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Learning Causal Structure of Time Series using Best Order Score Search

Irene Gema Castillo Mansilla, Urmi Ninad

TL;DR

Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and its results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.

Abstract

Causal structure learning from observational data is central to many scientific and policy domains, but the time series setting common to many disciplines poses several challenges due to temporal dependence. In this paper we focus on score-based causal discovery for multivariate time series and introduce TS-BOSS, a time series extension of the recently proposed Best Order Score Search (BOSS) (Andrews et al. 2023). TS-BOSS performs a permutation-based search over dynamic Bayesian network structures while leveraging grow-shrink trees to cache intermediate score computations, preserving the scalability and strong empirical performance of BOSS in the static setting. We provide theoretical guarantees establishing the soundness of TS-BOSS under suitable assumptions, and we present an intermediate result that extends classical subgraph minimality results for permutation-based methods to the dynamic (time series) setting. Our experiments on synthetic data show that TS-BOSS is especially effective in high auto-correlation regimes, where it consistently achieves higher adjacency recall at comparable precision than standard constraint-based methods. Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and our results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.

Learning Causal Structure of Time Series using Best Order Score Search

TL;DR

Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and its results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.

Abstract

Causal structure learning from observational data is central to many scientific and policy domains, but the time series setting common to many disciplines poses several challenges due to temporal dependence. In this paper we focus on score-based causal discovery for multivariate time series and introduce TS-BOSS, a time series extension of the recently proposed Best Order Score Search (BOSS) (Andrews et al. 2023). TS-BOSS performs a permutation-based search over dynamic Bayesian network structures while leveraging grow-shrink trees to cache intermediate score computations, preserving the scalability and strong empirical performance of BOSS in the static setting. We provide theoretical guarantees establishing the soundness of TS-BOSS under suitable assumptions, and we present an intermediate result that extends classical subgraph minimality results for permutation-based methods to the dynamic (time series) setting. Our experiments on synthetic data show that TS-BOSS is especially effective in high auto-correlation regimes, where it consistently achieves higher adjacency recall at comparable precision than standard constraint-based methods. Overall, TS-BOSS offers a high-performing, scalable approach for time series causal discovery and our results provide a principled bridge for extending sparsity-based, permutation-driven causal learning theory to dynamic settings.
Paper Structure (27 sections, 3 theorems, 6 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 27 sections, 3 theorems, 6 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Time Series Causal Discovery
  4. Preliminaries and Problem Setup
  5. Causal Discovery with Single versus Multiple Multivariate Time Series
  6. TS-BOSS
  7. Overview of BOSS
  8. TS-BOSS Pseudocode and Adaptation Challenges
  9. First Phase of TS-BOSS.
  10. Second Phase of TS-BOSS
  11. Theoretical Results
  12. Simulation Study
  13. Data Generation
  14. Models
  15. Evaluation Metrics
  16. ...and 12 more sections

Key Result

theorem 1

Let $P_{\mathcal{W}}$ be a graphoid over $\mathbf{S}_{{\mathcal{W}}}$ and ${\mathcal{G}}_{\mathcal{W}}^\pi$ be a window graph induced from an admissible permutation $\pi$. Then ${\mathcal{G}}_{\mathcal{W}}^\pi$ satisfies the window Markov property and is window subgraph minimal.

Figures (4)

  • Figure 1: An overview of the grow algorithm presented in BOSS. Given a target variable $a$ and a permutation $\pi$, the grow phase builds $a$'s candidate parent set by evaluating each available variable as a possible addition to the parent set and constructing branches on those that strictly improve $a$’s score. It then sorts these branches by the resulting (post-addition) grow score and chooses the best-ranked branch permissible by the order of $\pi$, that variable is added as a parent; the procedure continues until no further improving variable from the prefix of $\pi$ can can be added to $a$'s parent set.
  • Figure 2: An overview of the shrink algorithm presented in BOSS. After the grow phase, the parent set can include redundant variables, for instance, those that became redundant once other parents were added. The shrink phase prunes this set by iteratively removing any parent whose deletion improves the score, repeating until no further score-improving removals are possible.
  • Figure 3: Experimental results for TS-BOSS, TS-BOSS (i.i.d.), and PCMCI+ under varying parameter settings.
  • Figure 4: Results comparing the true data-generating DAG with the CPDAG returned by TS-BOSS, TS-BOSS (i.i.d.) and PCMCI+ under varying parameter settings.

Theorems & Definitions (10)

  • definition 1: Window causal graph
  • definition 2: Permutation-induced window graph
  • definition 3: Window Markov property
  • definition 4: Window subgraph minimality
  • theorem 1: Permutation-induced window graph minimality
  • lemma 1
  • lemma 2
  • proof
  • proof
  • proof