Distortion mismatch in the quantization of probability measures

Siegfried Graf; Harald Luschgy; Gilles Pagès

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 127-153
  • ISSN: 1292-8100

Abstract

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We elucidate the asymptotics of the Ls-quantization error induced by a sequence of Lr-optimal n-quantizers of a probability distribution P on ℝ d when s > r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s < r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on ℝ d and on the Wiener space.

How to cite

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Graf, Siegfried, Luschgy, Harald, and Pagès, Gilles. "Distortion mismatch in the quantization of probability measures." ESAIM: Probability and Statistics 12 (2008): 127-153. <http://eudml.org/doc/250412>.

@article{Graf2008,
abstract = { We elucidate the asymptotics of the Ls-quantization error induced by a sequence of Lr-optimal n-quantizers of a probability distribution P on $\mathbb\{R\}^d$ when s > r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s < r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on $\mathbb\{R\}^d$ and on the Wiener space. },
author = {Graf, Siegfried, Luschgy, Harald, Pagès, Gilles},
journal = {ESAIM: Probability and Statistics},
keywords = {Optimal quantization; Zador Theorem; Zador theorem},
language = {eng},
month = {1},
pages = {127-153},
publisher = {EDP Sciences},
title = {Distortion mismatch in the quantization of probability measures},
url = {http://eudml.org/doc/250412},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Graf, Siegfried
AU - Luschgy, Harald
AU - Pagès, Gilles
TI - Distortion mismatch in the quantization of probability measures
JO - ESAIM: Probability and Statistics
DA - 2008/1//
PB - EDP Sciences
VL - 12
SP - 127
EP - 153
AB - We elucidate the asymptotics of the Ls-quantization error induced by a sequence of Lr-optimal n-quantizers of a probability distribution P on $\mathbb{R}^d$ when s > r. In particular we show that under natural assumptions, the optimal rate is preserved as long as s < r+d (and for every s in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on $\mathbb{R}^d$ and on the Wiener space.
LA - eng
KW - Optimal quantization; Zador Theorem; Zador theorem
UR - http://eudml.org/doc/250412
ER -

References

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  11. G. Pagès, A space vector quantization method for numerical integration. J. Comput. Appl. Math.89 (1997) 1–38.  
  12. G. Pagès and J. Printems, Functional quantization for numerics with an application to option pricing. Monte Carlo Methods & Applications11 (2005) 407–446.  
  13. A. Sellami, Quantization based filtering method using first order approximation. Pré-pub. LPMA-1009 (2005). To appear in SIAM J. Numerical Analysis. 
  14. P.L. Zador, Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. thesis, Stanford University (1963).  
  15. P.L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory28, Special issue on quantization, A. Gersho & R.M. Grey Eds. (1982) 139–149.  

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