Estimation of Granger Causality of State-Space Models using a Clustering with Gaussian Mixture Model

An estimation of brain dynamical models not only can provide a characteristic of brain dynamics but estimated parameters can also infer brain networks that explains relationships among brain regions. This paper provides a scheme of discovering a brain connectivity using the Granger causality concept that is characterized on state-space models. A Granger inference matrix is implicitly derived from the state-space system matrices where its nonzero entries infer a directional dependence of the corresponding brain regions. Determining significant entries of a Granger matrix cannot be done in a straightforward way due to a lack of known statistical distribution of the inference measure. For this reason, we propose a scheme of clustering between significant and insignificant entries of a Granger matrix based on the use of Gaussian mixture models (GMMs). Our idea relies on the assumption of having a sufficient number of estimated Granger inference matrices and the use of central limit theorem to claim that the sample means of estimated Granger inference matrices converge to a Gaussian. As a result, the distribution of the vectorization of the sample mean matrices is similar to a mixture of Gaussians which can be clustered by the estimated posterior probabilities. Insignificant entries of Granger inference matrix is clustered to the Gaussian component with the lowest mean and then are inferred as no Granger causality. We illustrate the idea by simulation results based on several ground-truth Granger causality patterns derived from state-space equations.