Theory and Optimization of Generalized Maximum Intuitionistic Fuzzy Entropy Methods


Abstract


The studying and formulating of the generalized maximum fuzzy entropy methods for Intuitionistic Fuzzy Entropy is the focus of this manuscript. The methods were constructed by finding two generalized maximum fuzzy entropy distributions as MinMaxFE and MaxMaxFE, which gives the least and the greatest values of the entropy based on membership function values. We define the optimization problem and study the existence of the solution subject to moment constraints through Lagrange multiplier method. Real life application of data sets in medical fields and in image processing is studied to show whether the developed method can be applied successfully in fuzzy data analysis, and the performance of these distributions is measured using chi-square, RMSE, MFE criteria.

DOI Code: 10.1285/i20705948v16n2p294

Keywords: Maximum fuzzy entropy, Fuzzy set theory, Intuitionistic fuzzy entropy, Entropy optimization distributions, Lagrange multipliers

References


Akaike, H. (1983). On minimum information prior distributions. Annals of the Institute of Statistical Mathematics, 35.2.

Asadpour, A. and Saberi, A., 2010. An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods. SIAM Journal on Computing, 39.7, 2970-2989.

Atanassov, K. (1986). Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20.1, 96.

Atanassov, K. (1989). More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33.1, 37-46.

Cheeseman, P. and Stutz, J. (2005). Generalized maximum entropy. AIP Conference Proceedings. 803.1.

De, S. K., Biswas, R. and Roy, A. R. (2001) An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy sets and Systems, 117.2, 209-213.

De Luca, A. and Termini, S. (1972). A definition of a non-probabilistic entropy in the setting of fuzzy sets theory, Inform. and Control 20.1, 301- 312.

Gharan, S., Saberi, A. and Singh, M. (2011). A Randomized Rounding Approach to the Traveling Salesman Problem. 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, 550-559.

Jaynes, E. (1957). Information theory and statistical mechanics. Phys. Rev. 106.4, 620.

Jaynes, E. (1978). Where do We Stand on Maximum Entropy, in The Maximum Entropy Formalism, edited by R. D. Levine and M. Tribus, MIT Press, Cambridge, MA, USA 15–118.

Jaynes, E. (1982). On the rationale of maximum-entropy methods. Proc. IEEE 70.9, 939–952.

Jaynes, E. (2003). Probability theory: The logic of science. Cambridge University Press, Cambridge, 0-521-59271-2.

Li M, Sun H, Singh VP, Zhou Y, Ma M. Agricultural Water Resources Management Using Maximum Entropy and Entropy-Weight-Based TOPSIS Methods. Entropy. 2019; 21(4):364.

Lisman, J. and Zuylen, M. (1972). Note on the generation of most probable frequency distributions. Statistica Neerlandica, 26.1, 19-23.

Mohammad-Djafari A. and Demoment G. (1990). Estimating Priors in Maximum Entropy Image Processing. Proc. of ICASSP , 2069-2072.

Mohammad-Djafari, A. (1992). A Matlab program to calculate the maximum entropy distributions. In Maximum entropy and Bayesian methods, 221-233.

Mukherjee, D. and Hurst, D.C. (1984). Maximum Entropy Revisited. Statistica Neerlandica, 38.1, 1–12.

Radhika, C. and Parvathi, R. (2016). Defuzzification of intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets, 22.5, 19-26.

Shamilov A., Senturk, S., Yilmaz, N., (2016). Generalized Maximum Fuzzy Entropy Methods with Applications on Wind Speed Data. Journal of Mathematics and System Science Journal of Mathematics and System Science 6: 46-52.

Shamilov A., İnce, N. (2017). A New Method of Approximation for Fuzzy Membership Function with Application. Anadolu University Journal of Science and Technology B – Theoretical Sciences,5(1),1-12.

Şamilov, A., Şentürk, S., İnce, N. (2017). An Estimation Method of Membership Function for Given Fuzzy Data. ,Sigma Journal of Engineering and Natural Sciences,8(1),11-18.

Shamilov. A., İnce, N. (2016). Minimum Cross Fuzzy Entropy Problem, The Existence of Its Solution and Generalized Minimum Cross Fuzzy Entropy Problems, Journal of Mathematics and System Science 6: 315-322.

Shannon, C. E. (1948). A mathematical theory of communication. Reprinted with corrections from. The Bell System Technical Journal, 27, 379–423, 623–656.

Patrick Smadbeck and Yiannis N Kaznessis. A closure scheme for chemical master equations. Proceedings of the National Academy of Sciences of the United States of America, 110(35):14261–5, 2013.

Sutter, T., Sutter, D., Esfahani, P., M. and Lygeros. J. (2017). Generalized maximum entropy estimation. Journal of Machine Learning Research. 20, 138.

Szmidt, E, Kacprzyk, J. (2001). Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118, 467–477.

Vlachos, I. K. and Sergiadis, G. D. (2007). Intuitionistic fuzzy information—application to pattern recognition. Pattern Recognition Lett 28.2, 197–206.

Uddin, Z., Khan, M.B., Zaheer, M.H. et al. An alternate method of evaluating Lagrange multipliers of MEP. SN Appl. Sci. 1, 224 (2019).

Zellner, Arnold and Highfield, Richard A., (1988), Calculation of maximum entropy distributions and approximation of marginalposterior distributions, Journal of Econometrics, 37, issue 2, p. 195-209


Full Text: pdf
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.