Two Principal Points of Symmetric Distributions

Abstract

In the present article, we discuss some conditions on the existence of asymmetric 2-principal points of univariate symmetric distributions and show some examples of 2-principal points that are asymmetric.k-Principal points of a distribution, proposed by Flury(1990), mean k points that minimize the expected squared distance from the nearest of the point with respect to the distribution. The criterion of principal points is the almost same as that of k-means clustering. The k-means clustering is a method for a data set, but the concept of principal points is a theory for theoretical distributions.Many people may guess that principal points of symmetric distributions are symmetric but Flury shows counter examples. We study conditions for symmetric or asymmetric principal points. A characteristic function p(c) for 2- principal points is defined in the paper. We investigate 2-principal points of mixed normal distributions with the characteristic function.KeywordsCluster Analysisk-meansOptimal Allocation