Halfspace depth and floating body

Abstract

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

Depth and outliers for samples of sets and random sets distributions

“…The historically first and most popular one is the Tukey depth (also called the half-space depth): the depth of a point x with respect to a probability distribution in Euclidean space is the smallest probability content among all half-spaces having x on the boundary, see Tukey (1975) and Rousseeuw & Ruts (1999). Various interpretations of this depth can be found in the recent survey by Nagy, Schütt & Werner (2019). Hamel & Kostner (2018) discussed the quantile-based multivariate depth in relation to a partial order on the space.…”

Eikonal depth: an optimal control approach to statistical depths

“…Despite their central place in robust statistics, our understanding of many of their properties remains incomplete. For example, in the last five years there have been many works exploring analytical properties of halfspace depths, including connections to convex geometry [41] and differential equations [37]. Fundamental questions about whether depths characterize their distributions, and whether depth functions are smooth and well approximated by empirical approximations, are still of current interest [36,40,41].…”

“…For example, in the last five years there have been many works exploring analytical properties of halfspace depths, including connections to convex geometry [41] and differential equations [37]. Fundamental questions about whether depths characterize their distributions, and whether depth functions are smooth and well approximated by empirical approximations, are still of current interest [36,40,41]. Similar questions in the context of other depths have also been the topic of recent work [9,13].…”

Half-space depth of log-concave probability measures

“…A formal definition of the half-space depth as a way to distinguish points that fit the overall pattern of a multivariable probability distribution and to obtain an efficient description, visualization, and nonparametric statistical inference for multivariable data, was given by Donoho and Gasko in [9] (see also [20]). We refer the reader to the survey article of Nagy, Schütt and Werner [17] for an overview of this topic, with an emphasis on its connections with convex geometry, and many references. The purpose of this article is to study the expectation…”