Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions

Abstract

The k principal points ξ 1, ..., ξ k of a random vector X are the points that approximate the distribution of X by minimizing the expected squared distance of X to the nearest of the ξ j . A given set of k points y 1, ..., y k partition R p into domains of attraction D 1, ..., D k respectively, where D j ,consists of all points x ∈ R p such that ∥ x − y j ∥ < ∥ x − y l ∥, l ≠ j. If E[ X | X ∈ D j ] = y j for each j, then y 1, ..., y k are k self-consistent points of X (∥·∥ is the Euclidian norm). Principal points are a special case of self-consistent points. Principal points and sell-consistent points are cluster means of a distribution and represent a generalization of the population mean from one to several points. Principal points and self-consistent points are studied for a class of strongly symmetric multivariate distributions. A distribution is strongly symmetric if the distribution of the principal components ( Z 1, ..., Z p )′ is invariant up to sign changes, i.e., ( Z 1, ..., Z p )′ has the same distribution as (± Z 1, ..., ± Z p )′. Elliptical distributions belong to the class of strongly symmetric distributions. Several results are given for principal points and self-consistent points of strongly symmetric multivariate distributions. One result relates self-consistent points to principal component subspaces. Another result provides a sufficient condition for any set of self-consistent points lying on a line to be symmetric to the mean of the distribution.