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

arXiv: https://arxiv.org/abs/2310.05526 · HTML: https://mlsanity.com/paper/2310.05526/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 75 references that are in the map.
- First posted on arXiv 9 Oct 2023.

## Builds on (most cited first, 15 of 15)

- [Detecting causal associations in large nonlinear time series datasets](https://mlsanity.com/paper/1702.07007/) (2017)
- [Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets](https://mlsanity.com/paper/2003.03685/) (2020)
- [High-recall causal discovery for autocorrelated time series with latent confounders](https://mlsanity.com/paper/2007.01884/) (2020)
- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [Constraint-based Causal Discovery from Multiple Interventions over Overlapping Variable Sets](https://mlsanity.com/paper/1403.2150/) (2014)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles](https://mlsanity.com/paper/2005.00610/) (2020)
- [Causal Modeling of Dynamical Systems](https://mlsanity.com/paper/1803.08784/) (2018)
- [Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2102.10324/) (2021)
- [Identifying Causal Effects using Instrumental Time Series: Nuisance IV and Correcting for the Past](https://mlsanity.com/paper/2203.06056/) (2022)
- [Optimal model-free prediction from multivariate time series](https://mlsanity.com/paper/1506.05822/) (2015)
- [Characterization of causal ancestral graphs for time series with latent confounders](https://mlsanity.com/paper/2112.08417/) (2021)
- [A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum](https://mlsanity.com/paper/1610.04712/) (2016)
- [Causal Inference on Process Graphs, Part I: The Structural Equation Process Representation](https://mlsanity.com/paper/2305.11561/) (2023)

## Abstract (arXiv)

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.