Intrinsic Data Depth for Hermitian Positive Definite Matrices

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.

非退化协方差矩阵、相关矩阵与谱密度矩阵必然为对称或埃尔米特（Hermitian）正定矩阵。本文借助该空间作为黎曼流形（Riemannian manifold）的几何结构，针对埃尔米特正定矩阵集合构建了统计数据深度方法。该深度函数既可自然刻画绝大多数中心或离群矩阵，亦可为正定矩阵样本场景下的统计推断提供实用框架。首先，本文阐述了作用于埃尔米特正定矩阵空间的内蕴数据深度函数所需满足的核心性质；其次，本文提出两类分别满足上述要求的逐点型与积分型数据深度函数，并对其多项稳健性与有效性特征展开研究。作为应用实例，本文针对正定矩阵样本的内蕴均值构建了基于数据深度的置信区域，并将其应用于多中心临床试验（multicenter clinical trial）中协方差矩阵集合的探索性分析。本文的补充材料与配套R包（R-package）均可在线获取。