Projecting infinite time series graphs to finite marginal graphs using number theory

TL;DR

This paper introduces a novel algorithm to project infinite, stationary time series causal graphs onto finite marginal graphs, enabling finite graph-based causal inference in infinite settings.

Contribution

It develops an algorithmic method to convert infinite time series graphs into finite marginal graphs, facilitating causal discovery and effect estimation.

Findings

Algorithm successfully projects infinite graphs to finite marginals

Enables application of finite graph causal inference methods to infinite graphs

Addresses the challenge of infinite path sets via Diophantine equations

Abstract

In recent years, a growing number of method and application works have adapted and applied the causal-graphical-model framework to time series data. Many of these works employ time-resolved causal graphs that extend infinitely into the past and future and whose edges are repetitive in time, thereby reflecting the assumption of stationary causal relationships. However, most results and algorithms from the causal-graphical-model framework are not designed for infinite graphs. In this work, we develop a method for projecting infinite time series graphs with repetitive edges to marginal graphical models on a finite time window. These finite marginal graphs provide the answers to $m$-separation queries with respect to the infinite graph, a task that was previously unresolved. Moreover, we argue that these marginal graphs are useful for causal discovery and causal effect estimation in time series, effectively enabling to apply results developed for finite graphs to the infinite graphs. The projection procedure relies on finding common ancestors in the to-be-projected graph and is, by itself, not new. However, the projection procedure has not yet been algorithmically implemented for time series graphs since in these infinite graphs there can be infinite sets of paths that might give rise to common ancestors. We solve the search over these possibly infinite sets of paths by an intriguing combination of path-finding techniques for finite directed graphs and solution theory for linear Diophantine equations. By providing an algorithm that carries out the projection, our paper makes an important step towards a theoretically-grounded and method-agnostic generalization of a range of causal inference methods and results to time series.