Distances and Riemannian Metrics for Multivariate Spectral Densities

The paper is concerned with the practical problem of how to compare power spectral densities of multivariable time-series. For scalar time-series several notions of distance (divergences) have been proposed and studied starting from the early 1970s while multivariable ones have only recently began to receive any attention. In the paper, two classes of divergence measures inspired by classical prediction theory are introduced. These divergences naturally induce Riemannian metrics on the cone of multivariable densities. The metrics amount to the quadratic term in the divergence between “infinitesimally close to each other” power spectra. For one of the two we provide explicit formulae for the corresponding geodesics and geodesic distance. A close connection between the geometry of power spectra and the geometry of the Fisher-Rao metric is noted.