Halfspace depths for scatter, concentration and shape matrices

Abstract

We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren (2017) and Zhang (2002). Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Zuo and Serfling (2000), the structural properties a scatter depth should satisfy, and investigate whether or not these are met by scatter halfspace depth. Companion concepts of depth for concentration matrices and shape matrices are also proposed and studied. We show the practical relevance of the depth concepts considered in a real-data example from finance.

Depth and outliers for samples of sets and random sets distributions

“…Further generalisations of the concept of depth to other data types have been elaborated by Pandolfo, Paindaveine & Porzio (2018) for directional data (and so belonging to a non‐linear space), by Chen, Gao & Ren (2018) and Paindaveine & Van Bever (2018) for matrix‐valued data, and by Lafaye De Micheaux, Mozharovskyi & Vimond (2020) for curves. Already in the context of set‐valued data discussed in this manuscript, Whitaker, Mirzargar & Kirby (2013) extended the band constructions of López‐Pintado & Romo (2009) by considering intersections and unions of sets.…”

Generative Adversarial Nets for Robust Scatter Estimation: A Proper Scoring Rule Perspective

“…An important feature of the minimax rate is its dimensionfree dependence on the contamination proportion through the second term 2 . An estimator that can achieve the minimax rate is given by the maximizer of the covariance matrix depth function [59,9,46].…”

Tukey's Depth for Object Data

“…Though defining depth notions for non-Euclidean data has garnered wide interest, the literature has focused on specialized spaces, such as a unit sphere (Small, 1987;Liu and Singh, 1992;Pandolfo et al, 2018), positive definite matrices (Fletcher et al, 2011;Chau et al, 2019), networks (Fraiman et al, 2017), data on a graph (Small, 1997), and infinitedimensional functional data (Fraiman and Muniz, 2001;López-Pintado and Romo, 2009). Chen et al (2018) and Paindaveine and Van Bever (2018) considered halfspace depth for the scatter matrix of Euclidean data points. Fraiman et al (2019) proposed a spherical depth that applies to Riemannian manifold data.…”

“…An example of a halfspace H x 1 x 2 is shown in ??. Here we evaluate the depth of an SPD matrix with respect to a sample of SPD matrices as the data units, which is different from the scenario considered for scatter depth (Chen et al, 2018;Paindaveine and Van Bever, 2018) where the depth of an SPD matrix is evaluated with respect to Euclidean data units for estimating the covariance matrix.…”