Prof. Dr. Henry Wynn
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 trace(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 in 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 Waart (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.
Datum: 24.11.2018, Raum: G03-106, Zeit: 17:00