Simplicial variances, potentials and Mahalanobis distances

Abstract : 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.
Type de document :
Article dans une revue
Journal of Multivariate Analysis, Elsevier, In press, 〈10.1016/j.jmva.2018.08.002〉
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Contributeur : Luc Pronzato <>
Soumis le : mardi 28 août 2018 - 17:32:59
Dernière modification le : lundi 5 novembre 2018 - 15:52:02
Document(s) archivé(s) le : jeudi 29 novembre 2018 - 16:07:56


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Luc Pronzato, Henry Wynn, Anatoly Zhigljavsky. Simplicial variances, potentials and Mahalanobis distances. Journal of Multivariate Analysis, Elsevier, In press, 〈10.1016/j.jmva.2018.08.002〉. 〈hal-01863605〉



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