The thesis is motivated by the study of brain effective connectivity using neuroimaging data, in particular, functional magnetic resonance imaging (fMRI) data and electroencephalography (EEG) data. We focus on a largely applied methodology to study effective connectivity, the vector autoregressive (VAR) model, as it is closely related to the notion of Granger causality. Statistical challenges in inference with VAR models include the high dimension of the parameter space and the choice of the number of lags.
We address these challenges and propose a novel framework based on tensor decomposition to achieve dimension reduction. We adopt a Bayesian approach, which allows to incorporate information from experts and to give a formal quantification of uncertainty. We first develop a (static) Bayesian tensor VAR model with a careful choice of the prior distributions. However, the main objective of the thesis is to develop dynamic tensor VAR models, in order to take into account dynamic changing patterns of the brain connectivity and non-linearities. The thesis thus contributes to the established and still growing literature on dynamics in brain activities.
We propose a Bayesian time-varying tensor VAR model that employs a tensor decomposition for the VAR coefficient matrices at different lags. Dynamically varying connectivity patterns are captured by assuming a latent binary state process that selects the active components of the tensor decomposition at each time via a novel Ising prior specification in the time domain, and we use carefully designed sparsity-inducing priors that allow to ascertain model complexity through the posterior distribution. The model is studied on synthetic data and in a real fMRI study involving a book reading experiment.
We further explore a more direct specification of a time-varying tensor VAR model through dynamic shrinkage priors. While the above Ising prior specification essentially assumes transition in terms of discrete latent states, an alternative approach is to envisage smoother temporal transitions by modeling the time-varying coefficients as an autoregressive process. We pursue this approach with the additional objectives of dimension reduction and temporal dependent sparsity. Our contribution is to employ dynamic shrinkage priors, recently proposed for dynamic variable selection in a regression setting, for time-varying tensor VAR models. More specifically, we employ the dynamic spike and slab prior and the dynamic shrinkage process to define hierarchical Bayesian time-varying tensor VAR models for multiple homogeneous trials.
As an ongoing project, we aim to contribute to Bayesian statistical methodology for dynamic regression with multivariate time series by proposing a new process prior that has the generalized double Pareto (GDP) prior as the marginal distribution
