Castor: Causal Temporal Regime Structure Learning

We extend our gratitude to Reviewer Xsp2 for the very insightful comments. We are pleased to note the appreciation for the introduction of regime learning and the acknowledgment of our approach in handling both instantaneous and time-lagged relations.

The significance of the problem setting is unclear:

• We are committed to providing further motivation for the problem at hand. In our introduction, we highlighted the relevance of our framework to EEG recordings in epilepsy, where distinct regimes (non-seizure, pre-ictal, seizure) form Multivariate Time Series (MTS) sequences. Additionally, Saeed et al.'s work, on causal structure discovery from mixtures of Directed Acyclic Graphs (DAGs) presented at ICML 2020, shows the importance of this setting in learning gene regulatory networks from diverse subtypes in neurological diseases. We believe that there are several other settings where our framework is particularly relevant and attractive, and we will revise the text to clarify these points.
• Then, we clarify that each partition in our framework corresponds to a MTS block, representing a distinct regime. Visual aids in the form of figures have been added to the appendix to elucidate our setting.

Even under this setting, the problem can already be well solved by existing methods:

Our statement "CD-NOD cannot infer individual causal graphs and necessitates prior knowledge about the number of regimes."

• We sincerely appreciate the insightful remark. Our statement aims to clarify that CD-NOD provides a summary graph for all regimes, which is a more constrained output compared to the comprehensive representation offered by our framework, CASTOR (Full temporal causal graph per regime). As the reviewer correctly noted, CD-NOD can infer change points but lacks the ability to express if regimes are repeated in the Multivariate Time Series due to the absence of temporal causal graphs for each regime. In contrast, CASTOR does not only learn and present the temporal causal graph for each regime but also leverages this information to achieve more accurate causal graphs by grouping together time series blocks from the same causal graph. This approach precisely determines the total number of distinct regimes.
• We argue that learning a full temporal causal graph per regime is more powerful than acquiring a single summary graph. The full temporal causal graph does not only contain the information necessary to reconstruct the summary causal graph but also offers a more accurate evaluation metric. For instance, consider two simple settings with two variables: In setting 1, we have 1->2 in a time-lagged manner and 2->1 instantaneously, resulting in the summary causal graph 1<->2. Now, in setting 2, we have 2->1 in a time-lagged manner and 1->2 instantaneously, resulting in the summary causal graph 1<->2. However, evaluating based on the full temporal causal graph in each setting provides a nuanced and accurate assessment of the differences between them.
• what's the benefit of using the authors' proposed method than just using CD-NOD as above? Thank you for your valuable feedback. We initially considered a fair and challenging comparison with methods learning full temporal causal graphs through manual regime splitting. We want to clarify that we had presented comparisons to RPCMCI which is a model for MTS with different regimes. However, in response to your suggestion, we conducted further experiments, comparing with CD-NOD and using phases 1 and 2 from the official Python package causal-learn, providing regime indexes to CD-NOD. The results of these extended comparisons are detailed below. | | method | K=2 | K=3 | K=4 | |----|--------|---------|----------|----------| | 10 | CD-NOD | 20.2 | 11.4 | 38.8 | | | CASTOR | 100 | 100 | 97.9 | | 20 | CD-NOD | 25.2 | 23.7 | 12.7 | | | CASTOR | 100 | 97.2 | 93.4 | | 40 | CD-NOD | 0 | 11.3 | 5.57 | | | CASTOR | 100 | 99.8 | 99.2 |

The parametric assumption is quite limited:

• We want to thank you for this remark, part of our future work will be to extend to more complicated settings with more complicated noises.

The experimental comparisons are incomplete:

• We have incorporated additional experimental results, conducting comparisons with CD-NOD and Rhino (CASTOR surpasses both models in various settings, achieving over 93% compared to 23% for CD-NOD and 39% for Rhino.). Please refer to the appendix for detailed outcomes (Comparison with Rhino). Furthermore, we have included running times for all the models for comprehensive insights.

The presentation somewhat lacks clarity:

• We acknowledge the importance of clarity and have taken steps to address this concern. New figures have been added to the paper, available in the updated appendix, aiming to enhance the reader's understanding.

Thank the authors for your response. And especially thank you for running additional experiments in short time.

However, my two biggest concerns are still not very well addressed. Specifically,

1. The significance of the problem setting is unclear:

Motivative or real-world examples of such dataset, i.e., time series from several regimes where the relations inside each regime is stationary?

The authors mention EEG recordings in epilepsy (Tang et al. 2021; Rahmani et al. 2023) and DAG mixtures in gene data (Saeed et al. 2020) to show the importance of this setting. However, for the former, could the authors please refer to the specific paragraphs in the two cited papers that explicitly model the EEG data as this setting? For the latter, if I understand correctly, they care about mixture of different DAGs in a same dataset without temporal partition, which is quite different from the setting of this paper.

Usually it's more natural to think of nonstationarity in time series data as changing constantly but gradually.

By "time partition" (Definition 2), did the authors assume that each partition is a continent time series block? Also, did the authors assume/constrain any "similarities" between the causal graphs of different regimes?

It seems that the authors didn't answer these questions. The authors did add some "visual aids in the form of figures" to the appendix, but here are some nit questions: in e.g., Fig 2, in each regime there are two adjacency matrices - one for instantaneous relations and one for time-lagged ones. How are time-lagged ones plotted in one matrix? Are the lags the same? Which colors of light and dark blues denote 1 or 0? Also, the y-axis 0.0, 7.5,... are misleading. Please make them integers.

2. Even under this setting, the problem can already be well solved by existing methods (e.g., CD-NOD):

By using the time stamp IDs as the surrogate variable, CD-NOD's phases I and II can output the abstract causal graph. Then, using kernel non-stationary visualization of phase III, the different regime partition is readily available.

As mentioned earlier, it seems that what this paper does (identify different regimes, and then identify causal graph within each regime) can already be achieved by CD-NOD. So I am wondering what is the benefit of this paper.

The authors mention "CD-NOD can infer change points but lacks the ability to express if regimes are repeated" -- I don't quite get it. What do authors mean by "repeated regimes"? I searched "repeat" in your paper but didn't find anything related. It seems that once each regime is identified, whether regimes are repeated will also be trivially available?

The authors argue that "learning a full temporal causal graph per regime is more powerful than acquiring a single summary graph" -- that is unrelated. As mentioned above, the single summary graph is just a start point. CD-NOD can further identify the regimes partition and then temporal causal graphs within each regime can be estimated using many existing methods separately as well.

Moreover, some of my other concerns still exists, e.g.,

3. The parametric assumption is quite limited: the authors have to assume the equal-variance of exogenous noise components, only with which the DAG identifiability under instantaneous relations is possible. However, this assumption is generally untestable. I don't put it as a major concern for this specific paper only because this assumption is widely used now by optimization based methods.

In summary, my major concerns are not well addressed. So I keep my score of rejection.

We express our sincere gratitude to Reviewer VxPv for the dedicated effort and valuable comments on our manuscript. We are pleased that the reviewer acknowledges the quality of our paper's presentation and recognizes its strengths, particularly in regime learning and causal discovery from heterogeneous data.

By addressing the concern raised by the reviewer regarding the correctness of our theorem, we appreciate the opportunity to provide clarification. The example presented by the reviewer, featuring two components with the same mean but different variance, neglects a critical aspect: all our noises originate from $\mathcal{N}(0,1)$. Consequently, variations in the mean of different components of the mixture occur when changing the graphs. This nuanced point ensures that if the means are the same, the underlying graphs are also identical, indicating the same regime. We are hopeful that this clarification will alleviate any doubts about the validity of our theorem.

Concerning the inquiry about the choice the number of regimes, our paper explicitly addresses this problem in the learning algorithm. The detailed process is explained in the third section, involving the decomposition of the time series into equivalent windows. Subsequently, we iteratively update these windows by merging some, and canceling others. To enhance clarity, visual explanations, and figures detailing the learning process and CASTOR framework have been included in the appendix.

We extend our gratitude to the diligent efforts of the reviewer and value the insightful comments. We are grateful for Reviewer (4Xkb)'s recognition of the importance of our problem and the generality of our framework in addressing both linear and non-linear relations. The references cited by the author have been duly acknowledged in our paper, and they indeed represent special cases within our framework. We have added a discussion that permits to further clarify the novelty of the proposed framework.

Notably, the reviewer references DYNOTEARS and its extension to the non-linear setting using the extended version of NOTEARS. We outline that our method encompasses these as special instances. In Equation 3 on page 3 of our paper, we present SEM of our settings. If $K=1$, corresponding to a time series with only one regime, it essentially becomes an extension of DYNOTEARS to non-linear settings. However, it is crucial to highlight that our framework operates in a setting where time series comprise multiple consecutive regimes, with each regime could be seen as MTS with a different causal graph (We employed the EM algorithm to iteratively alternate between regime learning and graph learning). CASTOR learns together the number of regimes, their respective indexes (indicating when each regime starts and ends), and the associated causal graph - none of the methods in the literature is able to address this joint learning problem, to the best of our knowledge.

Thank you for the clarification. In our framework, heterogeneous time series refers to the fact that the time series data has multiple regimes. Based on our understanding, the term "heterogeneous data" or "heterogeneous time series" is used to denote data from different causal models in causal discovery literature (we provided some references). Your insight is valuable, and we have therefore refrained from using this terminology in the updated version of our manuscript to enhance clarity.

• Saeed, Basil, et al. "Causal Structure Discovery from Distributions Arising from Mixtures of DAGs." International Conference on Machine Learning. PMLR, 2020.
• Huang, Biwei et al. "Causal Discovery from Heterogeneous/Nonstationary Data." The Journal of Machine Learning Research, 21

To address concerns regarding runtime efficiency, we have incorporated comprehensive running time experiments in the appendix. Additionally, new models have been introduced, and all pertinent results are presented in the supplementary materials.

Setting: 10 nodes and different number of regimes (2,3,4)

VARLINGAM and CD-NOD (employing Fisher’s Z conditional independence test for faster runs; note that KCI CD-NOD takes over 2000 seconds for 2 regimes, with F1 scores in a similar range) exhibit remarkable speed compared to other methods (less than 4 seconds). However, in terms of scores, both models encounter challenges in effectively learning causal graphs. Notably, CASTOR (around 200 seconds for 4 regimes) runs faster than Rhino (around 2000 seconds for 4 regimes), even though CASTOR learns both temporal causal graphs and regime indexes.

A fair comparison arises when comparing RPCMCI and CASTOR, as both models learn regime indexes and temporal causal graphs. It's essential to mention that RPCMCI necessitates specifying the number of regimes and the maximum number of transitions, producing only time-lagged relations. CASTOR (200 seconds for 4 regimes) converges more rapidly than RPCMCI (more than 2300 seconds for 4 regimes). This difference in convergence time becomes more pronounced as the settings become more complex.

We sincerely appreciate the time and effort invested by the reviewer in evaluating our manuscript, and we are grateful for the insightful comments. We concur with the reviewer's observation that the introduction of multiple regimes adds a novel dimension to existing causal discovery frameworks.

• Regarding the identified weakness in the paper's presentation, we acknowledge the importance of clarity and have taken steps to address this concern. New figures have been added to the paper, available in the updated appendix, aiming to enhance the reader's understanding. Additionally, we are refining the overall presentation and rectifying any typographical errors in the appendix.

• In response to the specific query about $\gamma_{t,u}$, we want to clarify that we learn it, as detailed in Equation 11( E-step). Furthermore, in each iteration, the regime with maximum probability is assigned a value of 1, while others are set to 0.

• In response to your observation concerning lag and regimes, our work operates under the assumption that the duration of the regimes is greater than the number of lags.

• $p^{(\ell)}$ is the optimal distribution of the regime $\ell$.

To address the comparison with the Rhino framework, we conducted rigorous evaluations. This involved training Rhino on 50 time series, each with a length of 200, for every regime. The regime indexes were provided to Rhino, and the entire process was carried out following the guidelines outlined in the authors' GitHub repository. We adhered to their README description to ensure accuracy and consistency in our approach.