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Simplicial variances, potentials and Mahalanobis distances

Pronzato, Luc, Wynn, Henry P. and Zhigljavsky, Anatoly A. ORCID: 2018. Simplicial variances, potentials and Mahalanobis distances. Journal of Multivariate Analysis 168 , pp. 276-289. 10.1016/j.jmva.2018.08.002

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The average squared volume of simplices formed by independent copies from the same probability measure on defines an integral measure of dispersion , which is a concave functional of after suitable normalization. When it corresponds to and when we obtain the usual generalized variance , with the covariance matrix of . The dispersion generates a notion of simplicial potential at any , dependent on . We show that this simplicial potential is a quadratic convex function of , with minimum value at the mean for , and that the potential at defines a central measure of scatter similar to , thereby generalizing results by Wilks (1960) and van der Vaart (1965) for the generalized variance. Simplicial potentials define generalized Mahalanobis distances, expressed as weighted sums of such distances in every -margin, and we show that the matrix involved in the generalized distance is a particular generalized inverse of , constructed from its characteristic polynomial, when . Finally, we show how simplicial potentials can be used to define simplicial distances between two distributions, depending on their means and covariances, with interesting features when the distributions are close to singularity.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Elsevier
ISSN: 0047-259X
Date of First Compliant Deposit: 12 October 2018
Date of Acceptance: 16 August 2018
Last Modified: 24 Oct 2022 16:25

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