Kernelizing Geweke's measures of granger causality

In this paper we extend Geweke's approach of Granger causality by deriving a nonlinear framework based on functional regression in reproducing kernel Hilbert spaces (RKHS). After giving the definitions of dynamical and instantaneous causality in the Granger sense, we review Geweke's measures. These measures quantify improvement in predicting a time series when the past of another one is taken into account. Geweke's measures are based on linear prediction, and we present an alternative using nonlinear prediction implemented using regularized regression in RKHS. We develop the approach and describe the cross-validation step implemented to optimize the hyperparameters (kernel and regularization parameters). We illustrate the approach on two examples. The first one shows the importance of taking into account side information and possible nonlinear effects. The second one is an illustration of the complete inference problem: surrogate data are generated to create the null hypothesis and the nonlinear measures of causal influence are presented in a test framework.