Title: Data depth for covariance and spectral density matrices
Authors: Joris Chau - Université catholique de Louvain (Belgium) [presenting]
Rainer von Sachs - Université catholique de Louvain (Belgium)
Hernando Ombao - King Abdullah University of Science and Technology (KAUST) (Saudi Arabia)
Abstract: In multivariate time series analysis, objects of primary interest to study cross-dependences in the time series are the autocovariance matrices in the time domain or spectral density matrices in the frequency domain. Non-degenerate covariance and spectral density matrices are necessarily Hermitian and positive definite. We introduce the concept of a statistical data depth for data observations in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to collections of observed covariance or spectral density matrices. This allows one to characterize central points or regions of the data, detect outlying observations, 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 introduced. Second, we propose two computationally efficient pointwise and integrated data depth functions that satisfy each of these requirements. Several applications of the new data depth concepts are illustrated by the analysis of multivariate brain signal time series datasets.