Kernel Methods for Nonlinear Connectivity Detection

In this paper, we show that the presence of nonlinear coupling between time series may be detected using kernel feature space [Formula: see text] representations while dispensing with the need to go back to solve the pre-image problem to gauge model adequacy. This is done by showing that the kernelized auto/cross sequences in [Formula: see text] can be computed from the model rather than from prediction residuals in the original data space [Formula: see text]. Furthermore, this allows for reducing the connectivity inference problem to that of fitting a consistent linear model in [Formula: see text] that works even in the case of nonlinear interactions in the [Formula: see text]-space which ordinary linear models may fail to capture. We further illustrate the fact that the resulting [Formula: see text]-space parameter asymptotics provide reliable means of space model diagnostics in this space, and provide straightforward Granger connectivity inference tools even for relatively short time series records as opposed to other kernel based methods available in the literature.