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Orientability of Causal Relations in Time Series using Summary Causal Graphs and Faithful Distributions

Timothée Loranchet, Charles K. Assaad

Abstract

Understanding causal relations between temporal variables is a central challenge in time series analysis, particularly when the full causal structure is unknown. Even when the full causal structure cannot be fully specified, experts often succeed in providing a high-level abstraction of the causal graph, known as a summary causal graph, which captures the main causal relations between different time series while abstracting away micro-level details. In this work, we present conditions that guarantee the orientability of micro-level edges between temporal variables given the background knowledge encoded in a summary causal graph and assuming having access to a faithful and causally sufficient distribution with respect to the true unknown graph. Our results provide theoretical guarantees for edge orientation at the micro-level, even in the presence of cycles or bidirected edges at the macro-level. These findings offer practical guidance for leveraging SCGs to inform causal discovery in complex temporal systems and highlight the value of incorporating expert knowledge to improve causal inference from observational time series data.

Orientability of Causal Relations in Time Series using Summary Causal Graphs and Faithful Distributions

Abstract

Understanding causal relations between temporal variables is a central challenge in time series analysis, particularly when the full causal structure is unknown. Even when the full causal structure cannot be fully specified, experts often succeed in providing a high-level abstraction of the causal graph, known as a summary causal graph, which captures the main causal relations between different time series while abstracting away micro-level details. In this work, we present conditions that guarantee the orientability of micro-level edges between temporal variables given the background knowledge encoded in a summary causal graph and assuming having access to a faithful and causally sufficient distribution with respect to the true unknown graph. Our results provide theoretical guarantees for edge orientation at the micro-level, even in the presence of cycles or bidirected edges at the macro-level. These findings offer practical guidance for leveraging SCGs to inform causal discovery in complex temporal systems and highlight the value of incorporating expert knowledge to improve causal inference from observational time series data.

Paper Structure

This paper contains 5 sections, 7 theorems, 3 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Implications on Causal Effects Identification
  5. Discussion

Key Result

Lemma 1

Let $\mathcal{G}^{\mathbbl{s}}$ be an SCG and $\mathbb{P}^*$ the set of distributions compatible with $\mathcal{G}^{\mathbbl{s}}$. Under Assumptions assumption:causal_sufficiency, assumption:faithfulness and assumption:stationarity, if $S_\mathbb{X} \to S_\mathbb{Y} \in \mathcal{G}^{\mathbbl{s}}$, t

Figures (4)

  • Figure 1: An SCG $\mathcal{G}^{\mathbbl{s}}$ and two FT-DAGs ${\mathcal{G}}_1,{\mathcal{G}}_2\in compatible(\mathcal{G}^{\mathbbl{s}})$: The blue edge is oriented differently in the two FT-DAGs.
  • Figure 2: Visual illustration of Lemma \ref{['lemma:bidir_singleSL']}, with the top FT-MPDAGs based on the top SCG $\mathcal{G}^{\mathbbl{s}}_1$ (no self-loop) and the bottom FT-MPDAGs based on the bottom SCG $\mathcal{G}^{\mathbbl{s}}_2$ (with a self-loop). Red illustrates the edges used to orient the edge $X_t-Y_t$, blue is the repetition of this edge by stationarity.
  • Figure 3: Visual illustration of Lemma \ref{['lemma:bidir+UC']}, with the top FT-MPDAGs based on the top SCG $\mathcal{G}^{\mathbbl{s}}_1$ (unshielded collider) and the bottom FT-MPDAGs based on the bottom SCG $\mathcal{G}^{\mathbbl{s}}_2$ (shielded collider but with $S_\mathbb Z$ not a parent of $S_\mathbb Y$). Red illustrates the edges used to orient the edge $X_t-Y_t$, blue is the repetition of this edge by stationarity.
  • Figure 4: SCG $\mathcal{G}^{\mathbbl{s}}$ with two examples of FT-MPDAG illustrating Theorem \ref{['th:theorem1']}. Orange and purple edges are $s$-orientable and thus oriented in the FT-MPDAGs, while blue edge is non-$s$-orientable and may remain unoriented as in (b).

Theorems & Definitions (20)

  • Definition 1: Causal Markov condition, Spirtes_2000
  • Definition 2: Summary Causal Graph (SCG), Assaad_2024
  • Definition 3: Self-loop
  • Definition 4: FT-MPDAG using SCG
  • Definition 5: Orientability of an edge from SCGs and faithful distributions compatible with an FT-DAG
  • Definition 6: $s$-orientability of an edge from an SCG and all compatible faithful distributions
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 10 more