Bayesian Covariance Regression and Autoregression

Many inferential tasks, such as analyzing the functional connectivity of the brain via coactivation patterns or capturing the changing correlations amongst a set of assets for portfolio optimization, rely on modeling a covariance matrix whose elements evolve as a function of time. A number of multivariate heteroscedastic time series models have been proposed within the econometrics literature, but are typically limited by lack of clear margins, computational intractability, and curse of dimensionality. In this talk, we first introduce and explore a new class of time series models for covariance matrices based on a constructive definition exploiting inverse Wishart distribution theory. The construction yields a stationary, first-order autoregressive (AR) process on the cone of positive semi-definite matrices.

We then turn our focus to more general predictor spaces and scaling to high-dimensional datasets. Here, the predictor space could represent not only time, but also space or other factors. Our proposed Bayesian nonparametric covariance regression framework harnesses a latent factor model representation. In particular, the predictor-dependent factor loadings are characterized as a sparse combination of a collection of unknown dictionary functions (e.g, Gaussian process random functions). The induced predictor-dependent covariance is then a regularized quadratic function of these dictionary elements. Our proposed framework leads to a highly-flexible, but computationally tractable formulation with simple conjugate posterior updates that can readily handle missing data. Theoretical properties are discussed and the methods are illustrated through an application to the Google Flu Trends data and the task of word classification based on single-trial MEG data.

Joint work with Mike West and David Dunson.