Single Parameter Entropy of Uncertain Variables
ABSTRACT
Uncertainty theory is a new branch of axiomatic mathematics for studying the subjective uncertainty. In uncertain theory, uncertain variable is a fundamental concept, which is used to represent imprecise quantities (unknown constants and unsharp concepts). Entropy of uncertain variable is an important concept in calculating uncertainty associated with imprecise quantities. This paper introduces the single parameter entropy of uncertain variable, and proves its several important theorems. In the framework of the single parameter entropy of uncertain variable, we can obtain the supremum of uncertainty of uncertain variable by choosing a proper q. The single parameter entropy of uncertain variable makes the computing of uncertainty of uncertain variable more general and flexible.
Uncertainty theory is a new branch of axiomatic mathematics for studying the subjective uncertainty. In uncertain theory, uncertain variable is a fundamental concept, which is used to represent imprecise quantities (unknown constants and unsharp concepts). Entropy of uncertain variable is an important concept in calculating uncertainty associated with imprecise quantities. This paper introduces the single parameter entropy of uncertain variable, and proves its several important theorems. In the framework of the single parameter entropy of uncertain variable, we can obtain the supremum of uncertainty of uncertain variable by choosing a proper q. The single parameter entropy of uncertain variable makes the computing of uncertainty of uncertain variable more general and flexible.
Cite this paper
J. Liu, L. Lin and S. Wu, "Single Parameter Entropy of Uncertain Variables," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 888-894. doi: 10.4236/am.2012.38131.
J. Liu, L. Lin and S. Wu, "Single Parameter Entropy of Uncertain Variables," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 888-894. doi: 10.4236/am.2012.38131.
References
[1] C. Shannon, “The Mathematical Theory of Communication,” The University of Illinois Press, Urbana, 1949. [2] A. De Luca and S. Termini, “A Definition of Nonprobabilitistic Entropy in the Setting of Fuzzy Sets Theory,” Information and Control, Vol. 20, 1972, pp. 301-312. [3] B. Liu, “Some Research Problems in Uncertainty Theory,” Journal of Uncertain Systems, Vol. 3, No. 1, 2009, pp. 3-10. [4] A comprehensive list of references can currently be obtained from http://tsallis.cat.cbpf.br/biblio.htm [5] C. Tsallis, “Possible Generalization of Boltzmann-Gibbs,” Statistics, Vol. 52, No. 1-2, 1988, pp. 479-487. doi:10.1007/BF01016429 [6] C. Tsallis, “Non-Extensive Thermostatistics: Brief Review and Comments,” Physica A, Vol. 221, No. 1-3, 1995, pp. 277-290. doi:10.1016/0378-4371(95)00236-Z [7] S. Abe, “Axiom and Uniqueness Theorem for Tsallis Entropy,” Physics Letters A, Vol. 271, No. 1-2, 2000, pp. 74-79. doi:10.1016/S0375-9601(00)00337-6 [8] S. Abe and Y. Okamoto, “Nonextensive Statistical Mechanics and Its Applications, Lecture Notes in Physics,” Springer-Verlag, Heidelberg, 2001. doi:10.1007/3-540-40919-X [9] R. J. V. dos Santos, “Generalization of Shannon’s Theorem for Tsallis Entropy,” Journal of Mathematical Physics, Vol. 38, No. 8, 1997, pp. 4104-4107. doi:10.1063/1.532107 [10] B. Liu, “Uncertainty Theory,” 2nd Edition, Springer-Verlag, Berlin, 2007. [11] B. Liu, “Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,” Springer-Verlag, Berlin, 2010. doi:10.1007/978-3-642-13959-8 [12] B. Liu, “Fuzzy Process, Hybrid Process and Uncertain Process,” Journal of Uncertain Systems, Vol. 2, No. 1, 2008, pp. 3-16. http://orsc.edu.cn/process/071010.pdf [13] X. Li and B. Liu, “Hybrid Logic and Uncertain Logic,” Journal of Uncertain Systems, Vol. 3, No. 2, 2009, pp. 83-94. [14] B. Liu, “Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Control,” Journal of Uncertain Systems, Vol. 4, No. 2, 2010, pp. 83-98. [15] B. Liu, “Uncertain Risk Analysis and Uncertain Reliability Analysis,” Journal of Uncertain Systems, Vol. 4, No. 3, 2010, pp. 163-170. [16] B. Liu, “Theory and Practice of Uncertain Programming,” 2nd Edition, Springer-Verlag, Berlin, 2009. doi:10.1007/978-3-540-89484-1 [17] W. Dai and X. Chen, “Entropy of Function of Uncertain Variables,” Technical Report, 2009. http://orsc.edu.cn/online/090805.pdf [18] X. Chen and W. Dai, “Maximum Entropy Principle for Uncertain Variables,” Technical Report, 2009. http://orsc.edu.cn/online/090618.pdf [19] X. Chen, “Cross-Entropy of Uncertain Variables and Its Applications,” Technical Report, 2009. http://orsc.edu.cn/online/091021.pdf [20] W. Dai, “Maximum Entropy Principle of Quadratic Entropy of Uncertain Variables,” Technical Report, 2010. http://orsc.edu.cn/online/100314.pdf [21] Z. X. Peng and K. Iwamura, “A Sufficient and Necessary Condition of Uncertainty Distribution,” Journal of Interdisciplinary Mathematics, Vol. 13, No. 3, 2010, pp. 277-285.
[1] C. Shannon, “The Mathematical Theory of Communication,” The University of Illinois Press, Urbana, 1949. [2] A. De Luca and S. Termini, “A Definition of Nonprobabilitistic Entropy in the Setting of Fuzzy Sets Theory,” Information and Control, Vol. 20, 1972, pp. 301-312. [3] B. Liu, “Some Research Problems in Uncertainty Theory,” Journal of Uncertain Systems, Vol. 3, No. 1, 2009, pp. 3-10. [4] A comprehensive list of references can currently be obtained from http://tsallis.cat.cbpf.br/biblio.htm [5] C. Tsallis, “Possible Generalization of Boltzmann-Gibbs,” Statistics, Vol. 52, No. 1-2, 1988, pp. 479-487. doi:10.1007/BF01016429 [6] C. Tsallis, “Non-Extensive Thermostatistics: Brief Review and Comments,” Physica A, Vol. 221, No. 1-3, 1995, pp. 277-290. doi:10.1016/0378-4371(95)00236-Z [7] S. Abe, “Axiom and Uniqueness Theorem for Tsallis Entropy,” Physics Letters A, Vol. 271, No. 1-2, 2000, pp. 74-79. doi:10.1016/S0375-9601(00)00337-6 [8] S. Abe and Y. Okamoto, “Nonextensive Statistical Mechanics and Its Applications, Lecture Notes in Physics,” Springer-Verlag, Heidelberg, 2001. doi:10.1007/3-540-40919-X [9] R. J. V. dos Santos, “Generalization of Shannon’s Theorem for Tsallis Entropy,” Journal of Mathematical Physics, Vol. 38, No. 8, 1997, pp. 4104-4107. doi:10.1063/1.532107 [10] B. Liu, “Uncertainty Theory,” 2nd Edition, Springer-Verlag, Berlin, 2007. [11] B. Liu, “Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,” Springer-Verlag, Berlin, 2010. doi:10.1007/978-3-642-13959-8 [12] B. Liu, “Fuzzy Process, Hybrid Process and Uncertain Process,” Journal of Uncertain Systems, Vol. 2, No. 1, 2008, pp. 3-16. http://orsc.edu.cn/process/071010.pdf [13] X. Li and B. Liu, “Hybrid Logic and Uncertain Logic,” Journal of Uncertain Systems, Vol. 3, No. 2, 2009, pp. 83-94. [14] B. Liu, “Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Control,” Journal of Uncertain Systems, Vol. 4, No. 2, 2010, pp. 83-98. [15] B. Liu, “Uncertain Risk Analysis and Uncertain Reliability Analysis,” Journal of Uncertain Systems, Vol. 4, No. 3, 2010, pp. 163-170. [16] B. Liu, “Theory and Practice of Uncertain Programming,” 2nd Edition, Springer-Verlag, Berlin, 2009. doi:10.1007/978-3-540-89484-1 [17] W. Dai and X. Chen, “Entropy of Function of Uncertain Variables,” Technical Report, 2009. http://orsc.edu.cn/online/090805.pdf [18] X. Chen and W. Dai, “Maximum Entropy Principle for Uncertain Variables,” Technical Report, 2009. http://orsc.edu.cn/online/090618.pdf [19] X. Chen, “Cross-Entropy of Uncertain Variables and Its Applications,” Technical Report, 2009. http://orsc.edu.cn/online/091021.pdf [20] W. Dai, “Maximum Entropy Principle of Quadratic Entropy of Uncertain Variables,” Technical Report, 2010. http://orsc.edu.cn/online/100314.pdf [21] Z. X. Peng and K. Iwamura, “A Sufficient and Necessary Condition of Uncertainty Distribution,” Journal of Interdisciplinary Mathematics, Vol. 13, No. 3, 2010, pp. 277-285.