Reliability of the Granger causality inference

Douglas Zhou,Yanyang Xiao,David Cai,Yaoyu Zhang

doi:10.1088/1367-2630/16/4/043016

Abstract

How to characterize information flows in physical, biological, and social systems remains a major theoretical challenge. Granger causality (GC) analysis has been widely used to investigate information flow through causal interactions. We address one of the central questions in GC analysis, that is, the reliability of the GC evaluation and its implications for the causal structures extracted by this analysis. Our work reveals that the manner in which a continuous dynamical process is projected or coarse-grained to a discrete process has a profound impact on the reliability of the GC inference, and different sampling may potentially yield completely opposite inferences. This inference hazard is present for both linear and nonlinear processes. We emphasize that there is a hazard of reaching incorrect conclusions about network topologies, even including statistical (such as small-world or scale-free) properties of the networks, when GC analysis is blindly applied to infer the network topology. We demonstrate this using a small-world network for which a drastic loss of small-world attributes occurs in the reconstructed network using the standard GC approach. We further show how to resolve the paradox that the GC analysis seemingly becomes less reliable when more information is incorporated using finer and finer sampling. Finally, we present strategies to overcome these inference artifacts in order to obtain a reliable GC result.

Highlights

Information flow plays a central role in physical and biological systems, such as in gene regulation and cortical computation in the brain
We study the reliability of Granger causality (GC) inference for both linear and nonlinear systems and demonstrate that there are inference hazards in the GC analysis arising from the manner in which a continuous dynamical process is projected to a discrete process
Because most dynamical quantities are continuous in time and GC values are generally evaluated using discrete time series sampled from these continuous time processes, we investigate the reliability of GC evaluation by studying the GC value as a function of the sampling interval length τ

Summary

Introduction

Information flow plays a central role in physical and biological systems, such as in gene regulation and cortical computation in the brain. For both linear and nonlinear dynamics, there are surprisingly common features in the GC sampling structure: (i) when by the design of our system there is a causal flow, yet the GC value can vanish over a certain set of τ, yielding an incorrect causal inference; or (ii) when there is no causal influence by construction, the GC value can become of appreciable size for some ranges of τ, yielding again a potentially erroneous causal inference These phenomena greatly complicate the interpretation of the GC inference and potentially produce opposing causality conclusions when using different sampling τs for empirical data. We will use idealized models to study the mechanisms underlying these phenomena and discuss an approach that can eliminate these artifacts to obtain a reliable GC relation

GC sampling structures

Small τ limit for GC

Procedure for reliable GC inference

An example: small-world networks

Discussion and conclusion