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The (discrete time) vector Multivariate Autoregressive (MAR) model is generalized as a stochastic process defined over a continuous spatial manifold. The underlying motivation is the study of brain connectivity via the application of Granger Causality measures to functional Neuroimages. Discretization of the spatial MAR (sMAR) leads to a densely sampled MAR for which the number of time series p is much larger than the length of the time series N. In this situation usual time series models work badly or fail. Previous approaches, reviewed here, involve the reduction of the dimensionality of the MAR, either by the selection of arbitrary regions of interest or by latent variable analysis. An example of the latter is given using a multi-linear reduction of the multichannel EEG spectrum into atoms with spatial, temporal and frequency signatures. Influence measures are applied to the temporal signatures giving an interpretation of the interaction between brain rhythms. However the approach introduced here is that of extending usual influence measures for Granger Causality to sMAR by defining "influence fields", that is the set of influence measures from one site (voxel) to the whole manifold. Estimation is made possible by imposing Bayesian priors for sparsity, smoothness, or both on the influence fields. In fact, a prior is introduced that generalizes most common priors studied to date in the literature for variable selection and penalization in regression. This prior is specified by defining penalties paired with a priori covariance matrices. Simple pairs of penalties/covariances include as particular cases the LASSO, Data Fusion and Ridge Regression. Double pairs encompass the recently introduced Elastic Net and Fussed Lasso. Quadruples of penalty/covariance combinations are also possible and used here for the first time. Estimation is carried out via the MM algorithm, a new technique that generalized the EM algorithm and allows efficient estimation even for massive time series dimensionalities. The proposed technique performs adequately for a simulated "small world" cortical network with linear dynamics, validating the use of the more complex penalties. Application of this model to fMRI data validate previous conceptual models for the brain circuits involved in the generation of the EEG alpha rhythm.
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