# Intrinsic data depth for Hermitian positive definite matrices

## TL;DR

This paper introduces new statistical data depth functions for collections of Hermitian positive definite matrices, leveraging Riemannian geometry to enable robust inference and characterization of centrality in such matrix data.

## Contribution

It develops the first intrinsic data depth functions for Hermitian positive definite matrices, with computationally efficient methods and practical applications in confidence region construction.

## Key findings

- Proposed two fast data depth functions satisfying key properties.
- Demonstrated robustness and efficiency of the depth functions.
- Applied depth-based methods to analyze covariance matrices from a clinical trial.

## Abstract

Nondegenerate covariance, correlation and spectral density matrices are necessarily symmetric or Hermitian and positive definite. The main contribution of this paper is the development of 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 computationally fast 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 associated to a multicenter research trial.