Beyond Stationarity in Time Series: Discovering Causal Structures and Latent Regimes via Markov Blankets

Lei Zan; Charles K. Assaad; Emilie Devijver; Eric Gaussier

cs.LGSep 4, 2026

Beyond Stationarity in Time Series: Discovering Causal Structures and Latent Regimes via Markov Blankets

Authors: Lei Zan, Charles K. Assaad, Emilie Devijver, Eric Gaussier

Organizations: Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, Grenoble, France · Sorbonne Université, INSERM, Institut Pierre Louis d’Epidémiologie et de Santé Publique, F75012, Paris, France

Abstract

This paper introduces Regime-aware Constraint-Based and Noise-Based causal discovery with Markov Blankets (RCBNB-MB), a novel causal discovery algorithm for time series that relaxes the common assumption of a single, time-consistent causal structure. Time series are typically observed at discrete time points and often exhibit regime changes that challenge the assumption of a static causal structure, a limitation in many real-world dynamic systems. To address this challenge, RCBNB-MB identifies latent causal regimes, defined as subsets of time points within which a stable causal structure holds. The algorithm follows an iterative strategy that segments the time series into regimes and discovers the causal graph within each regime. By leveraging the Markov blanket rather than direct parents, RCBNB-MB gains robustness to errors in causal discovery and preserves predictive information. We provide theoretical guarantees for RCBNB-MB's ability to recover both regime transitions and causal graphs under reasonable assumptions. Furthermore, we validate its effectiveness through extensive experiments on simulated datasets with known ground truth and real-world IT monitoring data, where taking into account regime shifts is critical. Empirical results show that RCBNB-MB systematically outperforms baseline approaches in accurately detecting regime changes and their associated causal graphs, positioning it as a robust and versatile framework for non-stationary time series analysis.

Figures & tables

Method	Regimes	Temporal	Contemp.	Fixed	Prediction
		Lags	Effects	Causal Order
CD-NOD [ 14 ]	Segmentation	✓	✓	×	×
J(oint)-PCMCI + [ 11 ]	Input	✓	✓	×	×
LoSST [ 16 ]	Mahalanobis distance	×	×	×	×
SPACETIME [ 18 ]	Learnt	✓	✓	✓	PA
SDCI [ 27 ]	Learnt	✓	×	×	PA

Table 1: Comparison of causal discovery methods for non-stationary time series. PA and MB denote the parent set and Markov blanket, respectively.

Panel of Figure 1 — Figure 1: Two graphic causal relation representations for time series with two distinct regimes: (a) full time causal graphs from two distinct regimes (b) window causal graphs from two distinct regimes. The variable $C_{t}$ is latent and has to be inferred.

Figure 2: Illustration of our method. Left: an example of a dataset with 6 time series, which violates the causal stationarity assumption. Middle: flowchart of our iterative method through the two steps: causal discovery and regime detection. Right: results of our method on this dataset: two regimes are considered, and we display the two causal graphs and the regime assignment.

	2 regimes					3 regimes
	MER	$Total$	$Lagged$	$Instant$	nSHD	MER	$Total$	$Lagged$	$Instant$	nSHD
RCBNB-MB	1.27 $\%$	0.73	0.77	0.67	0.52	0.81 $\%$	0.66	0.71	0.58	0.55
RCBNB-PA	1.27 $\%$	0.69	0.73	0.62	0.54	2.93 $\%$	0.65	0.69	0.57	0.57
RPCMCI-MB	0.24 $\%$	0.57	0.65	$\times$	1.03	0.39 $\%$	0.54	0.62	$\times$	1.07
RPCMCI-PA [ 32 ]	1.26 $\%$	0.57	0.66	$\times$	1.00	0.35 $\%$	0.55	0.63	$\times$	1.04
RPCMCI + -MB	1.27 $\%$	0.64	0.70	0.46	0.60	2.95 $\%$	0.55	0.62	0.39	0.56

Table 2: Results on generated data. The Mean Error Rate (MER), the mean F1-score, and the mean normalized Structural Hamming Distance (nSHD) across two scenarios: 2 regimes and 3 regimes . The F1-score is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

(only instantaneous edges considered). Reported values are means over 50 repetitions, with time series of length 600.

Figure 3: RCBNB-MB assignments assuming 3 regimes (top) and 2 regimes (bottom). Blue denotes normal regime, while rose and green denote abnormal regimes. The circle ( $\circ$ ) and cross ( $\times$ ) mark the expert identified anomaly start and end at timestamps 46,683 and 46,783, respectively.

	loss	F1-score in normal regimes
2 regimes	3.56	0.53
3 regimes	3.02	0.53

Table 3: Results on real data. The F1-score for each regime is computed for our method RCBNB-MB, with 2 and 3 regimes.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Sensitivity analysis of RCBNB-MB with respect to three hyperparameters: the number of initialization points $N_{i}$ (top row), the maximum number of optimization iterations $N_{o}$ (middle row), and the minimum size per regime $N_{\ell}$ (bottom row). The left and right columns report the mean F1-score with standard deviation and the mean MER, respectively, evaluated over 50 simulated datasets with a time series length of 600.

Table 4: The Mean Error Rate (MER), the F1-score, and the normalized Structural Hamming Distance (nSHD) across two scenarios: 2 regimes and 3 regimes . The F1-score is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

(only instantaneous edges considered). The reported values are based on 50 repetitions, with each time series having a length of 1,200.

Equal Regime Sizes
	2 regimes				3 regimes
	$Total$	$Lagged$	$Instant$	nSHD	$Total$	$Lagged$	$Instant$	nSHD
RCBNB-MB	0.01	0.01	0.02	0.04	0.01	0.01	0.03	0.01
RCBNB-PA	0.03	0.03	0.04	0.04	0.02	0.02	0.03	0.04
RPCMCI-MB	<0.01	0.01	$\times$	0.06	<0.01	<0.01	$\times$	0.02
RPCMCI-PA	0.01	0.01	$\times$	0.07	<0.01	<0.01	$\times$	0.02

Appendix

Table 5: Variance of the F1-score and the normalized Structural Hamming Distance (nSHD) across two scenarios: 2 regimes and 3 regimes . For each scenario, the F1-score is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

(only instantaneous edges considered). Additionally, only oriented edges are considered. The reported values are based on 50 repetitions. Each time series has a length of 600, with the disturbance term distributed as

\epsilon_{t}^{j}\sim\mathbf{U}(-0.1,0.1)

Table 6: Unequal Regime Sizes. The Mean Error Rate (MER), the F1-score, and the normalized Structural Hamming Distance (nSHD) across two scenarios: 2 regimes and 3 regimes . The F1-score is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

(only instantaneous edges considered). The reported values are based on 50 repetitions. The data generation process follows Equation 8 . The only difference compared to Section 5.1 is that the regime sizes are randomly determined, with the constraint that each regime occupies at least 10

\%

of the time series and remains contiguous (i.e., without interruptions).

Table 7: Unequal regime sizes. Variance of the F1-score and the normalized Structural Hamming Distance (nSHD) across two scenarios: 2 regimes and 3 regimes . The variance is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

\%

of the time series and remains contiguous (i.e., without interruptions).

Table 8: The Mean Error Rate (MER), the F1-score and the normalized Structural Hamming Distance (nSHD) in the 2 regimes scenario with 15 variables and unequal regime sizes. The F1-score is evaluated under three conditions:

Total

(all edges considered),

Lagged

(only lagged edges considered), and

Instant

(only instantaneous edges considered). The reported values are based on 50 repetitions, with each time series having a length of 2,000.

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We describe Causal-TS, an open-source Python library for causal discovery in high-dimensional and nonstationary multivariate time series. Causal-TS provides four specialized algorithms-CDNOTS, CDNOTS+, CEDAR, and GRACE-along with wrappers for GES, Granger, LASSO-VAR, and LGES, all sharing a unified conditional independence (CI) test layer with GPU acceleration via PyTorch. A regime discovery pipeline detects structural breaks via pluggable changepoint detectors and runs discovery per regime with regime-specific parameters. A command-line interface, synthetic data generators, and optional DoWhy integration provide an end-to-end pipeline from raw time series to causal effect estimates. The library is pip-installable, tested on Python 3.10--3.12, and available at https://github.com/bloomberg/causal-ts.

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