Rényi Entropy and Rényi Divergence in the Intuitionistic Fuzzy Case

Riečan, Beloslav; Markechová, Dagmar

doi:10.2478/tmmp-2018-0023

Abstract

Our objective in this paper is to define and study the Rényi entropy and the Rényi divergence in the intuitionistic fuzzy case. We define the Rényi entropy of order of intuitionistic fuzzy experiments (which are modeled by IF-partitions) and its conditional version and we examine their properties. It is shown that the suggested concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of IF-partitions. In addition, we introduce and study the concept of Rényi divergence in the intuitionistic fuzzy case. Specifically, relationships between the Rényi divergence and Kullback-Leibler divergence and between the Rényi divergence and the Rényi entropy in the intuitionistic fuzzy case are studied. The results are illustrated with several numerical examples.

References

[1] ARIMOTO, S.: Information Measures and Capacity of Order α for Discrete Memoryless Channels. In: Topics in Information Theory; Colloquia Mathematica Societatis Vol. 16 (János Bolyai, Csiszár, I., Elias, P—Eds.). János Bolyai Mathematical Society and North- Holland, Budapest, Hungary, 1977, pp. 493–519.
Search in Google Scholar Back to article
[2] ATANASSOV, K.: Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986), 87–96.10.1016/S0165-0114(86)80034-3
Search in Google Scholar Back to article
[3] ______ More on intuitionistic fuzzy sets, Fuzzy Sets Syst. 33 (1989), 37–45.10.1016/0165-0114(89)90215-7
Search in Google Scholar Back to article
[4] ______ New operations defined over the intuitionistic fuzzy sets, Fuzzy Set. Syst. 61 (1994), 137–142.10.1016/0165-0114(94)90229-1
Search in Google Scholar Back to article
[5] ______ Intuitionistic Fuzzy Sets: Theory and Applications, Physica Verlag, New York, NY, USA, 1999.
Search in Google Scholar Back to article
[6] ATANASSOV, K.—RIEČAN, B.: On two operations over intuitionistic fuzzy sets, J. Appl. Math. Stat. Inform. 2 (2006), 145–148.
Search in Google Scholar Back to article
[7] BAN, A.:Measurable entropy of intuitionistic fuzzy dynamical system, Notes Intuit. Fuzzy Sets 6 (2000), 35–47.
Search in Google Scholar Back to article
[8] BAN, A. I.: Intuitionistic Fuzzy Measures: Theory and Applications. Nova Science Publishers: New York, NY, USA, 2006.
Search in Google Scholar Back to article
[9] BURILLO, P.—BUSTINCE, H.: Entropy on intuitionistic fuzzy sets and on intervalvalued fuzzy sets, Fuzzy Sets Syst. 78 (1996), 305–316.10.1016/0165-0114(96)84611-2
Search in Google Scholar Back to article
[10] CHEN, T.Y.—LI, C.H.: Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis, Inf. Sci. 180 (2010), 4207–4222.10.1016/j.ins.2010.07.009
Search in Google Scholar Back to article
[11] CIUNGU, L.—RIEČAN, B.: Representation theorem of probabilities on IFS-events, Inf. Sci. 180 (2010), 793–798.10.1016/j.ins.2009.11.003
Search in Google Scholar Back to article
[12] ČUNDERLÍKOVÁ, K.: The individual ergodic theorem on the IF-events with product, Soft Comput, 14 (2010), 229–234. Doi: 10.1007/s00500-008-0396-7.10.1007/s00500-008-0396-7
Open DOI Search in Google Scholar Back to article
[13] DE, S.K.—BISWAS, R.—ROY, A. R.: Some operations on intuitionistic fuzzy sets, Fuzzy Sets Syst. 114 (2000), 477–484.10.1016/S0165-0114(98)00191-2
Search in Google Scholar Back to article
[14] ĎURICA, M.: Entropy on IF-events, Notes Intuit. Fuzzy Set 13 2007, 30–40.
Search in Google Scholar Back to article
[15] FARNOOSH, R.—RAHIMI, M.—KUMAR, P.: Removing noise in a digital image using a new entropy smethod based on intuitionistic fuzzy sets. In: Proc. of the International Conference on Fuzzy Systems, Vancouver, BC, Canada, 24–29 July 2016; Institute of Electrical and Electronics Engineers Inc., New York, NY, USA, 2016, pp. 1328–1332.
Search in Google Scholar Back to article
[16] FEHR, S.—BERENS, S.: On the conditional Rényi entropy, IEEE Trans. on Information Theory 60 (2014), no. 11, 6801–6810.10.1109/TIT.2014.2357799
Open DOI Search in Google Scholar Back to article
[17] GRAY, R.M.: Entropy and Information Theory. Springer-Verlag, Berlin, Heidelberg, Germany, 2009.
Search in Google Scholar Back to article
[18] GUTIERREZ GARCIA, J.—RODABAUGH, S.E.: Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies, Fuzzy Sets Syst. 156 (2005), 445–484.10.1016/j.fss.2005.05.023
Search in Google Scholar Back to article
[19] HE, X.—WU, Y.: Global Research Trends of Intuitionistic Fuzzy Set: A Bibliometric Analysis, J. Intelligent Systems (Published Online: 2017-09-11), Doi: org/10.1515/jisys-2017-0240. https://www.degruyter.com/downloadpdf/j/jisys.ahead-of-print/jisys-2017-0240/jisys-2017-0240.pdf
Search in Google Scholar Back to article
[20] ILIČ, V. M.—DJORDJEVIČ, I.B.—STANKOVIČ, M.: On a General Definition of Conditional Rényi Entropies, he 4th International Electronic Conference on Entropy and Its Applications (ECEA 2017), 21 November–1st December 2017; Sciforum Electronic Conference Series, Vol. 4, 2017. https://sciforum.net/paper/view/conference/503010.3390/ecea-4-05030
Search in Google Scholar Back to article
[21] JIZBA, P.—ARIMITSU, T.: The world according to Rényi: thermodynamics of multifractal systems, Ann. Phys. 312 (2004), 17–59.10.1016/j.aop.2004.01.002
Search in Google Scholar Back to article
[22] KULLBACK, S.—LEIBLER, R.A.: On information and sufficiency, Ann. Math. Stat. 22 (1951), 79–86.10.1214/aoms/1177729694
Open DOI Search in Google Scholar Back to article
[23] MARKECHOVÁ, D.—RIEČAN, B.: Logical Entropy of Experiments in the Intuitionistic Fuzzy Case, Entropy 19 (2017), no. 8,10.3390/e19080429
Search in Google Scholar Back to article
[24] ______ K-L Divergence, Entropy and Mutual Information of Experiments in the Intuitionistic Fuzzy Case, J. Intelligent Fuzzy Syst. (2018) (to appear).
Search in Google Scholar Back to article
[25] RENNER, R.—WOLF, S.: Simple and tight bounds for information reconciliation and privacy amplification. In: Advances in Cryptology—ASIACRYPT 2005, Proc. of the 11th International Conference on the Theory and Application of Cryptology and Information Security, Chennai, India, December 4–8, 2005; Lecture Notes in Comput. Sci. Vol. 3788, Springer-Verlag, Berlin, Germany, 2005, pp. 199–216.
Search in Google Scholar Back to article
[26] RÉNYI, A.: On measures of entropy and information. In: Proc. of the fourth Berkeley Symposium on Mathematics, Statistics and Probability Vol. I Univ. California Press, Berkeley, Calif, 1961, pp. 547–561.
Search in Google Scholar Back to article
[27] RIEČAN, B.: On a problem of Radko Mesiar: general form of IF-probabilities, Fuzzy Sets Syst. 152 (2006), 1485–1490.10.1016/j.fss.2005.12.005
Search in Google Scholar Back to article
[28] ______ Probability theory on IF events. (Aguzzoli, Stefano et al., eds.) In: Algebraic and Proof-Theoretic Aspects of Non-Classical Logics, (Papers in Honor of Daniele Mundici on the Occasion of his 60th Birthday), Lecture Notes in Computer Science Vol. 4460, Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin. 2007, pp. 290–308.
Search in Google Scholar Back to article
[29] RIEČAN, B.—ATANASSOV, K.: Some properties of operations conjunction and disjunction from Łukasiewicz type over intuitionistic fuzzy sets (Part 1), Notes Intuit. Fuzzy Sets 20 (2014), no. 3, 1–5.
Search in Google Scholar Back to article
[30] ______ Some properties of operations conjunction and disjunction from Łukasiewicz type over intuitionistic fuzzy sets. (Part 2), Notes Intuit. Fuzzy Sets 20 (2014), no. 4, 1–6.
Search in Google Scholar Back to article
[31] SHANNON, C.E.: A Mathematical Theory of Communication, Bell Syst. Tech. J. 27 (1948), 379–423.10.1002/j.1538-7305.1948.tb01338.x
Search in Google Scholar Back to article
[32] SZMIDT, E.—KACPRZYK, J.: Entropy for intuitionistic fuzzy sets, Fuzzy Sets Syst. 118 (2001), 467–477.10.1016/S0165-0114(98)00402-3
Search in Google Scholar Back to article
[33] TEIXEIRA, A.—MATOS, A.—ANTUNES, L.: Conditional Rényi entropies, IEEE Trans. on Information Theory 58 (2012), no. 7, 4273–4277. Doi: 10.1109/TIT.2012.2192713.10.1109/TIT.2012.2192713
Open DOI Search in Google Scholar Back to article
[34] ZADEH, L. A.: Fuzzy sets, Inf. Control 8 (1965), 338–358.10.1016/S0019-9958(65)90241-X
Search in Google Scholar Back to article
[35] ZENG, W.—LI, H.: Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy Set. Syst. 157, 2006, 1477–1484.10.1016/j.fss.2005.11.020
Search in Google Scholar Back to article

Rényi Entropy and Rényi Divergence in the Intuitionistic Fuzzy Case

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