# Halfspace depths for scatter, concentration and shape matrices

@article{Paindaveine2018HalfspaceDF, title={Halfspace depths for scatter, concentration and shape matrices}, author={Davy Paindaveine and Germain Van Bever}, journal={The Annals of Statistics}, year={2018} }

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…

## 21 Citations

### Scatter Halfspace Depth: Geometric Insights

- Mathematics
- 2020

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

- Computer Science, Mathematics
- 2022

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

- MathematicsBiometrika
- 2019

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

- Computer Science
- 2022

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

- MathematicsStatistics & Probability Letters
- 2019

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

- Mathematics
- 2017

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

- Computer ScienceJournal of the American Statistical Association
- 2021

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

- MathematicsStatistical Science
- 2022

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

- MathematicsJ. Comput. Graph. Stat.
- 2021

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

- MathematicsJournal of Computational and Graphical Statistics
- 2019

Abstract Nondegenerate covariance, correlation, and spectral density matrices are necessarily symmetric or Hermitian and positive definite. This article develops statistical data depths for…

## References

SHOWING 1-10 OF 80 REFERENCES

### From Depth to Local Depth: A Focus on Centrality

- Computer Science
- 2013

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

- Mathematics
- 2014

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

- Mathematics
- 2003

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

- Mathematics
- 2016

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

- Computer Science
- 2004

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

- Mathematics

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

- Mathematics
- 1999

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

- Mathematics
- 2014

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…

### SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE I. OPTIMAL RANK-BASED TESTS FOR SPHERICITY

- Mathematics
- 2006

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…