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Causal Temporal Regime Structure Learning

Abdellah Rahmani, Pascal Frossard

TL;DR

CASTOR tackles causal structure learning in multivariate time series that exhibit multiple, sequential regimes with regime-specific graphs. It casts the problem as learning a mixture over regimes and employs an EM algorithm to jointly infer the number of regimes, their segmentation, and the per-regime temporal DAGs, with identifiability guarantees up to permutation under Gaussian noise. The approach handles both instantaneous and time-lagged effects and supports linear and nonlinear relationships via parametric SEMs and neural networks, respectively. Empirical results on synthetic and real datasets show CASTOR outperforms existing methods in regime detection and DAG learning, confirming its practical value for non-stationary dynamic systems. The work provides a principled framework for regime-aware causal discovery with theoretical identifiability and scalable learning.

Abstract

Understanding causal relationships in multivariate time series is essential for predicting and controlling dynamic systems in fields like economics, neuroscience, and climate science. However, existing causal discovery methods often assume stationarity, limiting their effectiveness when time series consist of sequential regimes, consecutive temporal segments with unknown boundaries and changing causal structures. In this work, we firstly introduce a framework to describe and model such time series. Then, we present CASTOR, a novel method that concurrently learns the Directed Acyclic Graph (DAG) for each regime while determining the number of regimes and their sequential arrangement. CASTOR optimizes the data log-likelihood using an expectation-maximization algorithm, alternating between assigning regime indices (expectation step) and inferring causal relationships in each regime (maximization step). We establish the identifiability of the regimes and DAGs within our framework. Extensive experiments show that CASTOR consistently outperforms existing causal discovery models in detecting different regimes and learning their DAGs across various settings, including linear and nonlinear causal relationships, on both synthetic and real world datasets.

Causal Temporal Regime Structure Learning

TL;DR

CASTOR tackles causal structure learning in multivariate time series that exhibit multiple, sequential regimes with regime-specific graphs. It casts the problem as learning a mixture over regimes and employs an EM algorithm to jointly infer the number of regimes, their segmentation, and the per-regime temporal DAGs, with identifiability guarantees up to permutation under Gaussian noise. The approach handles both instantaneous and time-lagged effects and supports linear and nonlinear relationships via parametric SEMs and neural networks, respectively. Empirical results on synthetic and real datasets show CASTOR outperforms existing methods in regime detection and DAG learning, confirming its practical value for non-stationary dynamic systems. The work provides a principled framework for regime-aware causal discovery with theoretical identifiability and scalable learning.

Abstract

Understanding causal relationships in multivariate time series is essential for predicting and controlling dynamic systems in fields like economics, neuroscience, and climate science. However, existing causal discovery methods often assume stationarity, limiting their effectiveness when time series consist of sequential regimes, consecutive temporal segments with unknown boundaries and changing causal structures. In this work, we firstly introduce a framework to describe and model such time series. Then, we present CASTOR, a novel method that concurrently learns the Directed Acyclic Graph (DAG) for each regime while determining the number of regimes and their sequential arrangement. CASTOR optimizes the data log-likelihood using an expectation-maximization algorithm, alternating between assigning regime indices (expectation step) and inferring causal relationships in each regime (maximization step). We establish the identifiability of the regimes and DAGs within our framework. Extensive experiments show that CASTOR consistently outperforms existing causal discovery models in detecting different regimes and learning their DAGs across various settings, including linear and nonlinear causal relationships, on both synthetic and real world datasets.
Paper Structure (42 sections, 2 theorems, 43 equations, 16 figures, 15 tables, 1 algorithm)

This paper contains 42 sections, 2 theorems, 43 equations, 16 figures, 15 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Related works.
  3. FRAMEWORK
  4. CASTOR: CAUSAL TEMPORAL REGIME STRUCTURE LEARNING
  5. Challenges of the learning problem and EM choice justification
  6. Expectation step: Regime learning
  7. Maximization step: Graph learning
  8. Linear case
  9. Non-linear case
  10. IDENTIFIABILITY RESULTS
  11. EXPERIMENTS
  12. Synthetic data
  13. Web activity dataset
  14. Biosphere–Atmosphere data
  15. CONCLUSION
  16. ...and 27 more sections

Key Result

Theorem 1

Assume SEMs with Gaussian noise, presented in Eq(6), that satisfy the causal Markov property, stationarity, minimality and sufficiency. If each regime has enough data and the penalty coefficients in Eq (1313-eq1010) are sufficiently small, it holds asymptotically that for any estimation If any of the estimated graphs $\hat{\mathcal{G}}^u$ represents an edge disagreement with all the ground truth

Figures (16)

  • Figure 1: (a) An illustration of CASTOR processing an input MTS using an EM procedure to infer two regimes determining their partitions ($\mathcal{E}^{*}_1$ and $\mathcal{E}^{*}_2$) and learning the temporal causal graphs. Dashed edges represent time-lagged links; solid arrows indicate instantaneous links. (b) CASTOR's graphical model for lag $L=1$: observed variables are depicted in grey, and the latent variables are uncolored.
  • Figure 2: (a) Initialization with $N_w = 3$ windows: $\mathcal{E}_1$ and $\mathcal{E}_3$ are pure regimes; $\mathcal{E}_2$ is impure, containing samples from ground-truth regimes $\mathcal{E}^*_1$ and $\mathcal{E}^*_2$ ($K = 2$). (b) Illustration of $\pi\left(\boldsymbol{\alpha}^u, t\right)$ after CASTOR's first iteration with equal windows of 500 samples for an MTS of 1500 samples with two ground-truth regimes: $\mathcal{E}^*_1 = [|0:799|]$ and $\mathcal{E}^*_2 = [|800:1500|]$. (c) Comparison between CASTOR, CD-NOD and KCP on regime detection for a MTS of 10 nodes and 4 regimes using accuracy metric.
  • Figure 3: F1 scores by Models for 20 nodes and 4 regimes for non linear causal relationships. Orange indicates performance on instantaneous links, and sky-blue signifies performance on time-lagged relationships.
  • Figure 4: CASTOR's results on Biosphere-Atmosphere data.(a) CASTOR identifies two regimes, months with hot weather colored in yellow and other with cold weather colored in blue. (b) The instantaneous links are for the two regimes, with the blue graph corresponding to the blue regime, cold weather, and the yellow one to the yellow regime, hot weather.
  • Figure 5: Overview of CASTOR: This illustration demonstrates that CASTOR relies on the MTS to infer the number of regimes (equal to 2 in this figure), the regime partition ($\mathcal{E}_1$ for the first regime and $\mathcal{E}_2$ for the second) and learn the temporal causal graphs ($\mathcal{G}^1$ for the first regime and $\mathcal{G}^2$ for the second). Dashed edges symbolize time-lagged links, while normal arrows represent instantaneous links.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Definition 1: Temporal Causal Graph
  • Definition 2: Causal Stationarity, runge2018causal
  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6