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Discovering Mechanistic Causality from Time Series: A Behavioral-System Approach

Yingzhu Liu, Shengyuan Huang, Zhongkui Li, Xiaoguang Yang, Wenjun Mei

TL;DR

The paper tackles the problem of identifying mechanistic causality from time-series data, addressing the gap left by Granger causality which captures statistical dependencies rather than driver–response mechanisms. It introduces the BeCaus test based on behavioral-system theory, transforming causality discovery into fictitious control problems for deterministic LTI systems with input $u(t)$ and output $y(t)$, including open systems with unobserved inputs. The authors provide sufficiency conditions for causality-discoverable systems, requiring nonzero $D$-blocks and not-full-row rank of certain blocks to discriminate among independence, full causality, partial causality, and latent-common-cause structures, under identifiable time series. An exploratory nonlinear extension via data-enabled predictive control suggests potential applicability beyond LTI, with BeCaus demonstrating more reliable causal inferences than Granger in the studied scenarios.

Abstract

Identifying ``true causality'' is a fundamental challenge in complex systems research. Widely adopted methods, like the Granger causality test, capture statistical dependencies between variables rather than genuine driver-response mechanisms. This critical gap stems from the absence of mathematical tools that reliably reconstruct underlying system dynamics from observational time-series data. In this paper, we introduce a new control-based method for causality discovery through the behavior-system theory, which represents dynamical systems via trajectory spaces and has been widely used in data-driven control. Our core contribution is the \textbf{B}ehavior-\textbf{e}nabled \textbf{Caus}ality test (the BeCaus test), which transforms causality discovery into solving fictitious control problems. By exploiting the intrinsic asymmetry between system inputs and outputs, the proposed method operationalizes our conceptualization of mechanistic causality: variable $X$ is a cause of $Y$ if $X$ (partially) drives the evolution of $Y$. We establish conditions for linear time-invariant systems to be causality-discoverable, i.e., conditions for the BeCaus test to distinguish four basic causal structures (independence, full causality, partial causality, and latent-common-cause relation). Notably, our approach accommodates open systems with unobserved inputs. Moreover, an exploratory case study indicates the new method's potential extensibility to nonlinear systems.

Discovering Mechanistic Causality from Time Series: A Behavioral-System Approach

TL;DR

The paper tackles the problem of identifying mechanistic causality from time-series data, addressing the gap left by Granger causality which captures statistical dependencies rather than driver–response mechanisms. It introduces the BeCaus test based on behavioral-system theory, transforming causality discovery into fictitious control problems for deterministic LTI systems with input and output , including open systems with unobserved inputs. The authors provide sufficiency conditions for causality-discoverable systems, requiring nonzero -blocks and not-full-row rank of certain blocks to discriminate among independence, full causality, partial causality, and latent-common-cause structures, under identifiable time series. An exploratory nonlinear extension via data-enabled predictive control suggests potential applicability beyond LTI, with BeCaus demonstrating more reliable causal inferences than Granger in the studied scenarios.

Abstract

Identifying ``true causality'' is a fundamental challenge in complex systems research. Widely adopted methods, like the Granger causality test, capture statistical dependencies between variables rather than genuine driver-response mechanisms. This critical gap stems from the absence of mathematical tools that reliably reconstruct underlying system dynamics from observational time-series data. In this paper, we introduce a new control-based method for causality discovery through the behavior-system theory, which represents dynamical systems via trajectory spaces and has been widely used in data-driven control. Our core contribution is the \textbf{B}ehavior-\textbf{e}nabled \textbf{Caus}ality test (the BeCaus test), which transforms causality discovery into solving fictitious control problems. By exploiting the intrinsic asymmetry between system inputs and outputs, the proposed method operationalizes our conceptualization of mechanistic causality: variable is a cause of if (partially) drives the evolution of . We establish conditions for linear time-invariant systems to be causality-discoverable, i.e., conditions for the BeCaus test to distinguish four basic causal structures (independence, full causality, partial causality, and latent-common-cause relation). Notably, our approach accommodates open systems with unobserved inputs. Moreover, an exploratory case study indicates the new method's potential extensibility to nonlinear systems.
Paper Structure (17 sections, 3 theorems, 23 equations, 2 figures, 1 table)

This paper contains 17 sections, 3 theorems, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Literature Review
  4. Statement of Contributions
  5. Organization
  6. Notations, Definitions, and Preliminaries
  7. Basic Notations
  8. Granger Causality
  9. Behavioral System Theory
  10. Conceptualization of Causality and Problem Statement
  11. Conceptualization of Causality
  12. Problem Formulation
  13. Causal Inference for Deterministic LTI systems
  14. Comparisons with Granger Causality and Further Discussions
  15. Comparing with Granger causality test via examples
  16. ...and 2 more sections

Key Result

Lemma 1

( nonpersistentexcitation2022): Let the $T$-length offline data ${w_{\mathrm{d}}}$ be generated by ${\mathscr{B}} \in \partial \mathscr{L}^{q,n}_{k,l}$. Then, image$\mathscr{H}_L({w_{\mathrm{d}}})$ equals ${\mathscr{B}}_L$, for $L > \ell$ if and only if rank$(\mathscr{H}_L({w_{\mathrm{d}}}) = mL + n

Figures (2)

  • Figure 1: Four possible causal relations between variables $\theta$ and $\psi$, and the associated six possible relations. (I) independence; (II) full causality; (III) partial causality; (IV) latent-common-cause relation.
  • Figure 2: Detailed test results for Examples 1 4.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2: Behavior systems
  • Lemma 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5: BeCaus Test
  • Theorem 3
  • proof
  • ...and 6 more