Data depth and rank-based tests for covariance and spectral density matrices

Abstract

In multivariate time series analysis, objects of primary interest to study cross-dependences in the time series are the autocovariance or spectral density matrices. Non-degenerate covariance and spectral density matrices are necessarily Hermitian and positive definite, and our primary goal is to develop new methods to analyze samples of such matrices. The main contribution of this paper is the generalization of the concept of statistical data depth for collections of covariance or spectral density matrices by exploiting the geometric properties of the space of Hermitian positive definite matrices as a Riemannian manifold. This allows one to naturally characterize most central or outlying matrices, but also provides a practical framework for rank-based hypothesis testing in the context of samples of covariance or spectral density matrices. First, the desired properties of a data depth function acting on the space of Hermitian positive definite matrices are presented. Second, we propose two computationally efficient pointwise and integrated data depth functions that satisfy each of these requirements. Several applications of the developed methodology are illustrated by the analysis of collections of spectral matrices in multivariate brain signal time series datasets.