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Simplicial variances, potentials and Mahalanobis distances

Author

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  • Pronzato, Luc
  • Wynn, Henry P.
  • Zhigljavsky, Anatoly A.

Abstract

The average squared volume of simplices formed by k independent copies from the same probability measure μ on Rd defines an integral measure of dispersion ψk(μ), which is a concave 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 Σμ the covariance matrix of μ. The dispersion ψk(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 of x, with minimum value at the mean aμ for μ, and that the potential at aμ defines a central measure of scatter similar to ψk(μ), 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 k-margin, and we show that the matrix involved in the generalized distance is a particular generalized inverse 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 two distributions, depending on their means and covariances, with interesting features when the distributions are close to singularity.

Suggested Citation

  • Pronzato, Luc & Wynn, Henry P. & Zhigljavsky, Anatoly A., 2018. "Simplicial variances, potentials and Mahalanobis distances," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 276-289.
  • Handle: RePEc:eee:jmvana:v:168:y:2018:i:c:p:276-289
    DOI: 10.1016/j.jmva.2018.08.002
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    References listed on IDEAS

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    1. Pronzato, L., 1998. "On a property of the expected value of a determinant," Statistics & Probability Letters, Elsevier, vol. 39(2), pages 161-165, August.
    2. SenGupta, Ashis, 1987. "Tests for standardized generalized variances of multivariate normal populations of possibly different dimensions," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 209-219, December.
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    Cited by:

    1. Volodina, Victoria & Wheatcroft, Edward & Wynn, Henry, 2022. "Comparing district heating options under uncertainty using stochastic ordering," LSE Research Online Documents on Economics 114292, London School of Economics and Political Science, LSE Library.
    2. Jonathan Gillard & Emily O’Riordan & Anatoly Zhigljavsky, 2023. "Polynomial whitening for high-dimensional data," Computational Statistics, Springer, vol. 38(3), pages 1427-1461, September.
    3. Luc Pronzato & Henry P. Wynn & Anatoly Zhigljavsky, 2019. "Bregman divergences based on optimal design criteria and simplicial measures of dispersion," Statistical Papers, Springer, vol. 60(2), pages 545-564, April.

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