit about what the current state of the art is in that direction, and how the current work compares. You mention them in Section 5 (SVAR-FCI, NG-EM, Dynotears, PC-GCE) but I'd like to see a bigger discussion on the differences. Specifically, it would be great if we had a basic example in mind where \"naive\" approaches clearly fail, and make us understand how the proposed method avoids the issue intuitively.\n\nQUESTIONS:\n1. I feel like Definition 2.3 can be phrased more simply. The way you wrote it makes it look circular.\n2. I am a little confused by the figures. In Figure 2.c., I don't understand why there wouldn't be an arrow from A(t_3) to M(t_5). Or perhaps the faded arrows count? Why is there a difference between faded and non-faded arrows?\n3. In Figure 3, where are some vertices faded and some not? Is there a difference? It looks like it is meant that faded means unobserved, but I don't think that's what you meant.\n4. Put Remark 4.2 outside of the algorithm.\n5. Line 224: random is misspelled.\n\nLIMITATIONS:\nNone.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n# Summary\nIn this paper, the authors address the problem of inferring causal structures from subsampled time series data, where the frequency of measurement is much lower than that of causal influence. This presents challenges in identifying the causal structure, as hidden variables at unobserved time steps can induce bias. Existing methods that tackle this problem are limited to linear cases or fail to achieve identifiability.\n\nThe main contribution of this paper is a constraint-based algorithm that can identify the entire causal structure from subsampled time series without any parametric constraints. The authors propose a proxy-based causal discovery algorithm that leverages the temporal structure of time series data to remove the bias induced by hidden variables. The algorithm is nonparametric and can achieve full causal identification. Specifically, the author leverages the proxy variables to test the edge directions of the summary DAG from the uniquely identified MAG. \n\nThe authors demonstrate the theoretical advantages of their method and provide experiments on both synthetic and real-world data, showcasing improved performance over existing methods.\n\nSTRENGTHS:\n# Originality\n\nThe paper presents a novel approach to causal discovery in subsampled time series data by proposing a constraint-based algorithm that leverages proxy variables to handle the challenges posed by hidden variables at unobserved time steps. The originality of the method lies in its ability to identify the entire causal structure without any parametric constraints, setting it apart from existing methods that are limited to linear cases or fail to achieve full identifiability. The authors draw inspiration from the recent progress in proximal causal discovery and adapt it to the subsampled time series setting.\n\n# Quality\nThe quality of the paper is high, as it presents a well-formulated methodology with solid theoretical foundations. The authors provide rigorous proofs for their proposed algorithm's identifiability properties, ensuring that the algorithm is grounded in strong theoretical underpinnings. Although I have some questions regarding its generality and required assumptions, I will elaborate about them later.\n\nAs for the experiments, the author conduct 1 synthetic and 1 real-world experiment, which demonstrates its effectiveness against the baselines.\n\n# Clarity\n\nThe paper is well-written and clear in its presentation. The methodogy is clear, thanks to the intuitive explanation provided by the authors. They also provide necessary background information on causal discovery and related literature. \n\n# Significance\n\nThe problem that this paper aims to talks is significant in the field of tmeproal causal discovery. Although I don't think this paper fully addresses it, it made a reasonable progress towards the final destination.\n\nWEAKNESSES:\n# Weakness\nAlthough the paper provide a solid progress towards the final destination, here are some limitations I am a bit worried about: \n\n## Assumptions are too strong:  \nOne assumption is that only 1 step window is considered in this paper. However, is this true in practice? Is it possible that some factors has long-standing effect (higher-order Markovian) to the target variable? Can your method handle such scenario? \n\nFor summary DAG, does it has to be a DAG? Since from general full-time DAG, the summary graph may not be a DAG and can contain cycles. Does your method handle such cases? For example, A(1) -> B(2) and B(1) -> A(2), and the summary graph is A<->B. \n\nWhat are the consequence if some variables are not self-caused? That means the proxy variables may not exists. In that case, full identifiablity cannot be establish, right? You may want to add something like to what extend of identifiability your method can establish without such assumption. \n\n## Comparison with baselines\nThe author includes some of the well-known baselines. Surprisingly, Dynotears performes reasonably ok considering that it is a linear model (in its original form). However, after Dynotears, several state-of-the-art baselines are proposed, like Rhino [1]. Can you compare your method with this nonlinear model?\n\n[1]Gong, Wenbo, et al. \"Rhino: Deep Causal Temporal Relationship Learning With History-dependent Noise.\" arXiv preprint arXiv:2210.14706 (2022). Code at https://github.com/microsoft/causica/tree/v0.0.0 \n\n## Scalability\n\nHow expensive it is to run such algorithm, since you need to test every pair. For real-world experiment, you scale it to 90 variables, how long does it need to run it? If your method can be scaled to higher-order SVAR, how does it scales with the window size?\n\nQUESTIONS:\n1. In figure 4 proxy model, do you mean $M(t_2)$?\n\nLIMITATIONS:\nThe author did discuss its limitations but they only discuss the limitations of conditional independence test. Howver, in the weakness section, I have raised several improvements the author can consider. I suggest including these discussions in the limitations as well.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-29"}

comments: SUMMARY:
This paper proposed a non-parametric constraint-based algorithm that can identify the entire causal structure from subsampled data, which leverages the proxy variable to adjust the bias induced by the hidden variable.

STRENGTHS:
- Concise and clear theoretical derivation. Introduce the proposed method by discussing the connections between different graphs step by step.

WEAKNESSES:
- The paper is rather incremental as the core of the method is based on [19] to use proxy variables to detect and eliminate the confounding effect brought by the subsampling.  
- Moreover, the method proposed by [19] seems to require more assumptions but is not disclosed in this paper. And it is unclear whether those specific assumptions can be satisfied in this work.
- In addition, this work assumes the self-exciting property of the time-series which is not necessarily held and it is interesting to see how to find the proxy variables and what would be the result in this case.
- It could be better to provide more than 5 vertex numbers in the experiment.
- What would be the result if the causal graph becomes denser?

QUESTIONS:
See the weakness above.

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-04

comments: SUMMARY:
In this paper, the author(s) propose a new technique to learn the summary graph of time-series data. As a motivation for their work, the author(s) discuss the interesting application of learning causal pathways in Alzheimer’s disease. The time series model studied in this work is quite general, and suitable for many applications. The author(s) provide a simple algorithmic approach to learn the summary graph. This algorithm is essentially based on verifying d-separation, by testing conditional independence. The author(s) conclude with an experimental comparison, showing that their method achieves superior performance than a baseline, using various synthetic datasets and a real-world medical application.

STRENGTHS:
This paper studies an very important and difficult problem. The application of learning causal pathways in Alzheimer’s disease is very relevant. The paper is also well-written and easy to follow.

WEAKNESSES:
My main concern pertains the faithfulness assumption. Faithfulness it is useful for getting identifiability results, and it allows to recover the causal structure by testing conditional independence. However, in many scenarios there is no practical reason to assume faithfulness. The use of faithfulness significantly limits the novelty of their work. If I understand correctly, the implementation of their algorithm essentially uses conditional independence to recover the causal structure. Testing conditional independence is practically problematic. Hence, I also have doubts on the scalability of their method.

QUESTIONS:
Can you please describe exactly step 1 in your algorithm? How do you test conditional independence?

LIMITATIONS:
Yes, the author address limitations in Section 6.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-04

comments: SUMMARY:
This paper studies the problem of subsampled time series in causal discovery, in which the unobserved time steps may lead to the existence of latent confounders. To this end, this paper proposes a constraint-based algorithm by leveraging proxy variables to remove the bias induced by latent confounders. The experimental results verify the effectiveness of the proposed algorithm.

STRENGTHS:
1. This paper addresses the problem of subsampled time series in causal discovery, which is important but challenging.

2. The paper is well-structured and written.

3. The experimental results show that the proposed method outperforms several representative baselines.

WEAKNESSES:
1. In Theorem 2.8, extra assumptions are required for testing conditional independent relations in related literature, but they are not discussed in this paper. 

2. How to search the proxy variable of target hidden variables. If an invalid proxy variable is selected, what is the output of the proposed algorithm?


3. In simulation data, the dimension of the random graph is only five. Can you show the performance of the proposed method in a larger-scale network?

QUESTIONS:
Refer to Weaknesses

LIMITATIONS:
Refer to Weaknesses
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2023-07-15