DyCAST: Learning Dynamic Causal Structure from Time Series

We thank the reviewer for the valuable comments. Below we address your concerns and questions point by point.

Weaknesses Part

1.The concept of ... need refinement.

Response: We address your concern as follows:

• Clarity and Novelty: We emphasized the unique integration of latent Neural ODEs and constrained optimization for dynamic causal discovery. All revisions clarifying this are highlighted in blue.

• Method Explanation: To enhance understanding, we provided a more detailed description of the methodology, including the key steps and intuition behind the approach, in Section 3.3, page 6. Revisions to improve clarity are also marked in blue.

• Supplement of Experiments: Additional numerical experiments have been included to validate the proposed approach. These include evaluations on scalability, efficiency, and robustness, as well as experiments with dynamic intra-slice graphs to demonstrate the method's effectiveness. The results are shown in Appendix C (pages 15 to 22) and Appendix E(page 23).

Questions Part

1.On the large-scale systems. First, we would like to clarify that 300 variables are already considered large-scale in causal discovery tasks. Prominent methods such as DYNOTEARS [1], NTSNOTEARS [2], TECDI [3], CUTS [4], CUTS+ [5], NGM [6], NGC [7], and LCCM [8] typically benchmark their main experiments on datasets with around 20 variables. Our method, however, has been tested with variable counts exceeding 100, 200, and even 300, maintaining an impressive F1 score of 0.56. In contrast, similar approaches often fail to produce meaningful results with as few as 50 variables, underscoring the robustness and scalability of our approach.

Real-world causal discovery datasets with ground truth typically involve fewer than 50 variables. For instance, NetSim[9] ($d=5, 10, 15$), CausalTime [10] ($d=20, 20, 36$), and RiverFlow [11] ($d=3$) are among the standard benchmarks. Of these, CausalTime, the most recent and mainstream dataset published in ICLR2024, exemplifies this trend.

To further validate our method's effectiveness, we conducted additional experiments on DREAM3 [12], a gene expression and regulation dataset with $d=100$, as shown in the following Table:

The results demonstrate that DyCAST achieved an AUROC of 0.55, DyCAST (Not DAG) achieved 0.57, and DyCAST-CUTS+ achieved 0.65, outperforming CUTS+ alone. These findings illustrate the ability of DyCAST to handle complex, nonlinear real-world datasets, achieving competitive performance even at larger scales.

2.On the time interval between data points. Regarding the time interval between data points, we adhere to the same approach as DYNOTEARS [1], NTSNOTEARS [2], TECDI [3], CUTS [4], CUTS+ [5], NGM [6], NGC [7], and LCCM [8]. Specifically, we utilize a coherent time series without interval sampling between data points for synthetic datasets.

For real-world datasets, we also utilize a coherent time series without interval sampling between data points, and we maintain the inherent interval sampling of the respective datasets:

• NetSim: No interval sampling is applied between time intervals.
• CausalTime:
  • In the AQI subset, the interval between data points is hourly.
  • In the Traffic subset, the interval is 5 minutes.
  • In the Medical subset, the interval is 2 hours.
• Human3.6: The interval between data points corresponds to one frame.

We follow the established settings outlined in the NTSNOTEARS and CUTS+ papers, ensuring consistent comparisons when running experiments on all algorithms. As we do not incorporate interval sampling into our method, our experiments remain aligned with the natural intervals of the datasets or the standard practices of prior works. We will include the corresponding description in the revision.

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Reference

[1] Pamfil et al. (2020) DYNOTEARS: Structure Learning from Time-Series Data.

[2] Sun et al. (2023) NTS-NOTEARS: Learning Nonparametric DBNs With Prior Knowledge.

[3] Li et al. (2023) Causal Discovery in Temporal Domain from Interventional Data.

[4] Cheng et al. (2023) CUTS: Neural Causal Discovery from Irregular Time-Series Data.

[5] Cheng et al. (2024) CUTS+: High-Dimensional Causal Discovery from Irregular Time-Series.

[6] Bellot et al. (2022) Neural graphical modelling in continuous-time: consistency guarantees and algorithms.

[7] Tank et al. (2021) Neural Granger Causality.

[8] Brouwer et al. (2021) Latent convergent cross mapping.

[9] Smith et al. (2011) Network modelling methods for FMRI.

[10] Cheng et al. (2024) CausalTime: Realistically Generated Time-series for Benchmarking of Causal Discovery.

[11] Rohekar et al. (2023) From temporal to contemporaneous iterative causal discovery in the presence of latent confounders.

Quick comments:

1. The condition of a directed acyclic graph (DAG) should be much easier to deal with. Why, then, does even a basic, more general causality method like Granger causality fail in your setting? And also attentive transfer entropy?
2. If I was wrong, please clarify: I do not fully understand the phrase "continuous numerical time series." Authors have specified the time length and time interval, which suggests discrete time points rather than truly continuous time.

Importantly, I am strongly concerned about the accuracy and seriousness of this manuscript. The authors have surprisingly miswritten reference details, including the year and journal name, even after I pointed this out in earlier comments ALREADY!

So, I cannot trust any results of this manuscript.

We would like to clarify that the key contribution of DyCAST lies in providing a DBN-based full-time causal graph that captures both intra-slice and inter-slice relationships for each time point in a multivariate time series. Importantly, the dynamic intra-slice graphs always satisfy the directed acyclic graph (DAG) constraint, distinguishing our work from approaches focused on Granger Causality [1-3]. Granger Causality typically provides a static summary graph and does not enforce the DAG constraint.

We acknowledge that the cited work [3] can handle $10^4$ variables; however, its primary focus is on inferring coupling relationships between observed variables and reconstructing the coupled network from time series data. In contrast, our approach emphasizes inferring causal relationships between variables while strictly adhering to DAG constraints. This fundamental distinction explains why the two methods are designed to handle different scales of variables. Furthermore, the listed literature [1] and [2] primarily addresses causality in discrete event sequences, where causal relationships are easier to discover due to explicit event representations and clear timestamps. In contrast, continuous numerical time series introduces greater complexity, making causal discovery significantly more challenging. This complexity is especially pronounced when scaling to large systems, where dynamic causal discovery remains a demanding task.

we thank the reviewer for pointing out these papers, and we will add an additional related works section in the revision to discuss the papers you mentioned.

Reference

[1] Zhang, Wei, et al. (2020) "Cause: Learning granger causality from event sequences using attribution methods." ICML.

[2] Xiao, Shuai, et al. (2019) "Learning time series associated event sequences with recurrent point process networks. TNNLS.

[3] Ru, Xiaolei, et al. (2023) "Attentive transfer entropy to exploit transient emergence of coupling effect." NeurIPS.

2.If ... truly continuous time.

Response: In observed event sequences, each node represents a discrete event type (e.g., A, B, C). For instance, the ATMs dataset (which is used in reference [6]) contains event logs from 1,554 ATMs, including failure tickets (TIKTs) and error reports. These logs capture discrete states such as device identity, timestamps, message content, priorities, codes, and actions. This discrete nature focuses on direct connections between events, facilitating causal reasoning.

In contrast, the time series data we handle consists solely of continuous real values. For example, the CausalTime dataset records traffic flows in real values at 20 intersections, and the Human3.6 dataset provides three-dimensional spatial coordinates for human joints. This type of data is inherently noisy, continuously variable, and influenced by complex periodicities, trends, and multivariate dependencies, making it harder to define causal relationships. These complexities often obscure the underlying causality, significantly increasing the difficulty of causal discovery.

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Reference

[1] Zhang, Wei, et al. (2020) "Cause: Learning granger causality from event sequences using attribution methods." International Conference on Machine Learning.

[2] Ru, Xiaolei, et al. (2023) "Attentive transfer entropy to exploit transient emergence of coupling effect." Advances in Neural Information Processing Systems.

[3] Zheng, Xun, et al. (2018) "Dags with no tears: Continuous optimization for structure learning." Advances in neural information processing systems.

[4] Ng, Ignavier, et al. (2020) "On the role of sparsity and dag constraints for learning linear dags." Advances in Neural Information Processing Systems.

[5] Ng, Ignavier, et al. (2022) "On the convergence of continuous constrained optimization for structure learning." International Conference on Artificial Intelligence and Statistics.

[6] Xiao, Shuai, et al. (2019) "Learning time series associated event sequences with recurrent point process networks. IEEE transactions on neural networks and learning systems.

1.The condition of a directed acyclic graph (DAG) ... attentive transfer entropy?

Response: We sincerely apologize for our lack of rigor in checking reference details. It was indeed due to our negligence that the journal number of reference [1] and the year number of reference [2] were incorrectly marked. We have modified the corresponding answers above.

Granger causality-based models typically assume that each sampled variable $x_{i,t}$ is generated by a Structural Causal Model (SCM) with additive noise:

$X_{i,t} = f_i(X_{1, t-p:t-1}, X_{1, t-p:t-1}, .... ,X_{d, t-p:t-1})$

where any variable in $\{\boldsymbol{X}_{t-p}, \ldots, \boldsymbol{X}_{t-1}\}$ may effectively influence $\boldsymbol{X}_t$. Under this assumption, no additional constraints are imposed on the summary causal graph. To account for sparsity, regularization terms (e.g., $L_1$ or $L_2$) are applied.

Furthermore, Granger causality-based methods do not model the intra-slice causal term $\boldsymbol{X}_t \boldsymbol{W}_t$, thereby completely avoiding the DAG constraint. This omission ignores causal relationships within slices, limiting their ability to capture the full dynamics of $\boldsymbol{W}_t$. At the same time, these methods provide only a summary causal graph, identifying whether a causal relationship exists between $\boldsymbol{X}_i$ and $\boldsymbol{X}_j$, but not specifying if $\boldsymbol{X}_{i,t}$ is causally influenced by a lagged value $\boldsymbol{X}_{j,t-k}$. Consequently, even advanced approaches like Attentive transfer entropy cannot recover the full-time causal graphs.

In contrast, our model further incorporates the data $\boldsymbol{X}_t$ at time $t$ and assumes that when $(\boldsymbol{W}_t$ satisfies the DAG constraint, thus the model becomes:

$X_{i,t} = X_t W_t + X_{t-1} A_{1} + ... + X_{t-p}A_{p}$

Here, the intra-slice matrix $\boldsymbol{W}_t$ must satisfy the DAG constraint at all times. While full-time DAG causal graphs are generally sparser than directed graphs (as they exclude bidirectional edges and cycles), enforcing the DAG constraint is computationally challenging. The widely-used acyclicity constraint was introduced by Zheng et al [3]. defines $h(\boldsymbol{W}) = \text{Tr}(e^{\boldsymbol{W} \odot \boldsymbol{W}}) - d$, where $\text{Tr}$ denotes the matrix trace . In practice, enforcing this constraint requires careful tuning of the augmented Lagrangian parameters [4]. Moreover, as the penalty coefficient approaches infinity to ensure acyclicity, numerical instabilities, and ill-conditioning issues arise, as shown empirically by Ng et al. [5].

Our model further requires that $\boldsymbol{W}_t$ dynamically satisfy the DAG constraint at every time step, making optimization significantly more challenging. This is a key reason our method cannot scale to $10^4$ variables.

We thank the reviewer so much for the insightful comments and appreciation of our work. We address your concerns and questions in the following point by point.

Weaknesses Part

1.While DyCAST ... experiments.

Response: Actually, in the original version of the manuscript, we have already introduced a synthetic dataset with dynamic intra-slice graphs to evaluate our model quantitatively, as described in (Figure 3) and Section 4.1 (pages 7–8). This dataset features 7 dynamic intra-slice graphs, 5 variables, and an autoregressive order of 1. DyCAST achieves results closely matching the ground truth. Additionally, extensive experiments on datasets with varying numbers of variables, compared to DYNOTEARS and its variants (Figure 4), demonstrate DyCAST's effectiveness in capturing dynamic intra-slice graphs.

Given the limited availability of real-world dynamic causal graph datasets, we run DyCAST on a subset of scenes from the Human3.6M dataset and analyze the resulting dynamic intra-slice data, as shown in Figure 17 (Appendix E, page 23). Our experiments demonstrated that DyCAST was able to effectively capture the causal relationships between body joints as they change over time. During the smoking motion, CUTS+ captures the causal link between the hands and head but keeps it static, even after smoking ends. DYNOTEARS identifies the lagged influence of the left elbow and its link to the head but misses the dynamic relationship with the shoulder. In contrast, DyCAST dynamically captures and adjusts the causal relationships, reflecting the natural dynamics of smoking, and demonstrating its superiority in real-world scenarios.

2.Several baseline methods ... recent works would be acceptable.

Response:We agree that clarifying the choice of baselines is essential for a thorough evaluation of DyCAST.

In our synthetic experiments, the ground truth provided includes both intra-slice matrices at each time step and static inter-slice matrices. Since only DYNOTEARS and its variants are capable of outputting this type of ground truth, we focused our comparisons on these methods, namely DYNOTEARS, NTS-NOTEARS, and TECDI. This is why we limited our baseline comparisons to these methods in the synthetic dataset experiments.

For the CausalTime and NetSim datasets, the ground truth is in the form of summary graphs, which makes it possible to compare against a wider range of causal discovery methods. In the revised manuscript, we have aligned the baseline comparisons for these datasets by adding results from other top-performing methods. Specifically:

• For the NetSim dataset, we have added the results for GC, CUTS+, LCCM, DyCAST (Not DAG) and DyCAST-CUTS+. We report the results in Table 2, Section 4.2 (pages 9) and Table 6, Appendix C.10 (pages 21).

• For the CausalTime dataset, we have included the results for TECDI. We report the results in Table 3, Section 4.3 (pages 10).

• These updates ensure that we now provide a more comprehensive comparison of DyCAST's performance on both synthetic and real-world datasets. We appreciate your suggestion, as it has helped us improve the clarity and depth of our baseline analysis.

We thank the reviewer for the valuable comments. Below we address your concerns and questions point by point. We also make the corresponding changes in the revision following your suggestions.

Weaknesses Part

1.The combination of DyCAST ... using CUTS or CUTS+.

On the combination with other methods: We sincerely thank the reviewer for recognizing the flexibility of DyCAST. In its foundational model (Eq. (3)), DyCAST assumes that inter-slice dependencies are linear and fixed. While this is a reasonable simplification in many cases, real-world data often exhibit complex non-linear relationships.

To handle this, we extend Eq. (3) by replacing the linear inter-slice term with a more flexible function $\varphi$, allowing for non-linear modeling:

\boldsymbol{X}_t = \boldsymbol{X}_t\boldsymbol{W}_t+\varphi(\boldsymbol{X}_{t-p:t-1})+\boldsymbol{Z}_t $$ where $\varphi$ parameterizes the **non-linear inter-slice causal relationships**.This framework can accommodate a wide range of models specifically designed to capture such relationships. Therefore, with such an extension of DyCAST, we can easily integrate it into CUTS+, a method that focuses on learning inter-slice causal relationships, characterization of $\varphi (\boldsymbol{X}_{t-p:t-1})$ in **Eq(20) (Section 3.3, page 6)**. While CUTS+ is a Granger causality-based model and produces an inter-slice **summary graph**, it is capable of approximating a **full-time graph** when the lag order **p=1**, which aligns with our requirements (this implies that when the autoregressive order **p>1**, we can only derive the summary graph for the lagged relationships of $\boldsymbol{X}_t$ with respect to $\boldsymbol{X}_t$). A detailed explanation of this modification and its integration into DyCAST is now provided in **Section 3.3, Extension of DyCAST, page 6**. - The mention of "CUTS" instead of "CUTS+" in the main text and tables was an error. We confirm that all experiments were conducted using **CUTS+**, which is both more effective and efficient than its predecessor. We have thoroughly reviewed the manuscript and corrected this inconsistency throughout the revised submission.

Thanks a lot for your detailed reply, which I think addressed most of my concerns. I have adjusted my rating accordingly.

On the Size of Tested Datasets I agree with the authors that $d=300$ datasets are already fairly large. I do not think that the authors should make claims about "very large scale systems" or the like (and to my knowledge they don't), but find the inclusion of several real-world benchmarks to be more persuasive than larger synthetic datasets would be.

On the combination with other methods Thanks for clarifying that CUTS+ is used everywhere. So, once again to clarify: DyCAST-CUTS+ can only recover the full graph in the $p=1$ case, right? This is okay, but is only really implicitly mentioned in Section 3 ("Notably, when the preceding variables span only a single time step, the summary graph is equivalent to the full-time DAG"); I think it would improve clarity to mention at the start (e.g., line 291) that "When $p=1$, we may take advantage of the summary graph coinciding with the inter-slice graph, using Granger-causal methods to learn inter-slice graphs."

Ablations Thanks for including these. Regarding DyCAST w/o $S_0$: does this just mean that $z_0 \coloneqq \mathrm{Flatten}(\mathrm{ReLU}(W_0 P))$? I think this should be clarified.

On the Computational Cost of DyCAST+ Thanks for this table. Please consider adding this to an appendix.

Some Remaining Typos

• Line 275: "layres" should be "layers".
• Throughout: Sometimes "DyCAST(Not DAG)" is written (no space between "DyCAST" and "(Not DAG)").
• Line 461: "NTS-NOTEATS" should be "NTS-NOTEARS".
• Line 461: Use \citep for (Gong et al., 2023)
• Figure 9: The second plot is mislabeled as "F1 Score"; I think it's the SHD.
• Figure 11: Once again I don't think either of these are showing F1 score, but the caption labels them as such.
• Line 1082: "Obvious" should not be capital.
• Appendix C.9: The title "Evaluation on Static Synthetic Data Across Algorithm with Separate Learning Tricky." doesn't make much sense, please consider revising it.

We thank the reviewer so much for the reply and we really appreciate the reviewer for raising score.

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Comment

1.I agree with the authors that ... than larger synthetic datasets would be.

On the size of tested datasets. We thank the reviewer for the suggestion. We conduct a new experiment on the DREAM-3 dataset, comprising five independent datasets, each with 10 time-series recordings from 100 genes across 21 time steps (see Table 11 in Appendix F). The results demonstrate that DyCAST remains highly competitive, significantly outperforming comparable methods such as DYNOTEARS and NTS-NOTEARS. Leveraging the nonlinear CUTS+ integration further boosts performance, we also find that integrating the nonlinear CUTS+ further improves performance leading to enhanced results.

2.Thanks for clarifying that CUTS+ is used everywhere ... to learn inter-slice graphs.

On the combination with other methods. That's right that DyCAST-CUTS+ can only recover the full graph in the $p=1$ case. We now mention at the start (line 291) that "When $p = 1$, the summary graph coincides with the inter-slice graph, allowing the use of Granger-causal methods for learning inter-slice structure".

3.Thanks for including these ... I think this should be clarified.

Ablations. We thank the reviewer for the suggestion. DyCAST without $S_0$ simply corresponds to $\boldsymbol{z}_0 := Flatten(ReLU( \boldsymbol{W}_0 \boldsymbol{P}))$. We have now included this clarification in line 413 of Section 4.1 (Ablation Study) on page 8.

4.Thanks for this table. Please consider adding this to an appendix.

On the computational cost of DyCAST+ We thank the reviewer for the nice suggestion. We add an additional discussion on running time in Table 6 (Appendix C.7, page 20).

5.Some Remaining Typos

We thank the reviewer so much for pointing these typos out. For each of them,

• We fix this as "layers" in line 275, page 6.
• We add the missing space between "DyCAST" and "(Not DAG)".
• We fix this as "NTS-NOTEARS" in line 464, page 9.
• We add the "\citep" for (Gong et al., 2023) in line 464, page 9.
• We fix the second and fourth plots for the mislabeled "F1 score" to "SHD" in Figure 9, Appendix C.4, page 17.
• We fix this in Figure 11, Appendix C.5, page 18.
• We fix this in lines 1082, page 21.
• We revise the title of Appendix C.9 to "Challenges of evaluating separate learning on dynamic synthetic data".

Questions Part

1.The authors used traffic ... than a DAG?

Response: We agree that traffic networks and brain networks might not be DAGs in some cases. Fortunately, our method can be easily extended to handle directed graphs (non-DAGs). By simply removing the manifold constraint terms in the Eq.(9) (line 201, page 4), DyCAST can infer the dynamics of directed graphs effectively.

We conducted experiments using DyCAST (Not DAG) on the NetSim and CausalTime datasets. These additional experiments are presented in Table 2 (Section 4.1, page 9), Table 3 (Section 4.2, page 10), and Table 6 (Appendix C.10, page 21).We show that DyCAST (Not DAG) achieves impressive AUROC values for both datasets. However, removing the DAG constraint introduces spurious bidirectional edges in the inferred causal graph, slightly reducing the AUPRC metric.

2.How should the value ... inference process?

Response: We sincerely appreciate the reviewer's insightful comment regarding the selection of the time-lag order $p$. In fact, DyCAST does not require prior knowledge of the true lag order in the data. To demonstrate this, we evaluated DyCAST with $p=3$ on simulated data where the true lag order was $p_\text{true}=1$ (Figure 10, Appendix C.5, page 18). The results showed that DyCAST identified only three very small spurious entries (less than 0.1) in the estimated $A_2$, and no spurious entries in $A_3$. This suggests that DyCAST is robust to the choice of $p$, and selecting a slightly larger lag order does not significantly affect the final inference results.

Further details of this experiment can be found in the Appendix C.5. Additionally, we provide two practical strategies for determining $p$ in the Appendix: (1). Using the Bayesian Information Criterion (BIC) based on the model loss to select $p$as shown in **left panel** of Figure 11, **Appendix C.5, page 18**. (2). Estimating$p$by analyzing the maximum entry of the inferred$A_p$matrices ($\max(A_p) $) as shown in **right panel** of **Figure 11, Appendix C.5, page 18**). These approaches can guide users to select an appropriate lag order without prior knowledge of$p_\text{true}$.

We thank the reviewer so much for the valuable comments and appreciation of our work and efforts. Below we address your concerns and questions point by point.

Weaknesses Part

1.The authors tested their DyCAST algorithm ... in real-world scenarios.

Response: We acknowledge the importance of evaluating our method on large-scale variables of datasets to enhance the robustness of our results. Since real-world large-scale dynamic causal datasets for time series with ground truth are scarce, we conduct experiments on simulated data to ensure a fair comparison. Specifically, we evaluate datasets with $d = \{50, 100, 200, 300 \}$ variables, $T = 8$ time steps, and $n = 500$ samples in Table 5 (Appendix C.1, page 15).

DYNOTEARS, NTS-NOTEARS, and TECDI fail to converge with 50 variables, misidentifying all causal edges, while DyCAST achieves an average F1 score of 0.73. At 300 variables, the compared baselines remain non-convergent, but DyCAST maintains an F1 score of 0.56 and a low SHD (49.11±7.14), demonstrating scalability and robustness in large-scale causal structures.

2.While incorporating inter-slice ... to be effective.

Response: We thank the reviewer so much for this insightful question. To address this concern, we also conduct an experiment on datasets with $d = 20$ variables, $T = 8$ time steps, and over different sample sizes ($N = {400, 300, 200, 100, 50, 20}$). DyCAST uses observation data from the current moment ($t$) and the past $p$ ($\boldsymbol{X}{t-p:t-1}$) time points, consistent with other baselines. The corresponding discussion is presented in Figure 7 (Appendix C.2, page 16). Quantitative experiments show that DyCAST achieves an F1 score of ~0.4 and SHD <30 with as few as 20 samples, whereas baselines fail to identify any valid causal relationships, demonstrating DyCAST's ability to handle limited data effectively.

3.The algorithm relies on ... single trajectory of each variable.

Response: To address your concern, we experiment to examine the scalability and efficiency. We run DyCAST across datasets over different sample sizes (N = {400, 300, 200, 100, 50, 20}) and length (T=8, 16, 32, 64, 128). These additional experiments are presented in Appendix C.2 and C.3 (pages 16 to 17). Specifically,

• we show that DyCAST achieve an average F1 score of approximately 0.4 even with as few as 20 samples in Figure 7. In contrast, baseline models were unable to identify any valid causal relationships under similar low-sample conditions. The potential of DyCAST to effectively handle limited trajectories has been thoroughly demonstrated.

• We show that, despite the growing complexity of internal dynamics with longer time sequences, DyCAST consistently achieved an F1 score above 0.6 across all tested time step lengths, with a fixed sample size of 500 in Figure 8.The potential of DyCAST to effectively handle long temporal dynamics has been thoroughly demonstrated.

• Through extensive experiments, we determined that the model achieves 80% accuracy when the ratio of parameters to data is approximately 0.025%. In practice, we observed that for sufficiently simple causal relationships, the model performs well even with a lower ratio.

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Extension of DyCAST

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As suggested by Reviewer AYt4, we discuss the extension of DyCAST in Section 3.3 (starting from line 289, page 6 of the revision). Specifically, we show that:

1. Our DyCAST modeling framework is flexible, which can be easily incorporated into other causal discovery approaches, particularly those for time series, with slightly modifying our Eq.(3) into:

$\boldsymbol{X}^i_t = \boldsymbol{X}_t \boldsymbol{W}_t + \varphi(\boldsymbol{X}{t-p:t-1}) + \boldsymbol{Z}^i_t$

where $\varphi$ can be a complex nonlinear function.

Following this extension, we show that DyCAST can be introduced to CUTS+, which is a nonlinear model specifically designed to construct $\varphi$. To examine the effects of DyCAST-CUTS+ on complex real-world datasets in the causal discovery task, we also run DyCAST-CUTS+ on NetSim and CausalTime datasets. DyCAST-CUTS+ achieves the highest average performance on both NetSim and CausalTime, consistently providing performance improvement over the original CUTS+. We report the results in Table 2 (Section 4.2, page 9), Table 3 (Section 4.3, page 10), and Table 6 (Appendix C.10, page 21).