Simplicial variances, potentials and Mahalanobis distances - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Multivariate Analysis Année : 2018

Simplicial variances, potentials and Mahalanobis distances

Résumé

The average squared volume of simplices formed by k independent copies from the same probability measure µ on R d defines an integral measure of dispersion ψ k (µ), which is a concave functional of µ after suitable normalisation. When k = 1 it corresponds to Tr(Σ µ) and when k = d we obtain the usual generalised variance det(Σ µ), with Σ µ the covariance matrix of µ. The dispersion ψ k (µ) generates a notion of simplicial potential at any x ∈ R d , dependent on µ. We show that this simplicial potential is a quadratic convex function of x, with minimum value at the mean a µ for µ, and that the potential at a µ defines a central measure of scatter similar to ψ k (µ), thereby generalising results by Wilks (1960) and van der Vaart (1965) for the generalised variance. Simplicial potentials define generalised Mahalanobis distances, expressed as weighted sums of such distances in every k-margin, and we show that the matrix involved in the generalised distance is a particular generalised inverse of Σ µ , constructed from its characteristic polynomial, when k = rank(Σ µ). 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.
Fichier principal
Vignette du fichier
LP_JMA-2017-R1.pdf (687.24 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01863605 , version 1 (28-08-2018)

Identifiants

Citer

Luc Pronzato, Henry P. Wynn, Anatoly A. Zhigljavsky. Simplicial variances, potentials and Mahalanobis distances. Journal of Multivariate Analysis, 2018, 168, pp.276-289. ⟨10.1016/j.jmva.2018.08.002⟩. ⟨hal-01863605⟩
125 Consultations
512 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More