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Quantization for Infinite Affine Transformations
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DOI：10.3390/fractalfract6050239.png)
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Abstract
En 中文
Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations {S-ij} on R-2 with associated probabilities {p(ij)} such that p(ij) > 0 for all i, j is an element of N and Sigma(infinity)(i,j=1) p(ij) = 1. For such a probability measure P, the optimal sets of n-means and the nth quantization error are calculated for every natural number n. It is shown that the distribution of such a probability measure is the same as that of the direct product of the Cantor distribution. In addition, it is proved that the quantization dimension D (P) exists and is finite; whereas, the D(P)-dimensional quantization coefficient does not exist, and the D(P)-dimensional lower and the upper quantization coefficients lie in the closed interval [1/12, 5/4].
Keywords:
affine transformations
affine set
affine measure
optimal quantizers
quantization error
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