Quantization for a probability distribution generated by an infinite iterated function system
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Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.
Abstract
Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.
참고문헌 (16)
- E. F. Abaya, "Some remarks on the existence of optimal quantizers", Stat. Probab. Lett, 2(6), pp.349~351 (1984)
- D. Cömez, "Quantization for uniform distributions of Cantor dusts on R2", Topology Proc, 56, pp.195~218 (2020)
- Qiang Du, "Centroidal Voronoi Tessellations: Applications and Algorithms", SIAM Review, 41(4), pp.637~676 (1999), Society for Industrial & Applied Mathematics (SIAM)[DOI]
- A. Gersho, "Vector Quantization and Signal Compression" (1992), Kluwer Academy publishers
- R.M. Gray, "Locally optimal block quantizer design", Information and Control, 45(2), pp.178~198 (1980), Elsevier BV[DOI]
- S. Graf, "The Quantization of the Cantor Distribution", Mathematische Nachrichten, 183(1), pp.113~133 (1997), Wiley[DOI]
- S. Graf, "Foundations of quantization for probability distributions" (2000), Springer-Verlag
- R.M. Gray, "Quantization", IEEE Transactions on Information Theory, 44(6), pp.2325~2383 (1998), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- A. Gyorgy, "On the structure of optimal entropy-constrained scalar quantizers", IEEE Transactions on Information Theory, 48(2), pp.416~427 (2002), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- John Hutchinson, "Fractals and self-similarity", Indiana University Mathematics Journal, 30(5), pp.713~747 (1981), Indiana University Mathematics Journal[DOI]
- R. Daniel Mauldin, "Dimensions and Measures in Infinite Iterated Function Systems", Proceedings of the London Mathematical Society, s3-73(1), pp.105~154 (1996), Wiley[DOI]
- M. Moran, "Hausdorff measure of infinitely generated self-similar sets", Monatshefte für Mathematik, 122(4), pp.387~399 (1996), Springer Science and Business Media LLC[DOI]
- L. Roychowdhury, "Optimal quantization for nonuniform Cantor distributions", J. Interdiscip. Math, 22, pp.1325~1348 (2019)
- Mrinal Kanti Roychowdhury, "Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets", Journal of Fractal Geometry, 4(2), pp.127~146 (2017), European Mathematical Society - EMS - Publishing House GmbH[DOI]
- Mrinal Kanti Roychowdhury, "Optimal quantization for the Cantor distribution generated by infinite similutudes", Israel Journal of Mathematics, 231(1), pp.437~466 (2019), Springer Science and Business Media LLC[DOI]
- P. Zador, "Asymptotic quantization error of continuous signals and the quantization dimension", IEEE Transactions on Information Theory, 28(2), pp.139~149 (1982), Institute of Electrical and Electronics Engineers (IEEE)[DOI]