Causal Structure Recovery of Linear Dynamical Systems: An FFT based Approach

TL;DR

This paper introduces an FFT-based method for efficiently recovering causal structures in linear dynamical systems, significantly reducing computational complexity and enabling frequency-domain causal inference.

Contribution

The paper presents a novel FFT-based approach that reduces the complexity of causal structure recovery in VAR models and extends do-calculus to the frequency domain for LTI systems.

Findings

Reduced complexity from $O(Tn^3N^2)$ to $O(Tn^3 ext{log} N)$ for causality recovery.

Efficient causal inference in the frequency domain using FFT and Wiener projections.

Significant computational advantage over traditional algorithms like PC due to phase response properties.

Abstract

Learning causal effects from data is a fundamental and well-studied problem across science, especially when the cause-effect relationship is static in nature. However, causal effect is less explored when there are dynamical dependencies, i.e., when dependencies exist between entities across time. Identifying dynamic causal effects from time-series observations is computationally expensive when compared to the static scenario. We demonstrate that the computational complexity of recovering the causation structure for the vector auto-regressive (VAR) model is $O(Tn^3N^2)$, where $n$ is the number of nodes, $T$ is the number of samples, and $N$ is the largest time-lag in the dependency between entities. We report a method, with a reduced complexity of $O(Tn^3 \log N)$, to recover the causation structure to obtain frequency-domain (FD) representations of time-series. Since FFT accumulates all the time dependencies on every frequency, causal inference can be performed efficiently by considering the state variables as random variables at any given frequency. We additionally show that, for systems with interactions that are LTI, do-calculus machinery can be realized in the FD resulting in versions of the classical single-door (with cycles), front and backdoor criteria. We demonstrate, for a large class of problems, graph reconstruction using multivariate Wiener projections results in a significant computational advantage with $O(n)$ complexity over reconstruction algorithms such as the PC algorithm which has $O(n^q)$ complexity, where $q$ is the maximum neighborhood size. This advantage accrues due to some remarkable properties of the phase response of the frequency-dependent Wiener coefficients which is not present in any time-domain approach.