On Applications of a Generalized Hyperbolic Measure of Entropy

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Author(s)

P.K Bhatia 1,* Surender Singh 2 Vinod Kumar 3

1. Department of Mathematics, DCR University of Science and Technology, Murthal-131039 (Haryana), India

2. School of Mathematics, Shri Mata Vaishno Devi University, Sub post office, Katra-182320 (J & K) India

3. Department of Mathematics, Govt. Women College, Madlauda -132113 (Haryana) India

* Corresponding author.

DOI: https://doi.org/10.5815/ijisa.2015.07.05

Received: 21 Oct. 2014 / Revised: 21 Jan. 2015 / Accepted: 13 Mar. 2015 / Published: 8 Jun. 2015

Index Terms

Probabilistic Entropy, Fuzzy Entropy, Super Additive Entropy, Multi Attribute Decision

Abstract

After generalization of Shannon’s entropy measure by Renyi in 1961, many generalized versions of Shannon measure were proposed by different authors. Shannon measure can be obtained from these generalized measures asymptotically. A natural question arises in the parametric generalization of Shannon’s entropy measure. What is the role of the parameter(s) from application point of view? In the present communication, super additivity and fast scalability of generalized hyperbolic measure [Bhatia and Singh, 2013] of probabilistic entropy as compared to some classical measures of entropy has been shown. Application of a generalized hyperbolic measure of probabilistic entropy in certain situations has been discussed. Also, application of generalized hyperbolic measure of fuzzy entropy in multi attribute decision making have been presented where the parameter affects the preference order.

Cite This Paper

P.K Bhatia, Surender Singh, Vinod Kumar, "On Applications of a Generalized Hyperbolic Measure of Entropy", International Journal of Intelligent Systems and Applications(IJISA), vol.7, no.7, pp.36-43, 2015. DOI:10.5815/ijisa.2015.07.05

Reference

[1]C.E. Shannon, “The mathematical theory of communications,” Bell Syst. Tech. Journal, Vol 27, pp. 423–46, 1948.
[2]A. Rényi, “On measures of entropy and information,” Proc. 4th Berk. Symp. Math. Statist. and Probl., University of California Press, pp. 547-461, 1961.
[3]P.K. Bhatia and S. Singh, “On a New Csiszar’s f-Divergence Measure,” Cybernetics and Information Technologies, Vol.13, No. 2, pp. 43-57, 2013.
[4]L.A. Zadeh, “Fuzzy Sets,” Information and Control, Vol.8, pp. 338-353, 1965.
[5]A. De Luca and S. Termini, “A definition of non-probabilistic entropy in the settings of fuzzy set theory,” Information and Control, Vol. 20, pp. 301-312, 1971.
[6]P.K. Bhatia, S. Singh and V. Kumar, “On a Generalized Hyperbolic Measure Of Fuzzy Entropy,” International Journal of Mathematical Archives, Vol.4, No. 12,pp.136-142, 2013.
[7]W. LinLin and C. Yunfang, “Diversity Based on Entropy: A Novel Evaluation Criterion in Multi-objective Optimization Algorithm,” International Journal of Intelligent Systems and Applications, Vol. 4, No. 10, pp. 113-124, 2012.
[8]L. Abdullah and A. Otheman, “A New Entropy Weight for Sub-Criteria in Interval Type-2 Fuzzy TOPSIS and Its Application,” International Journal of Intelligent Systems and Applications, Vol.5, No. 2, 2013,
[9]J. Havrda and F. Charvát, “Quantification method of classification processes: Concept of structrual α-entropy,” Kybernetika, Vol.3, pp.30-35, 1967.
[10]C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics,” J. Statist. Phys., Vol.52, pp. 479–487, 1988.
[11]E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev., Vol.106, pp.620-630, 1957.
[12]P. Maji, “f-Information measures for efficient selection of discriminative genes from microarray data,” IEEE Transactions On Biomedical Engineering, Vol.56, No.4, pp.1063-1069, 2009.
[13]P.K. Sahoo, C.Wilkins and R. Yager, “Threshold selection using Renyi’s entropy,” Pattern Recognition,Vol.30: pp.71-84, 1997.
[14]P. K. Sahoo and G. Arora, “A thresholding method based on two dimensional Renyi’s entropy,” Patterm Recognition, Vol. 37, pp.1149-1161, 2004.
[15]P. K. Sahoo and G. Arora, “Image thresholding method using two dimensional Tsallis-Havrda-Charvat entropy,” Patterm Recognition Letters, Vol. 27, pp.520-528, 2006.
[16]L.K. Huang and M. J. Wang, “Image thresholding by minimizing the measure of fuzziness,” Pattern Recognition, Vol. 28, No.1, pp.41-51, 1995.
[17]Z.-W. Ye, et al., Fuzzy entropy based optimal thresholding using bat algorithm, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016 /j.asoc.2015.02.012.