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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.
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Contributor : Luc Pronzato <>
Submitted on : Tuesday, August 28, 2018 - 5:32:59 PM
Last modification on : Monday, October 12, 2020 - 10:30:33 AM
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Luc Pronzato, Henry Wynn, Anatoly Zhigljavsky. Simplicial variances, potentials and Mahalanobis distances. Journal of Multivariate Analysis, Elsevier, 2018, 168, pp.276-289. ⟨10.1016/j.jmva.2018.08.002⟩. ⟨hal-01863605⟩



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