Cookies?
Library Header Image
LSE Research Online LSE Library Services

Simplicial variances, potentials and Mahalanobis distances

Pronzato, Luc, Wynn, Henry P. and Zhigljavsky, Anatoly A. (2018) Simplicial variances, potentials and Mahalanobis distances. Journal of Multivariate Analysis, 168. pp. 276-289. ISSN 0047-259X

[img]
Preview
Text - Accepted Version
Download (864kB) | Preview

Identification Number: 10.1016/j.jmva.2018.08.002

Abstract

The average squared volume of simplices formed by k independent copies from the sameprobability measure µ on Rddefines an integral measure of dispersion k(µ), which is aconcave functional of µ after suitable normalization. When k = 1 it corresponds to tr(Σµ)and when k = d we obtain the usual generalized variance det(Σµ), with Σµthe covariancematrix of µ. The dispersion k(µ) generates a notion of simplicial potential at any x ∈ Rd,dependent on µ. We show that this simplicial potential is a quadratic convex function ofx, with minimum value at the mean aµfor µ, and that the potential at aµdefines a centralmeasure of scatter similar to k(µ), thereby generalizing results by Wilks (1960) andvan der Vaart (1965) for the generalized variance. Simplicial potentials define generalizedMahalanobis distances, expressed as weighted sums of such distances in every k-margin,and we show that the matrix involved in the generalized distance is a particular generalizedinverse of Σµ, constructed from its characteristic polynomial, when k = rank(Σµ). Finally,we show how simplicial potentials can be used to define simplicial distances between twodistributions, depending on their means and covariances, with interesting features whenthe distributions are close to singularity

Item Type: Article
Official URL: https://www.journals.elsevier.com/journal-of-multi...
Additional Information: © 2018 Elsevier Inc.
Divisions: Centre for Analysis of Time Series
Subjects: Q Science > QA Mathematics
Sets: Research centres and groups > Centre for the Analysis of Time Series (CATS)
Date Deposited: 26 Sep 2018 13:11
Last Modified: 20 Jun 2020 00:32
URI: http://eprints.lse.ac.uk/id/eprint/90265

Actions (login required)

View Item View Item

Downloads

Downloads per month over past year

View more statistics