Kernel-Granger causality and the analysis of dynamical networks

Abstract

We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel-Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: ͑i͒ the nonlinearity of the regression model can be controlled by choosing the kernel function and ͑ii͒ the problem of false causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no a priori assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of the bivariate Granger causality is a better choice than methods using L1 minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to wellknown transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

Investigation of nonlinear granger causality in the context of epilepsy

“…A second way to calculate these linear indices can be found in [8], where GC was equivalently reformulated and included a testing statistical procedure SP detailed in [8] to decide if GCI equals zero or not and to handle overfitting, i.e. to eliminate possible parasitic redundancies in the predictive variables set.…”

“…(2) respectively, aside from the redundancies correction effect. A generalization of the method was extended to a nonlinear case based on kernel methods [8]. It uses the inhomogeneous polynomial (IP) kernel of order p and the Gaussian kernel of width σ .…”

“…The GC approach is mainly based on linearregression models; however, interactions between or within systems are frequently nonlinear. For this reason, several attempts of nonlinear extensions of GC measures have been proposed including the nonlinear model identification [7], local linear predictors and Kernel Granger Causality [8], [9]. As previously mentioned, the effectiveness of GC algorithms depends on a proper selection of the model order, meaning the maximum time lag to consider so as to get an approximate Markov representation order for the observed time series data.…”

Synergy, redundancy and unnormalized Granger causality

“…A convenient nonlinear generalization of GC has been implemented in [14] by exploiting the kernel trick, which makes computation of dot products in high-dimensional feature spaces possible using simple functions (kernels) defined on pairs of input patterns. This trick allows the formulation of nonlinear variants of any algorithm that can be cast in terms of dot products, for example Support Vector Machines [15].…”

“…This trick allows the formulation of nonlinear variants of any algorithm that can be cast in terms of dot products, for example Support Vector Machines [15]. Thus, although the aim in [14] is still to perform linear GC, it does it within a space defined by the nonlinear features of the data. This projection is conveniently and implicitly performed through kernel functions [16] in addition to use a statistical procedure to avoid over-fitting.…”