Principal Points and Self-Consistent Points of Elliptical Distributions

The k principal points of a p-variate random vector X are those points ξ₁, …, ξₖ ∈ Rᵖ which approximate the distribution of X by minimizing the expected squared distance of X from the nearest of the ξⱼ. Any set of k points y₁, …, yₖ partitions Rᵖ into "domains of attraction" D₁, …, Dₖ according to minimal distance; following Hastie and Stuetzle we call y₁, …, yₖ self-consistent if E ∈ Dⱼ = yⱼ for j = 1, …, k. Principal points are a special case of self-consistent points. In this paper we study principal points and self-consistent points of p-variate elliptical distributions. The main results are the following: (1) If k self-consistent points of X span a subspace of dimension $q < p$, then this subspace is also spanned by q principal components, that is, self-consistent points of elliptical distributions exist only in principal component subspaces. (2) The subspace spanned by k principal points of X is identical with the subspace spanned by the principal components associated with the largest roots. This proves a conjecture of Flury. We also discuss implications of our results for the computation and estimation of principal points.