High Precision Numerical Computation of Principal Points For Univariate Distributions

TL;DR

This paper computes the principal points and their mean squared distances for various univariate distributions, providing precise numerical results to enhance understanding of optimal quantization.

Contribution

It introduces a method to accurately determine principal points and their mean squared distances for different univariate distributions, expanding the practical application of Flury's concept.

Findings

Computed principal points for several distributions

Provided numerical values of mean squared distances

Enhanced understanding of optimal quantization in univariate cases

Abstract

Principal points were first introduced by Flury: for a positive integer $n$, $n$ principal points of a random variable are the $n$ points that minimize the mean squared distance between the random variable and the nearest of the $n$ points. In this paper, we determine the $n$ principal points and the corresponding values of mean squared distance for different values of $n$ for some univariate absolutely continuous distributions.