آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی
محورهای موضوعی : آمار
محمدحسین زارع‌نژاد
1
(گروه ریاضی، واحد زاهدان، دانشگاه آزاد اسلامی، زاهدان، ایران)
ابوالفضل ابراهیم‌زاده
2
(گروه ریاضی، واحد زاهدان، دانشگاه آزاد اسلامی، زاهدان، ایران)
کلید واژه: fuzzy partition, Tsallis entropy, conditional Tsallis entropy,
چکیده مقاله :
هدف این پژوهش این است که مفاهیم آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی را تعریف کرده و نتایجی در مورد این نوع آنتروپی بدست آوریم. نشان می‌دهیم آنتروپی تیسالیس افرازهای فازی دارای ویژگی‌های زیرجمعی و تقعر می‌باشد. این اندازه اطلاعات را تحت روابطه تظریف و زیرمجموعه به مد صفر مورد مطالعه قرار می‌دهیم. قوانین زنجیره‌ای را برای این اندازه اطلاعات بررسی کرده و خواصی در مورد آنتروپی افرازهای فازی مستقل اثبات می‌نماییم. نتایجی درباره‌ی رابطه‌ی بین آنتروپی تیسالیس و آنتروپی تیسالیس شرطی افرازهای فازی بدست آورده و به کمک آنتروپی تیسالیس شرطی افرازهای فازی، نشان می‌دهیم که ویژگی زیرجمعی برای آنتروپی تیسالیس افرازهای فازی در حالتی که پارامتر این آنتروپی از یک کوچکتر است، برقرار نمی‌باشد. به طور کلی، آنتروپی تیسالیس افرازهای فازی در حالتی که پارامتر آنتروپی تیسالیس از یک بزرگتر است دارای خواصی شبیه به آنتروپی شانون افرازهای فازی می‌باشد و بنابراین می‌تواند علاوه بر آنتروپی شانون، برای اندازه‌گیری مقدار اطلاعات مستخرج از یک آزمایش فازی مورد استفاده قرار گیرد.
The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for this information measure and prove some properties about the entropy of independent fuzzy partitions. Some results of the relationship between the Tsallis entropy and conditional Tsallis entropy of fuzzy partitions are obtained and, by using conditional Tsallis entropy of fuzzy partitions, we show that the subadditivity property for Tsallis entropy of fuzzy partitions is not established in the case that the parameter of this entropy is smaller than one. In general, the Tsallis entropy of fuzzy partitions has similar properties to the shannon entropy, where the parameter of this entropy is larger than one, and therefore can be used in addition to the Shannon entropy, to measure the amount of information to be extracted from a fuzzy experiment.
[1] Abe, S., Rajagopal, K., Nonadditive Conditional Entropy and Its Significance for Local Realism, Physica A, 289, 157 (2001).
[2] Arimoto, S., Information-theoretical considerations on estimation problems, Inf. and Control 19, 181 (1971).
[3] Asadian, M.H., Ebrahimzadeh A., Entropy of Dynamical Systems from the Observer's Viewpoint, with Countable sigma-algebras, Journal of Uncertain Systems, 11(3) (2017) 197-204.
[4] Butnariu, D., Additive fuzzy measures and integrals, J. Math. Anal. Appl. 93 (1983) 436-452.
[5] Bertoluzza, C., Viviana, V., Naval, G., Uncertainty measure on fuzzy partitions, Fuzzy Sets Syst. 142 (2004) 105–116.
[6] Daroczy, Z., General information functions, Information and Control, 16 (1970) 36-51.
[7] Dubois, D., Prade, H., The logical view of conditioning and its application to possibility and evidence theories, Int. J Approx. Reason. 4 (1990) 23–46.
[8] Dumitrescu, D., A note on fuzzy information theory, Stud. Univ. Babes-Bolyai, Math. 33
[9] Dumitrescu, D., Measure-preserving transformation and the entropy of a fuzzy partition, in: 13th Linz Seminar on Fuzzy Set Theory, Linz, (1991) 25–27.
[10] Dumitrescu, D., Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl. 176 (1993) 359–373.
[11] Dumitrescu, D., Entropy of a fuzzy
process, Fuzzy Sets Syst. 55 (1993) 169–177.
[12] Dumitrescu, D., Hierarchical pattern classification, Fuzzy Sets Syst. 28 (1998) 145–162.
[13] Ebrahimzadeh, A., Logical entropy of quantum dynamical systems, Open Physics, 14 (2016) 1-5.
[14] Ebrahimzadeh, A., Eslami Giski Z., and Markechova, D., logical Entropy of Dynamical Systems - A General Model, $sum$ Mathematics, 5(1) (2017), DOI:10.3390/math 5010006.
[15] Ebrahimzadeh A., Jamalzadeh J., Conditional logical entropy of fuzzy $sigma$- algebras, Journal of intelligent and fuzzy systems, 33 (2017) 1019-1026.
[16] Ellerman D., An introduction to logical entropy and its relation to Shannon Entropy. Int. J. Semantic Comput. 7 (2013) 121–145.
[17] Eslami Giskia, Z., Ebrahimzadeh, A., An introduction of logical entropy on sequential effect algebra, Indagationes Mathematicae 28 (2017) 928-937.
[18] Furuichi, S., Information theoretical properties of Tsallis entropies, Journal of Mathematical Physics, 47 (2006), DOI: 10.1063/1.2165744.
[19] Furuichi, S., On uniqueness theorem for Tsallis entropy and Tsallis relative entropy, IEEE Trans. on Information Theory 51 (2005) 3638-3645.
[20] Gini, C., Variabilita e Mutabilita, Tipograa di Paolo Cuppini, Bologna, (1912).
[21] Good, I. J., Comment (on Patil and Taillie: Diversity as a concept and its measurement). Journal of the American Statistical Association. 77 (1982) 561- 563.
[22] Havrda, J., Charvat, F., Quantification methods of classification processes: Concept of structural alpha-entropy, Kybernetika, 3, 30 (1967).
[23] Jamaati, M., Mehri, A., Text mining by Tsallis entropy, Physica A: Statistical Mechanics and its Applications, 490 (2018) 1368-1376.
[24] Khosravi Tanak, A., Mohtashami Borzadaran G.R., Ahmadi J., Maximum Tsallis entropy with generalized Gini and Gini mean difference indices constraints, Physica A: Statistical Mechanics and its Applications, 471 (2017) 554-560.
[25] Kumar, V., Some results on Tsallis entropy measure and k-record values, Physica A: Statistical Mechanics and its Applications, 462 (2016) 667-673.
[26] Markechová, D., Entropy and mutual information of expriments in the fuzzy case, Neural Network World, 23 (2013) 339-349.
[27] Markechova, D., Logical entropy of fuzzy dynamical systems, Entropy, 18 (2016) 157 171.
[28] Mesiar, R., Rybárik, J., Entropy of fuzzy partitions: a general model, Fuzzy Sets Syst. 99 (1998) 73–79.
[29] Nguyen, H.T., Walker, E.A., A First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1997.
[30] Patil G.P., Taillie, C., Diversity as a concept and its measurement, Journal of the American Statistical Association 77 (1982) 548-561.
[31] Rao, C.R., Diversity and dissimilarity coecients: A unifin`1ed approach, Theoretical Population Biology 21 (1982) 24-43.
[32] Rahimi, M., Riazi, A., On local entropy of fuzzy partitions. Fuzzy Sets Syst. 234 (2014) 97-108.
[33] R´enyi, A., On measures of entropy and information, in: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1 (1961) 547-561.
[34] Rieˇcan, B., Markechová, D., The entropy of fuzzy dynamical systems, general scheme and generators, Fuzzy Sets Syst. 96 (1998) 191–199.
[35] Shannon, C.E., A Mathematical Theory of Communication. Bell Syst. Tech. J., 27 (1948). 379-423.
[36] Sinai, Y.G., Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics; Springer: Berlin/Heidelberg, Germany, (1990).
[37] Sinai, Y.G., On the Notion of Entropy of a Dynamical System. Dokl. Russ. Acad. Sci. 124 (1959) 768–771.
[38] Tsallis, C., Possible generalization of Bolzmann-Gibbs statistics, J.Stat.Phys., 52 (1988) 479-487.
[39] Wondie, L., Kumar, S., Some Inequalities in Information Theory Using Tsallis Entropy, International Journal of Mathematics and Mathematical Sciences, (2018) DOI: 10.1155/2018/2861612.