Escaping the Curse of Dimensionality in Estimating Multivariate Transfer Entropy

TLDR

Multivariate transfer entropy is a model‑free method that can distinguish direct from indirect causality and common drivers, yet it is rarely applied beyond bivariate settings because reliable estimation in high dimensions is difficult. The authors aim to overcome this limitation by embedding transfer entropy within graphical models and deriving a decomposed transfer entropy that sums finite‑dimensional contributions. Graphical models not only decompose transfer entropy but also reveal causal coupling delays, and the authors estimate the model with a low‑dimensional iterative algorithm based on a modified PC‑algorithm. They validate the approach with a significance test and show its effectiveness on nonlinear stochastic delay‑differential equations and sea‑level pressure climate data.

Abstract

Multivariate transfer entropy (TE) is a model-free approach to detect causalities in multivariate time series. It is able to distinguish direct from indirect causality and common drivers without assuming any underlying model. But despite these advantages it has mostly been applied in a bivariate setting as it is hard to estimate reliably in high dimensions since its definition involves infinite vectors. To overcome this limitation, we propose to embed TE into the framework of graphical models and present a formula that decomposes TE into a sum of finite-dimensional contributions that we call decomposed transfer entropy. Graphical models further provide a richer picture because they also yield the causal coupling delays. To estimate the graphical model we suggest an iterative algorithm, a modified version of the PC-algorithm with a very low estimation dimension. We present an appropriate significance test and demonstrate the method's performance using examples of nonlinear stochastic delay-differential equations and observational climate data (sea level pressure).