Directed information graphs for the Granger causality of multivariate time series

•A definition of Granger causality is proposed.•A decomposition of conditional directed information is proved.•The directed information graphs are presented to describe the Granger causality.•The method for structure learning of the graph models is investigated.

In this paper, we investigate the links between (strong) Granger causality and directed information theory for multivariate time series. Based on the decomposition of conditional directed information, we propose a definition of Granger causality including instantaneous variables in the conditional set, which can avoid the spurious causality. The directed information graphs are presented to describe the Granger causality and instantaneous coupling. The structure learning of the graph models is based on the Leonenko's k-nn estimator of the statistics and a permutation test of the significant. Finally, we demonstrate the numerical implementation of these techniques on linear and nonlinear time series.