  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 components of a
multivariate time series. To handle the intractability of the resulting
posterior, we introduce the adaptive $\Delta$-Spherical Hamiltonian Monte
Carlo. We demonstrate the validity and flexibility of our proposed framework in
a simulation study of periodic processes and an analysis of rat's local field
potential activity in a complex sequence memory task.
