Multivariate Granger causality and generalized variance

Adam B. Barrett, Lionel Barnett, and Anil K. Seth
Phys. Rev. E 81, 041907 – Published 12 April 2010

Abstract

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables but may occur among groups or “ensembles” of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke’s seminal 1982 work, we offer additional justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define “partial” Granger causality in the multivariate context and we also motivate reformulations of “causal density” and “Granger autonomy.” Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.

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  • Received 1 February 2010

DOI:https://doi.org/10.1103/PhysRevE.81.041907

©2010 American Physical Society

Authors & Affiliations

Adam B. Barrett*

  • Sackler Centre for Consciousness Science, School of Informatics, University of Sussex, Brighton BN1 9QJ, United Kingdom

Lionel Barnett

  • Centre for Computational Neuroscience and Robotics, School of Informatics, University of Sussex, Brighton BN1 9QJ, United Kingdom

Anil K. Seth

  • Sackler Centre for Consciousness Science, School of Informatics, University of Sussex, Brighton BN1 9QJ, United Kingdom

  • *adam.barrett@sussex.ac.uk
  • l.c.barnett@sussex.ac.uk
  • a.k.seth@sussex.ac.uk

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Issue

Vol. 81, Iss. 4 — April 2010

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