Bayesian Time-Varying Tensor Vector Autoregressive Models for Dynamic Effective Connectivity

Recent developments in functional magnetic resonance imaging (fMRI) investigate how some brain regions directly influence the activity of other regions of the brain {\it dynamically} throughout the course of an experiment, namely dynamic effective connectivity. Time-varying vector autoregressive (TV-VAR) models have been employed to draw inferencesfor this purpose, but they are very computationally intensive, since the number of parameters to be estimated increases quadratically with the number of time series. In this paper, we propose a computationally efficient Bayesian time-varying VAR approach for modeling high-dimensional time series. The proposed framework employs a tensor decomposition for the VAR coefficient matrices at different lags. Dynamically varying connectivity patterns are captured by assuming that at any given time only a subset of components in the tensor decomposition is active. Latent binary time series select the active components at each time via a convenient Ising prior specification. The proposed prior structure encourages sparsity in the tensor structure and allows to ascertain model complexity through the posterior distribution. More specifically, sparsity-inducing priors are employed to allow for global-local shrinkage of the coefficients, to determine automatically the rank of the tensor decomposition and to guide the selection of the lags of the auto-regression. We show the performances of our model formulation via simulation studies and data from a real fMRI study involving a book reading experiment.