Intrinsic Data Depth for Hermitian Positive Definite Matrices

Abstract

Nondegenerate covariance, correlation, and spectral density matrices are necessarily symmetric or Hermitian and positive definite. This article develops statistical data depths for collections of Hermitian positive definite matrices by exploiting the geometric structure of the space as a Riemannian manifold. The depth functions allow one to naturally characterize most central or outlying matrices, but also provide a practical framework for inference in the context of samples of positive definite matrices. First, the desired properties of an intrinsic data depth function acting on the space of Hermitian positive definite matrices are presented. Second, we propose two pointwise and integrated data depth functions that satisfy each of these requirements and investigate several robustness and efficiency aspects. As an application, we construct depth-based confidence regions for the intrinsic mean of a sample of positive definite matrices, which is applied to the exploratory analysis of a collection of covariance matrices in a multicenter clinical trial. Supplementary materials and an accompanying R-package are available online.

Keywords:

• Affine-invariant metric
• Confidence regions
• Covariance matrices
• Data depth
• Hermitian positive definite matrices
• Riemannian manifold

Acknowledgments

The authors gratefully acknowledge financial support from the following agencies and projects: the Belgian Fund for Scientific Research FRIA/FRS-FNRS (J. Chau), the contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique” (R. von Sachs), IAP research network P7/06 of the Belgian government (R. von Sachs), the US National Science Foundation and KAUST (H. Ombao). We thank Lieven Desmet and the SMCS/UCL for providing access to the clinical trial data, and two anonymous referees for their suggestions that helped to improve the presentation of this work. Computational resources have been provided by the CISM/UCL and the CÉCI funded by the FRS-FNRS.

Supplementary Materials

The detailed proofs of the results in this article and additional figures for the multicenter clinical trial data are available in the online supplementary materials, and the accompanying R-package pdSpecEst is publicly available on CRAN (Chau 2017).