Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach

Abstract

Many Bayesian nonparametric approaches to multivariate time series rely on Whittle’s Likelihood, involving the second order structure of a stationary time series by means of its spectral density matrix. In this work, we model the spectral density matrix by means of random measures that are constructed in such a way that positive definiteness is ensured. This is in line with existing approaches for the univariate case, where the normalized spectral density is modeled similar to a probability density, e.g. with a Dirichlet process mixture of Beta densities. We present a related approach for multivariate time series, with matrix-valued mixture weights induced by a Hermitian positive definite Gamma process. The latter has not been considered in the literature, allows to include prior knowledge and possesses a series representation that will be used in MCMC methods. We establish posterior consistency and contraction rates and small sample performance of the proposed procedure is shown in a simulation study and for real data.