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Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves

Author

Listed:
  • Gabriela Pena

    (University of Texas Rio Grande Valley)

  • Hansapani Rodrigo

    (University of Texas Rio Grande Valley)

  • Mrinal Kanti Roychowdhury

    (University of Texas Rio Grande Valley)

  • Josef Sifuentes

    (University of Texas Rio Grande Valley)

  • Erwin Suazo

    (University of Texas Rio Grande Valley)

Abstract

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form $$n=6k$$ n = 6 k for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n.

Suggested Citation

  • Gabriela Pena & Hansapani Rodrigo & Mrinal Kanti Roychowdhury & Josef Sifuentes & Erwin Suazo, 2021. "Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 113-142, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01771-1
    DOI: 10.1007/s10957-020-01771-1
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    References listed on IDEAS

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    1. Abaya, Efren F. & Wise, Gary L., 1984. "Some remarks on the existence of optimal quantizers," Statistics & Probability Letters, Elsevier, vol. 2(6), pages 349-351, December.
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    Cited by:

    1. Hansen, Joel & Marquez, Itzamar & Roychowdhury, Mrinal K. & Torres, Eduardo, 2021. "Quantization coefficients for uniform distributions on the boundaries of regular polygons," Statistics & Probability Letters, Elsevier, vol. 173(C).

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