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.

翻译：近年来，越来越多的方法与应用研究将因果图模型框架适配并应用于时间序列数据。其中许多工作采用时间解析的因果图，这些图无限延伸至过去和未来，且其边在时间上具有重复性，从而反映了因果关系的平稳性假设。然而，因果图模型框架的大部分结果与算法并非为无限图设计。本文提出一种方法，将具有重复边的无限时间序列图投影到有限时间窗口的边际图模型上。这些有限边际图能针对无限图提供$m$-分离查询的答案，这一问题此前未得到解决。此外，我们认为这些边际图对时间序列中的因果发现与因果效应估计具有实用价值，从而有效将有限图的结果推广至无限图。投影过程依赖于在待投影图中寻找共同祖先，这一思路本身并非创新。然而，由于时间序列图中的无限路径集可能产生共同祖先，该投影过程此前尚未针对时间序列图实现算法化。我们通过结合有限有向图的路径搜索技术与线性丢番图方程解理论，破解了对这些可能无限路径集的搜索难题。通过提供执行投影的算法，本文为一系列因果推断方法与结果在时间序列上的理论严谨且方法无关的泛化迈出了重要一步。