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Quantization for a probability distribution generated by an infinite iterated function system
Lakshmi Roychowdhury,Mrinal Kanti Roychowdhury 대한수학회 2022 대한수학회논문집 Vol.37 No.3
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 $n\geq 2$.
Uniform distributions on curves and quantization
Joseph Rosenblatt,Mrinal Kanti Roychowdhury 대한수학회 2023 대한수학회논문집 Vol.38 No.2
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise \cite{BW}, which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.