Physical Sciences and Mathematics Commons^™

Quantization For A Set Of Discrete Distributions On The Set Of Natural Numbers, Juan Gomez, Haily Martinez, Mrinal Kanti Roychowdhury, Alexis Salazar, Daniel J. Vallez

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a theorem, and then give a conjecture. We verify the conjecture by a few examples. Assuming that the conjecture is true, for a set of discrete distributions on the set of natural numbers we have calculated the optimal sets of n-means and the nth quantization errors for all positive integers n. In addition, the quantization dimension is also calculated.

Optimal Quantization For Some Triadic Uniform Cantor Distributions With Exact Bounds, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let {Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r) for all x∈R, and 1≤j≤3, where 0

Let {Sj:1≤j≤3}">{Sj:1≤j≤3}{Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r)">Sj(x)=rx+j−12(1−r)Sj(x)=rx+j−12(1−r) for all x∈R">x∈Rx∈R, and 1≤j≤3">1≤j≤31≤j≤3, where 0P has support the Cantor set generated by the similarity mappings Sj">SjSj for 1≤j≤3">1≤j≤31≤j≤3. Let r0=0.1622776602">r0=0.1622776602r0=0.1622776602, and r1=0.2317626315">r1=0.2317626315r1=0.2317626315 (which are ten digit rational approximations of two real numbers). In this paper, for 00n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers n≥2">n≥2n≥2. Previously, …

Optimal Quantization For Mixed Distributions, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 approximation of a continuous probability distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of n -means, the n th quantization errors, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more …

Quantization Coefficients For Uniform Distributions On The Boundaries Of Regular Polygons, Joel Hansen, Itzamar Marquez, Mrinal Kanti Roychowdhury, Eduardo Torres

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular m-sided polygons inscribed in a circle. The result shows that the quantization coefficient for the uniform distribution on the boundary of a regular m-sided polygon inscribed in a circle is an increasing function of m, and approaches to the quantization coefficient for the uniform distribution on the circle as m tends to infinity.

Quantization For A Mixture Of Uniform Distributions Associated With Probability Vectors, Mrinal Kanti Roychowdhury, Wasiela Salinas

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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 approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors

The Quantization Of The Standard Triadic Cantor Distribution, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. For a given k ≥ 2, let {Sj : 1 ≤ j ≤ k} be a set of k contractive similarity mappings such that Sj(x) = 1 2k−1x + 2(j−1) 2k−1 for all x ∈ R, and let P = 1 k Pk j=1 P ◦ S−1 j . Then, P is a unique Borel probability measure on R such that P has support the Cantor set generated by the similarity mappings …