<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all positive integers <italic>n</italic>. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all <italic>n</italic> ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of <italic>n</italic>-means for <italic>n ≥</italic> 2 one needs to know an optimal set of (<italic>n −</italic> 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of <italic>n</italic>-means and the <italic>n</italic>th quantization errors for all <italic>n</italic> ∈ N.</p></abstract>

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

Optimal Quantization for Piecewise Uniform Distributions

Indiana University-Purdue University Indianapolis

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Optimal Quantization for Piecewise Uniform Distributions"}

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of...

Optimal Quantization for Piecewise Uniform Distributions

Publié en ligne: 25 janv. 2019

Pages: 23 - 55

Reçu: 27 juil. 2017

Accepté: 30 nov. 2017

DOI: https://doi.org/10.2478/udt-2018-0009

Mots clés
Optimal quantizers, quantization error, uniform distribution

© 2018 Joseph Rosenblatt et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Optimal Quantization for Piecewise Uniform Distributions

Publié en ligne: 25 janv. 2019

Pages: 23 - 55

Reçu: 27 juil. 2017

Accepté: 30 nov. 2017

DOI: https://doi.org/10.2478/udt-2018-0009

Mots clésOptimal quantizers, quantization error, uniform distribution

© 2018 Joseph Rosenblatt et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
Optimal quantizers, quantization error, uniform distribution