Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions

The k principal points ξ1, ..., ξk 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 ξj. A given set of k points y1, ..., yk partition Rp into domains of attraction D1, ..., Dk respectively, where Dj,consists of all points x ∈ Rp such that ∥x − yj∥ < ∥x − yl∥, l ≠ j. If E[X | X ∈ Dj] = yj for each j, then y1, ..., yk 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 (Z1, ..., Zp)′ is invariant up to sign changes, i.e., (Z1, ..., Zp)′ has the same distribution as (±Z1, ..., ±Zp)′. 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.