Decomposing the transfer entropy to quantify lag-specific Granger causality in cardiovascular variability

We present a modification of the well known transfer entropy (TE) which makes it able to detect, besides the direction and strength of the information transfer between coupled processes, its exact timing. The approach follows a decomposition strategy which identifies--according to a lag-specific formulation of the concept of Granger causality--the set of time delays carrying significant information, and then assigns to each of these delays an amount of information transfer such that the total contribution yields the overall TE. We propose also a procedure for the practical estimation from time series data of the relevant delays and lag-specific TE in both bivariate and multivariate settings. The proposed approach is tested in simulations and in real cardiovascular time series, showing the feasibility of lag-specific TE estimation, the ability to reflect expected mechanisms of cardiovascular regulation, and the necessity of using the multivariate TE to properly assess time-lagged information transfer in the presence of multiple interacting systems