Projecting infinite time series graphs to finite marginal graphs using
  number theory
Andreas Gerhardus Jonas Wahl Sofia Faltenbacher Urmi Ninad Jakob Runge
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.