Optimal quantization for a probability measure on a nonuniform stretched Sierpi\'{n}ski triangle

TL;DR

This paper studies how to optimally approximate a probability measure supported on a nonuniform stretched Sierpiński triangle using a finite set of points, analyzing the best n-means and quantization errors.

Contribution

It provides the first detailed analysis of optimal quantization for a nonuniform stretched Sierpiński triangle measure, including explicit sets of n-means and quantization errors.

Findings

Determined optimal n-means for the measure.

Calculated the nth quantization errors.

Analyzed the structure of optimal quantizers.

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpi\'{n}ski triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.