In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.
This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce struc- tured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We dene this new class of models and their theoretical structure involving novel matrix-normal hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and compu- tational strategies for model tting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full se- quential and conjugate updating, ltering and forecasting analysis. The models are then applied in the context of nancial time series for predictive portfolio analysis. The improvements dened in optimal Bayesian decision analysis in this example context vividly illustrate the practical benets of the parsimony induced via appro- priate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.