# Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions

## Document Type

Article

## Publication Date

4-1995

## 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 ^{p} into domains of attraction *D*_{1}, ..., *D _{k}* respectively, where

*D*,consists of all points

_{j}**x**∈

^{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*)′ is invariant up to sign changes, i.e., (

_{p}*Z*

_{1}, ...,

*Z*)′ has the same distribution as (±

_{p}*Z*

_{1}, ..., ±

*Z*)′. 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.

_{p}## Repository Citation

Tarpey, T.
(1995). Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions. *Journal of Multivariate Analysis, 53* (1), 39-51.

https://corescholar.libraries.wright.edu/math/197

## DOI

10.1006/jmva.1995.1023