Two Time-Slices Help Topological Ordering for Learning Directed Acyclic Graphs

[Q13] You mention causal discovery algorithms identify the equivalence class, which is true for some (e.g. PC, FCI) but not all (e.g. NOTEARS aims to recover the whole DAG).

Response: Thank you for the correction. Most constraint-based methods (e.g. PC, FCI) typically find causal structures within an equivalence class, resulting in a limited understanding of the underlying causal relationships. Score-based methods (e.g. NOTEARS, GraNDAG) rely on local heuristics to enforce acyclicity constraints, which can be insufficient for effectively handling large datasets. Additionally, the causal graphs produced by minimizing a specific score function are not guaranteed to be entirely accurate. We have revised it in the revision.

[Q14] The time slice definition is confusing. You mention an initial slice from 1 to t-1 and the present is t. Where is the gap between them?

Response: In this context, a time slice denotes the cross-sectional observational data on units captured at a specific time point or time instant (within the given measurement accuracy) on the timeline of an event or process. 'Limited time slices' means that we can only access a few cross-sectional pieces of the samples at limited time points, which might be consecutive moments or non-consecutive moments.

In this paper, the 'initial record' and 'present record' are used in the healthcare example (Figure 1(b)), representing the patient's records from the first visit and second visit, respectively. Then, doctors typically compare the patient’s second follow-up visits to assess the patient's condition for more precise treatment. For further clarify, we would use $\mathcal{D} = \{ \boldsymbol{X}^{t_a}, \boldsymbol{X}^{t_b} \}_{t_a < t_b}$ to denote two time-slices, where $\boldsymbol{X}^{t_a}$ and $\boldsymbol{X}^{t_b}$ are observations of some system at time $t_a$ and $t_b$ ($1 < t_a < t_b$).

[Q15] Unclear why a time slice is an intervention. It seems a time slice just means recorded at different times (i.e. with a time gap)?

Response: As indicated by red marking in Figure 1(b), an intervention on the variable $X_2^{t-3}$ would transfer its effect to descendants ($X_3^{t-3}, X_4^{t-3}$), but not to non-descendants ($X_1^{t-3}, X_5^{t-3}$). This helps in swiftly identifying each variable's descendants, enhancing topological ordering. Inspired by this concept, as illustrated in Fig. 1(b), where each node is influenced by its ancestors and itself, each variable in the earlier time-slice would transmit self-perturbations to both itself and its descendants in the subsequent time-slice, serving as simulated interventions.

[Q16] Consider avoiding the use of notations that has not yet been introduced, such as $X_i^{t-3}$ in the introduction.

Response: Thanks for your advice. As shown in Figure 1(b), $X_i^{t-3}$ denotes the observations of i-th node $X_i$ at time $t-3$. We have added the corresponding description in the caption of Figure 1.

[Q17] You write "two time-slides data" at the end of the introduction. What is a "slide" here?

Response: This is a typo: "two time-slides data" → "two time-slices data".

[Q18] Defining X_i as a random variable does not make sense in the time series setting, where X_i^t is a random variable (with typically only one observation). See Section 10 of "Elements of Causal Inference".

Response: Thanks for your comment. In this paper, variable $X_i^t$ represents a measurement of the $i$-th observable $X_i$ of units at time $t$, where observable $X_i$ denotes the $i$-th feature or attribute of multiple samples in time series. We have revised it in the revision.

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According to your comments, our paper has been significantly improved in areas such as the definition of the problem setup, positioning in related work, discussions on baselines, theorem, and data generation function. We have uploaded the revised version of our paper (with track changes marked in blue). We welcome any further technical advice or questions on this work and we will do our best to address your concerns.

[Q7] You refer to "unique orderings" of the topology. However, this appears impossible since equivalent orderings exist (e.g. X → Y ← Z has completely equivalent orderings (X,Z,Y) and (Z,X,Y)). If the "uniqueness" refers to something else, then this should be clarified early on.

Response: The descendant hierarchical topology is unique and contains fewer non-essential edges when compared to non-unique complete topological ordering. In the descendant hierarchical topology, each node $X_i^t$ establishes direct edges pointing to its descendant nodes $\mathbf{de}_i^t$. In the case of X → Y ← Z, it becomes evident that only two descendant relationships can be established: de(X) = Y and de(Z) = Y. These relationships correspond to two direct edges, X → Y and Z → Y, within the unique descendant hierarchical topology (X → Y ← Z).

The complete topological orderings of summary graph are non-unique and may contains numerous spurious edges: The orderings (X,Z,Y) and (Z,X,Y) are complete topological orderings rather than our descendant hierarchical topology proposed in this paper.

[Q8] Assumption 1 seems flawed. X^t should be independent of X^t-i-k given X^t-1, ..., X^t-i since any previous lag could directly impact the current time lag (skip connection), rendering them dependent.

Response: Assumption 1 is the strict definition of the first-order Markov property for time series. It posits that the future state ($\boldsymbol{X}^{t+1}$) depends exclusively on the current state ($\boldsymbol{X}^{t}$), disregarding any influence from earlier history ($\boldsymbol{X}^{1 \cdots t-1}$). This definition precisely captures the essence of a first-order Markov process, where only the immediate past state is relevant for predicting the future state.

Furthermore, we can relax the first-order Markov Assumption to a high-order Markov Assumption. This means that the future time-slice $\boldsymbol{X}^{t+1}$ depends only on states $\boldsymbol{X}^{t \cdots t-q+1}$ and does not directly depend on states $\boldsymbol{X}^{1 \cdots t-q}$. Then, with $q+1$ time-slices ($\boldsymbol{X}^{t_a \cdots t_a-q+1}$ and $\boldsymbol{X}^{t_b}$), we can use $\mathbf{an}^{t_a \cdots t_a-q+1}$ to replace the condition set $\mathbf{an}^{t_a}$ to infer the Descendant-oriented Conditional Independence Criteria (Theorem 1). However, in this paper, we focus exclusively on the two-time-slices algorithm to demonstrate our theorem and algorithm. We have included a section to discuss the extension of the Markov Assumption to a high-order Markov Assumption in Appendix F.

[Q9] Definition 2 is unclear. What about X → Y → Z? It has 3 layers (X)_L1, (Y)_L2, (Z)_L3. By the definition, L1 > L3, so X has a direct edge to Z, which would be wrong?

Response: We have to emphasize the definition of Hierarchical Topological Ordering: If there is a directed edge from $X_{\pi_i}$ to $X_{\pi_j}$, then $l_{\pi_i} > l_{\pi_j}$. The statement "$X_{\pi_i} \rightarrow X_{\pi_j}$" is a sufficient but not necessary condition of "$l_{\pi_i} > l_{\pi_j}$". By the definition, X has a direct edge to Y and Y has a direct edge to Z, so $l_X > l_Y > l_Z$.

[Q10] When you mention 'DAG', do you refer to the summary graph? This should be clarified.

Response: Thanks for your feedback. In this paper, 'DAG' refers to the directed acyclic graph of the summary graph in Figure 1(a), and we have clarified this in the main text.

[Q5 & W2] Novelty is unclear. Theorem 1 seems to follow directly from the lagged structure and Markov property. A discussion comparing it to Section 10.3 of "Elements of Causal Inference" and the paper "Necessary and sufficient conditions for causal feature selection in time series with latent common causes" by Mastakouri et al. would be helpful.

Response: Thanks for your feedback. It seems there might be a misunderstanding about the challenges and setup of our problem. In this paper, we study the Directed Summary Causal Graph on Subsampled Time Series with instantaneous effects using only two time-slices, which differ from those addressed in the works of Section 10.3 of Peters et al. (2017) and Section 2 of Mastakouri et al. (2021). We appreciate your suggestion and incorporate an additional section to discuss these comparisons, thereby clarifying the novelty of our work.

Our theorem differs from that of Peters et al. (2017): Theorems 10.1, 10.2, 10.3, and 10.4 in Section 10 of \citet{peters2017elements} rely on all previous time-slices within the observation windows that are accessible, i.e., $\mathbf{X}_{\operatorname{past}(t)}$ could be observed, which are not satisfied in subsampled time-series. In Section 10.2.1, \citet{peters2017elements} also noted that current methods for subsampled time-series require well-modeled interventions. Without well-defined interventions, it remains challenging to learn causal structures from subsampled time series without efficient solutions.

In our Theorem 1, given two time-slice observations $\mathcal{D} = \{ \boldsymbol{X}^{t_a}, \boldsymbol{X}^{t_b} \}_{t_a < t_b}$, $X_j^{t_b}$ is a descendant node of $X_i^{t_b}$ iff $X_i^{t_a} \not \perp X_j^{t_b} \mid \mathbf{an}_i^{t_a}$. We rigorously prove that it is both sound and complete under some graph constraints, i.e., acyclic summary causal graph, stationary full-time graph, and Markov property.

Our theorem differs from that of Mastakouri et al. (2021): Mastakouri et al. (2021) focus on the detection of direct and indirect causes of a given target time series, rather than full graph discovery. Their theory requires that each observed candidate time series must be a non-descendant node of the target time series, and they need to conduct two conditional independence tests for each observed candidate to identify the direct causes of the target. In contrast, our theory not only constructs the Descendant Hierarchical Topology of the Directed Summary Causal Graph but also requires only one conditional independence test per component. Furthermore, the full graph studied by Mastakouri et al. (2021) does not include instantaneous effects.

[Q6 & W4] Equation (1) is more restrictive than an additive noise model. An additive noise model is defined as Y = f(instantaneous parents, all other lagged parents) + N in a time series setting. In your case, you restrict the lagged variables to be separate functions and has only an additive influence. This seems quite restrictive.

Response: Thanks for your insightful question. Our proposed theory and the DHT-CIT algorithm do not rely on additional structural or distributional assumptions, except for the faithfulness assumption. The causal structure-function (Eq. (1)) can be directly reformulated to the additive noise model $X_i^{\tau}=f_i (\mathbf{pa}_i^\tau, X_i^{\tau-1}, \mathbf{pa}_i^{\tau-1} )+\epsilon_i^{\tau}$ or more general function $X_i^\tau=f_i (\mathbf{pa}_i^\tau, X_i^{\tau-1}, \mathbf{pa}_i^{\tau-1}, \epsilon_i^\tau)$, where $f_i(\cdot)$ is a twice continuously differentiable function capturing the instantaneous effects from its parents $\mathbf{pa}_i^\tau$ at time $\tau$ and a non-zero time-lagged effect from variable $X_i^{\tau-1}$ at time $\tau-1$. We have revised the corresponding description in the Problem Setup and Experiments sections.

Dear reviewer,

Thank you for your insightful feedback and valuable suggestions, which have significantly enhanced our manuscript, especially in defining the problem setting, positioning this paper, discussing baselines, and clarifying the novelty and contributions of our theorem. We have posted point-to-point replies to each question/comment raised by you and uploaded the revised version of our paper (with track changes marked in blue). We sincerely hope that we could address your concerns. We look forward to any additional comments you might have that could further refine our paper.

[Q1] A much cleaner definition of the problem setting and introduction of "time slices" is needed, especially the connection between observations at different times as interventions.

Response: This paper studies the summary causal graph on subsampled time series with instantaneous effects using only two time-slices, in which time-slices (cross-sectional observations) are sampled at a coarser timescale than the causal timescale of the underlying system. In this paper, a time-slice denotes sampled cross-sectional observations on multiple samples captured at a specific time point or time instant (within the given measurement accuracy) on the timeline of some system.

Subsampled Time Series with Only Two Time-slices (Figure 1(b)) is a more challenging task than the Standard Subsampled Time Series problem (Figure 10.4 in Section 10.2.1 of "Elements of Causal Inference"). The challenges arise from: (1) We only observe two time-slices from the Subsampled Time Series, and numerous unmeasured time slices may be latent in the time series; (2) We focus on Subsampled Time Series with instantaneous effects, thus, the theorems of Granger causality, "Elements of Causal Inference" and "Necessary and sufficient conditions for causal feature selection in time series with latent common causes" are not applicable [Peters et al., 2017; Mastakouri et al., 2021]; (3) We cannot rely on full interventions to identify the Directed Summary Causal Graph, as full interventions are often expensive, unethical, or even infeasible. Therefore, we explore using two time-slices as a substitute for intervention data to improve causal ordering and construct a unique Descendant Hierarchical Topology.

The connection between observations at different times as interventions: As indicated by red marking in Figure 1(b), an intervention on the variable $X_2^{t-3}$ would transfer its effect to descendants ($X_3^{t-3}, X_4^{t-3}$), but not to non-descendants ($X_1^{t-3}, X_5^{t-3}$). This helps in swiftly identifying each variable's descendants, enhancing topological ordering. Inspired by this concept, as illustrated in Fig. 1(b), where each node is influenced by its ancestors and itself, each variable in the earlier time-slice would transmit self-perturbations to both itself and its descendants in the subsequent time-slice, serving as simulated interventions.

We have revised and rephrased the connection between our work and existing methods in the Introduction and Related Work sections, highlighted in blue, to further clarify our paper's position in the field.

Dear reviewer,

Thank you for raising the score. We appreciate your detailed feedback and understand your concerns regarding $X_i^t$ and the problem setup. However, there might still be some misunderstandings about these two aspects, and we are willing to explain and clarify them for you again.

The definition of $X_i^t$ in Problem Setup Section: "Let $\boldsymbol{X} = \\{X_i^{\tau}\\}_{d \times t}$ denote the full $d$-variate time series with all time slices $\boldsymbol{X}^{\tau}$ at $t$ time points, where ${X}_i^{\tau}$, $i \in \\{1,2,\cdots,d\\}$ and $\tau \in \\{1,2,\cdots,t\\}$, is a random vector comprising observations of $n$ samples. For simplicity of notation, we will not discuss each sample individually. Instead, we refer to ${X}_i^{\tau}$ as a random variable/vector when discussing the causal structure", which has been highlighted in blue in the updated revision. In the text, we never view $X_i^t$ as a sample, as it is a random vector containing observations of $n$ samples/repetitions. We have revised all corresponding descriptions about ${X}_i^{\tau}$, $\boldsymbol{X}_i$, and $\boldsymbol{X}^{\tau}$ in the last updated revision.

As for the cleaner definition of the problem-setting: "This paper studies the summary causal graph on subsampled time series with instantaneous effects using only two time-slices, in which time-slices (cross-sectional observations) are sampled at a coarser timescale than the causal timescale of the underlying system. Given the presence of unmeasured time-slices, conventional causal discovery methods designed for standard time series data would produce significant errors about the system’s causal structure", which has been highlighted in blue in the Abstract and Introduction Section.

In Section 10 of the book "Elements of Causal Inference", the sufficient condition for the identifiability of the direction of instantaneous effects in acyclic summary graphs has been given by Theorem 10.2. However, Theorems 10.1, 10.2, 10.3, and 10.4 in Section 10 of the book rely on all previous time-slices within the observation windows that are accessible, i.e., $X_{past(t)}$ could be observed, which are not satisfied in subsampled time-series. In Section 10.2.1, Peters et al. (2017) also noted that current methods for subsampled time-series require well-modeled interventions. Without well-defined interventions, it remains challenging to learn causal structures from subsampled time series without efficient solutions. In our Theorem 1, given two time-slice observations $\mathcal{D} = \{ \boldsymbol{X}^{t_a}, \boldsymbol{X}^{t_b} \}_{t_a < t_b}$, $X_j^{t_b}$ is a descendant node of $X_i^{t_b}$ iff $X_i^{t_a} \not \perp X_j^{t_b} \mid \mathbf{an}_i^{t_a}$. We rigorously prove that it is both sound and complete under some graph constraints, i.e., acyclic summary causal graph, stationary full-time graph, and Markov property. We have revised the corresponding descriptions in the last updated revision.

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Thank you for dedicating time to review our manuscript and providing valuable insights. We have carefully incorporated your suggestions into our revised paper, with changes marked in blue. We hope our responses address your concerns adequately. Please feel free to let us know if you believe the paper needs further changes. We would appreciate the opportunity to address them promptly.

Dear Reviewer,

We sincerely appreciate your constructive comments to improve our paper. Below, we address your concerns point by point. Please kindly let us know whether you have any further questions or suggestions.

[Q1] Does this paper rely on Gaussian Models? While the experimental results indicate that the proposed method performs well on some Non-Gaussian Models.

Response: Thanks for your question. Our DHT-CIT algorithm does not rely on Gaussian Models. In this paper, we use two time-slices as conditional instrumental variables to simulate the effects of exogenous interventions. When applied to a variable, these simulated interventions only affect the variable's value, and the permutation would propagate to its descendant nodes. Then, we can apply conditional independence tests to capture these intervention-related permutations for identifying each variable's descendants, without requiring any structural or distributional assumptions about the data.

[Q2] It would be time-consuming to conduct conditional independence tests for all variables from scratch. Why not use the proposed method directly, based on the Complete Topological Ordering learned by SCORE, to remove the unnecessary edges to identify the Descendant Hierarchical Topology? I think this would save at least half of the time.

Response: Thanks for your suggestion. Applying the DHT-CIT algorithm to the Complete Topological Ordering derived from SCORE is indeed an option that may reduce computational time. However, this approach does not improve the identification of the Descendant Hierarchical Topology. Additionally, SCORE relies on Gaussian Models, but our DHT-CIT algorithm does not require any structural or distributional assumptions about the data. Our DHT-CIT algorithm is an independent model that is capable of directly learning the descendant hierarchical topology of DAGs from observational data and does not rely on any other topology-based methods.

[Q3] Can the proposed DHT-CIT be regarded as a model-free plugin module that can be incorporated into any existing topology-based method to improve the learned topological graph? Additionally, can it function as a test tool for selecting the true causal graph from Markov equivalence classes?

Response: Yes, the proposed DHT-CIT can be integrated as a module into any existing topology-based method to enhance the topological ordering. Additionally, our DHT-CIT algorithm is capable of identifying the true causal graph from the Markov equivalence classes that are typically learned using traditional methods. For instance, our DHT-CIT algorithm can effectively use the Descendant Hierarchical Topology to orient the undirected edges outputted by a constraint-based algorithm such as PCMCI [Runge et al., 2019]. We have included an additional section to discuss the potential application of our DHT-CIT algorithm in Appendix H.

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Thank you for your constructive feedback which has been invaluable in enhancing our work. We have uploaded the revised version of our paper (with track changes marked in blue). We welcome any further technical advice or questions on this work and we will do our best to address your concerns.

Dear Reviewer,

We sincerely appreciate your constructive comments, which have been instrumental in enhancing our manuscript. Below, we have provided detailed explanations addressing your concerns and uploaded the revised version of our paper (with track changes marked in blue). We would be very grateful if you could review these responses and inform us of any additional issues or further feedback you may have.

[W1] The paper builds on existing methods and the main contribution is the use of instrumental variables to improve the topological ordering, but this idea seems have been explored in previous works.

Response: While using interventions (i.e., instrumental variables) can improve the topological ordering, full interventions to obtain instrumental variables are often expensive, unethical, or even infeasible. Therefore, we explore using conditional instrumental variables in two time-slices as a substitute for instrumental variables (interventions) to improve causal ordering in subsampled time series.

We have to emphasize that the main contribution of our paper lies in identifying the causal structure in subsampled time series with instantaneous effects using only two time-slices, in which time-slices (cross-sectional observations) are sampled at a coarser timescale than the causal timescale of the underlying system. This task presents more challenges compared to the Standard Time Series problem: (1) We only observe two time-slices from the Subsampled Time Series, and numerous unmeasured time slices may be latent in the subsampled time series; (2) We focus on subsampled time series with instantaneous effects, thus, the theorems proved by standard time-series assumption are not available, e.g., Granger causality, Peters et al., (2017) and Mastakouri et al., (2021); (3) We don't rely on full interventions to identify the Directed Summary Causal Graph, as full interventions are often expensive, unethical, or even infeasible. Given the presence of unmeasured time-slices, conventional causal discovery methods designed for standard time series data would produce significant errors about the system’s causal structure. We are the first to use just two time-slices to identify causal structure of subsampled time series, addressing the issue of missing time-slices in time series.

We have revised and rephrased the connection between our work and existing methods in the Introduction and Related Work sections, highlighted in blue, to further clarify our paper's position in the field.

[W2] The paper only evaluates the algorithm on synthetic data and one real-world dataset. The synthetic data are generated from linear Gaussian models, which may not reflect the complexity and diversity of real-world data. It would be better to conduct more experiments on real-world data from various fields.

Response: Thank you for your feedback. However, it appears there might be a misunderstanding regarding the scope of our experiments. As shown in Tables 1, 2, 4, and Figure 2, we have conducted extensive non-linear experiments with various noise types, which the reviewer may have overlooked. In the main text, similar to the experimental setups of previous studies (Rolland et al., 2022; Sanchez et al., 2022; Montagna et al., 2023), we have conducted comprehensive experiments to demonstrate the superiority of our DHT-CIT algorithm. To better reflect the complexity and diversity of real-world data, we generated multiple synthetic datasets not only from additive non-linear Gaussian models (Table 1) but also from non-linear Laplace models and non-linear Uniform models (Table 2). Additionally, we explored varying time-lagged edges and denser graphs in Figure 2 and included experiments with other complex non-linear relationships (such as Sin, Sigmoid, and Poly functions) in Table 4 in the Appendix. The application of our algorithm to the real PM-CMR dataset further showcases its superior performance and scalability in handling larger and denser graphs. We believe these experiments adequately demonstrate the efficacy of our method. While we are open to conducting more experiments with real-world data from various fields, obtaining such data presents challenges, and their use in evaluating baselines is limited due to the absence of known real causal graphs.

[Q3] How do you generalize your algorithm to other domains and applications, such as social sciences, economics, or biology? What are the challenges and opportunities for applying your algorithm to these fields?

Response: As long as the common time series causal assumptions in [Assaad et al., 2022] and the q-order Markov Assumption are satisfied, we can directly extend our algorithm to other domains and applications with q+1 time-slices ($\boldsymbol{X}^{t_a \cdots t_a-q+1}$ and $\boldsymbol{X}^{t_b}$). For example, we can analyze the causal graph of city status variables in PM-CMR [Wyatt et al., 2020], and explore the relationships between various factors affecting soil moisture. In human genomics and gene expression, we also can establish two-time-slices causal relationships (surjections: where each expression variable can find a corresponding conditional instrumental variable in the genomic sequence variables). The challenges arise as different time series data may adhere to various high-order Markov Assumptions, which we need to identify. Additionally, sometimes the temporal transfer of events/processes might conceal causal relationships, requiring further extraction, such as the two-time-slices causal relationships between genomic and gene expression data. The DHT-CIT algorithm provides an excellent tool and opportunity for identifying the topological ordering in the aforementioned forms of data. We do not present experiments for all fields within this paper, as data acquisition is an expensive process. We have included an additional section to discuss the future application in Appendix H, and if we have any updates on the above datasets, we will post these updates on our project pages.

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According to your comments, our paper has undergone significant improvements regarding the motivation, problem setup, differentiation from traditional methods, our contribution, and its potential application to real-world scenarios. We have uploaded the revised version of our paper (with track changes marked in blue). We welcome any further technical advice or questions on this work and we will make our best to address your concerns.

[Q1] How do you handle the cases where the previous time-slice data is missing or unreliable? How does this affect the performance of your algorithm?

Response: Thank you for your question, which highlights the core motivation and main contribution of this paper: we identify the summary causal graph from subsampled time series with instantaneous effects using only limited time-slices, in which numerous previous time-slices are missing or unreliable.

In many applications, the time series sampling process may be slower than the timescale of causal processes, resulting in numerous previous time-slices are missing or unreliable (Problem). In the presence of unmeasured time-slices, relying solely on a single time-slice is insufficient for identifying causal relations (Challenges). Therefore, our Motivation is to use just two reliable time-slices to explore the summary causal graph of subsampled time series, rather than depending on all previous time-slices that are available and reliable (the limitation of traditional methods). In this paper, we demonstrate that if two valid time slices at two arbitrary moments are available, the variables in the earlier slice can be used as conditional instrumental variables to replace interventions and improve topological ordering. This method significantly relaxes the assumption inherent in traditional time series studies that depend on modeling causal structures at the system timescale, causal sufficiency, and all time slices in the observation windows could be observed (our contribution)[Granger, 1969, 1980; Lutkepohl, 2005; Hyvarinen et al., 2010; Luo et al., 2015; Nauta et al., 2019; Runge et al., 2019; Runge et al., 2020; Bussmann et al., 2021; Lowe et al., 2022; Assaad et al., 2022]. However, it is important to note that if no previous time-slice data is available or reliable, our approach, like other causal discovery algorithms, will not produce identifiable results.

[Q2] What are the advantages and limitations of using two time-slices data?

Response: We have added an additional section to discuss the advantages and limitations of using two time-slices data in the Appendix G.

Advantage: (1) Enhanced Topological Ordering: Traditional topology-based methods typically produce non-unique topological orderings with numerous spurious edges, resulting in decreased accuracy and efficiency in downstream search tasks. By using two time-slices as auxiliary instrumental variables, we can learn causal relations more efficiently, with a reduced search space and fewer spurious edges. (2) Feasibility in Intervention-Limited Contexts: Using interventional data can quickly identify (non-)descendants for each node and construct a more precise topological ordering. In scenarios where interventions are infeasible, unethical, or too costly, using two time-slices to replace intervention can be a practical alternative. (3) Reduced Data Requirements: In time series scenarios, traditional methods depend on the modeling causal structures at the system timescale, causal sufficiency, and all time slices in the observation windows could be observed. In this paper, we propose exploring limited time-slices, i.e., two reliable time-slices, to ease the data requirements. (4) Scaling to Non-linear and Non-Gaussian Models: We use two time-slices as conditional instrumental variables to simulate exogenous interventions. When applied to a variable, these simulated interventions only affect the variable's value, and the permutation would propagate to its descendant nodes. Then, we can apply conditional independence tests to capture these intervention-related permutations for identifying each variable's descendants, without requiring any structural or distributional assumptions about the data.

Limitations: Our DHT-CIT algorithm strictly relies on the Acyclic Summary Causal Graph and Consistency Throughout Time Assumptions, which are common in time series studies [Assaad et al., 2022]. Additionally, if no previous time-slice data is available or reliable, both the two time-slices approach and other causal discovery algorithms will not yield identifiable results.

Notably, in this paper, we can relax the Markov Assumption to a high-order Markov Assumption. This means that the future time-slice $\boldsymbol{X}^{t+1}$ depends only on states $\boldsymbol{X}^{t \cdots t-q+1}$ and does not directly depend on states $\boldsymbol{X}^{1 \cdots t-q}$. Then, with $q+1$ time-slices ($\boldsymbol{X}^{t_a \cdots t_a-q+1}$ and $\boldsymbol{X}^{t_b}$), we can use $\mathbf{an}^{t_a \cdots t_a-q+1}$ to replace the condition set $\mathbf{an}^{t_a}$ to infer the Descendant-oriented Conditional Independence Criteria (Theorem 1). However, in this paper, we focus exclusively on the two-time-slices algorithm to demonstrate our theorem and algorithm. We have added a section to discuss the high-order Markov Assumption in Appendix F.

Dear Reviewer,

Thank you for dedicating your time to reviewing our manuscript and offering valuable insights. We look forward to any further feedback you may provide.

In response to your constructive feedback, we have made significant revisions to our manuscriptt. We have clarified the problem setting, established a clearer positioning for our work, elaborated on the advantages and limitations of using two time-slices in subsampled time series data, and discussed potential real-world applications in more detail. These changes, marked in blue for your convenience, aim to address the concerns you raised in your previous review. Provided that your concerns have been well-addressed, we would greatly appreciate it if you would consider raising your score.

Should you have any further comments or queries, we would greatly appreciate the opportunity to address them promptly.

Warm regards,

Authors