Halfspace depths for scatter, concentration and shape matrices

  title={Halfspace depths for scatter, concentration and shape matrices},
  author={Davy Paindaveine and Germain Van Bever},
  journal={The Annals of Statistics},
We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept extends the one from Chen, Gao and Ren (2015) to the non-centered case, and is in the same spirit as the one in 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… 

Figures from this paper

Scatter Halfspace Depth: Geometric Insights

Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth

Exact and approximate computation of the scatter halfspace depth

An exact algorithm for the computation of sHD in any dimension d is developed and implemented using C++ for d ≤ 5, and in R for any Dimension d ≥ 1 is proposed.

Tyler shape depth

In many problems from multivariate analysis, the parameter of interest is a shape matrix: a normalized version of the corresponding scatter or dispersion matrix. In this article we propose a notion

Depth profiles and the geometric exploration of random objects through optimal transport

The properties of transport ranks are studied and it is shown that they provide anective device for detecting and visualizing patterns in samples of random objects and establish the convergence of the empirical estimates to the population targets using empirical process theory.

Scatter halfspace depth for K-symmetric distributions

  • S. Nagy
  • Mathematics
    Statistics & Probability Letters
  • 2019

Data depth and rank-based tests for covariance and spectral density matrices

In multivariate time series analysis, objects of primary interest to study cross-dependences in the time series are the autocovariance or spectral density matrices. Non-degenerate covariance and

Tukey’s Depth for Object Data

The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking.

Choosing Among Notions of Multivariate Depth Statistics

Classical multivariate statistics measures the outlyingness of a point by its Mahalanobis distance from the mean, which is based on the mean and the covariance matrix of the data. A depth function is

Illumination Depth

Abstract The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the

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



From Depth to Local Depth: A Focus on Centrality

A local extension of depth is introduced at analyzing multimodal or nonconvexly supported distributions through data depth and has the advantages of maintaining affine-invariance and applying to all depths in a generic way.

Monge-Kantorovich Depth, Quantiles, Ranks and Signs

We propose new concepts of statistical depth, multivariate quantiles,ranks and signs, based on canonical transportation maps between a distributionof interest on IRd and a reference distribution on

Projection-based depth functions and associated medians

order √ n uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity

Geodesic Convexity and Regularized Scatter Estimators

As observed by Auderset et al. (2005) and Wiesel (2012), viewing covariance matrices as elements of a Riemannian manifold and using the concept of geodesic convexity provide useful tools for studying

Location–Scale Depth

A halfspace depth in the location–scale model is introduced that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions.

Some Perspectives on Location and Scale Depth Functions

Mizera and Müller are to be congratulated heartily for their thoughtful articulation of an intriguing new approach to univariate location and scale estimation. Applying notions of statistical depth

The depth function of a population distribution

Abstract. Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an

The spatial distribution in infinite dimensional spaces and related quantiles and depths

The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution


We propose a class of rank-based procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value V 0

The random Tukey depth