Advances in Data Science and Classification pp 171-176 | Cite as

# 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.

## Keywords

Cluster Analysis*k*-means Optimal Allocation

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