Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices

Author(s): Lan, S; Holbrook, A; Elias, GA; Fortin, NJ; Ombao, H; Shahbaba, B | Abstract: Modeling correlation (and covariance) matrices can be challenging due to the positive-definiteness constraint and potential high-dimensionality. Our approach is to decompose the covariance matrix into the correlation and variance matrices and propose a novel Bayesian framework based on modeling the correlations as products of unit vectors. By specifying a wide range of distributions on a sphere (e.g. the squared-Dirichlet distribution), the proposed approach induces flexible prior distributions for covariance matrices (that go beyond the commonly used inverse-Wishart prior). For modeling real-life spatio-temporal processes with complex dependence structures, we extend our method to dynamic cases and introduce unit-vector Gaussian process priors in order to capture the evolution of correlation among multiple time series. To handle the intractability of the resulting posterior, we introduce the adaptive $\Delta$-Spherical Hamiltonian Monte Carlo. Using an example of normal-Inverse-Wishart problem, a simulated periodic process, and an analysis of local field potential activity data (collected from the hippocampus of rats performing a complex sequence memory task), we demonstrate the validity and effectiveness of our proposed framework for (dynamic) modeling covariance and correlation matrices.