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Entropy 2013, 15(1), 113-143; doi:10.3390/e15010113

The Relation between Granger Causality and Directed Information Theory: A Review

1
GIPSAlab/CNRS UMR 5216/ BP46, 38402 Saint Martin d'Hères cedex, France
2
The University of Melbourne, Department of Mathematics and Statistics, Parkville, VIC, 3010, Australia
*
Author to whom correspondence should be addressed.
Received: 14 November 2012 / Revised: 19 December 2012 / Accepted: 19 December 2012 / Published: 28 December 2012
(This article belongs to the Special Issue Transfer Entropy)
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Abstract

This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The definitions of Granger causality based on prediction are recalled, and the importance of the observation set is discussed. We present the definitions based on conditional independence. The notion of instantaneous coupling is included in the definitions. The concept of Granger causality graphs is discussed. We present directed information theory from the perspective of studies of causal influences between stochastic processes. Causal conditioning appears to be the cornerstone for the relation between information theory and Granger causality. In the bivariate case, the fundamental measure is the directed information, which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling. We show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure, a relation known for the linear Gaussian case. We study the multivariate case, showing that the useful decomposition is blurred by instantaneous coupling. The links are further developed by studying how measures based on directed information theory naturally emerge from Granger causality inference frameworks as hypothesis testing.
Keywords: granger causality; transfer entropy; information theory; causal conditioning; conditional independence granger causality; transfer entropy; information theory; causal conditioning; conditional independence
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Amblard, P.-O.; Michel, O.J.J. The Relation between Granger Causality and Directed Information Theory: A Review. Entropy 2013, 15, 113-143.

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