不确定测度及其应用
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摘要
随机性是现实世界中最基本的客观不确定性,而概率论是用来处理随机现象的数学工具.随着人们对不确定现象了解的日益深入,一些学者开始挑战概率测度的可列可加性,先后提出了容度、模糊测度、可能性测度等完全非可加测度.完全非可加测度在否定可加性的同时,也否定了自对偶性,从而违背了数学科学中最基本的法则矛盾律和排中律.为了解决这一问题,Liu提出一类具有规范性、单调性、自对偶性、可列次可加性的部分可加的不确定测度,并发展出了一套与概率论平行的不确定理论.
本文首先证明了不确定测度连续性的判定条件,随后研究了不确定变量在测度连续情况下的性质,包括分布函数的连续性、关键值不等式以及期望值的单调收敛定理、Fatou引理、有界收敛定理等.其次,本文提出了不确定环境下的极大熵原理,并证明了一系列矩约束情况下不确定变量的极大熵定理.另外,为了度量一个不确定变量相对于另一个不确定变量的偏离程度,我们定义了不确定变量的互熵,并且把互熵的定义扩展到不确定向量上.进一步地,定义了不确定变量的广义互熵和广义熵,并分别研究了熵与互熵、以及广义熵与广义互熵之间的关系.在文章的最后,以条件不确定测度和辨识函数为基础,本文还建立了各种基于刘氏推理规则的不确定推理模型.
本文的创新点主要有:
1.研究了连续不确定测度及其数学性质;
2.研究了不确定变量的极大熵原理,得到矩约束情况下的极大熵定理;
3.研究了多个前件、多条准则的不确定推理规则,建立了相应的不确定推理模型.
Randomness is a fundamental type of uncertainty present in the real world, andprobability theory is a branch of mathematics for studying random phenomena. Asan improvement of our understanding about uncertain phenomena, researchers havechallenged the“countable additivity”condition, for example, in capacity theory, fuzzymeasure theory, possibility theory, and so on. However, since this class of non-additivemeasures does not consider the self-duality property, all of those measures are incon-sistent with the law of contradiction and law of excluded middle, which are basic rulesin mathematics. In order to solve this important problem, Liu (2007) founded an un-certainty theory that is a branch of mathematics based on normality, monotonicity,self-duality, countable subadditivity, and product measure axioms.
This dissertation first proposes the judgement conditions of continuous uncertainmeasures, then studies some important concepts about uncertain variables when the un-certain measure is continuous; more specifically, uncertainty distribution, critical value,expected value are studied, among others. Furthermore, this dissertation discusses themaximum entropy principle for uncertain variables and proves some maximum entropytheorems with moment constraints. Also defined and studied are cross-entropy of un-certain variables, generalized cross-entropy and generalized entropy. Finally, based ontools such as conditional uncertainty and identification functions, some expressions ofLiu’s inference rule for uncertain systems are derived.
In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects: 1. continuous uncertain measure is proposed and somebasic properties for uncertain variables in uncertainty space with continuous uncertainmeasures are studied; 2. maximum entropy principle for uncertain variables is investi-gated; 3. Liu’s inference rule with multiple antecedents and with multiple if-then rulesis considered.
本文首先证明了不确定测度连续性的判定条件,随后研究了不确定变量在测度连续情况下的性质,包括分布函数的连续性、关键值不等式以及期望值的单调收敛定理、Fatou引理、有界收敛定理等.其次,本文提出了不确定环境下的极大熵原理,并证明了一系列矩约束情况下不确定变量的极大熵定理.另外,为了度量一个不确定变量相对于另一个不确定变量的偏离程度,我们定义了不确定变量的互熵,并且把互熵的定义扩展到不确定向量上.进一步地,定义了不确定变量的广义互熵和广义熵,并分别研究了熵与互熵、以及广义熵与广义互熵之间的关系.在文章的最后,以条件不确定测度和辨识函数为基础,本文还建立了各种基于刘氏推理规则的不确定推理模型.
本文的创新点主要有:
1.研究了连续不确定测度及其数学性质;
2.研究了不确定变量的极大熵原理,得到矩约束情况下的极大熵定理;
3.研究了多个前件、多条准则的不确定推理规则,建立了相应的不确定推理模型.
Randomness is a fundamental type of uncertainty present in the real world, andprobability theory is a branch of mathematics for studying random phenomena. Asan improvement of our understanding about uncertain phenomena, researchers havechallenged the“countable additivity”condition, for example, in capacity theory, fuzzymeasure theory, possibility theory, and so on. However, since this class of non-additivemeasures does not consider the self-duality property, all of those measures are incon-sistent with the law of contradiction and law of excluded middle, which are basic rulesin mathematics. In order to solve this important problem, Liu (2007) founded an un-certainty theory that is a branch of mathematics based on normality, monotonicity,self-duality, countable subadditivity, and product measure axioms.
This dissertation first proposes the judgement conditions of continuous uncertainmeasures, then studies some important concepts about uncertain variables when the un-certain measure is continuous; more specifically, uncertainty distribution, critical value,expected value are studied, among others. Furthermore, this dissertation discusses themaximum entropy principle for uncertain variables and proves some maximum entropytheorems with moment constraints. Also defined and studied are cross-entropy of un-certain variables, generalized cross-entropy and generalized entropy. Finally, based ontools such as conditional uncertainty and identification functions, some expressions ofLiu’s inference rule for uncertain systems are derived.
In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects: 1. continuous uncertain measure is proposed and somebasic properties for uncertain variables in uncertainty space with continuous uncertainmeasures are studied; 2. maximum entropy principle for uncertain variables is investi-gated; 3. Liu’s inference rule with multiple antecedents and with multiple if-then rulesis considered.
引文
[1] Feller W. An Introduction to Probability Theory and its Applications. New York: Wiley,1950.
[2] Laha R, Rohatgi V. Probability Theory. New York: Wiley, 1979.
[3] Loe`ve M. Probability Theory I. 4th Edition. New York: Springer-Verlag, 1977.
[4] Kall P. Stochastic Linear Programming. Berlin: Springer-Verlag, 1976.
[5] Kall P, Wallace S. Stochastic Programming. Chichester: John Wiley and Sons, 1994.
[6] Shannon C. A mathematic theory of communication. Bell System Techical Journal, 1948,27:373–423, 623–656.
[7] Zadeh L. Fuzzy sets. Information and Control, 1965, 8:338–353.
[8] Zadeh L. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1978,1:3–28.
[9] Kaufmann A. Introduction to the Theory of Fuzzy Subsets. New York: Academic Press,1975.
[10] Nahmias S. Fuzzy variables. Fuzzy Sets and Systems, 1978, 1:97–110.
[11] Zadeh L. The concept of a linguistic variable and its application to approximate reasoning.Information Sciences, 1975, 8:199–251.
[12] Buckley J. Fuzzy complex numbers. Fuzzy Sets and Systems, 1989, 33:333–345.
[13] Dubois D, Prade H. Operations on fuzzy numbers. International Journal of System Science,1978, 9:613–626.
[14] Dubois D, Prade H. Operations in a fuzzy-valued logic. Information and Control, 1979,43:224–240.
[15] Dubois D, Prade H. Fuzzy Sets and Systems, Theory and Applications. New York: Aca-demic press, 1980.
[16] Yager R. A foundation for a theory of possibility. Journal of Cybernetics, 1980, 10:177–204.
[17] Dubois D, Prade H. The mean value of a fuzzy number. Fuzzy Sets and Systems, 1987,24:279–300.
[18] Dubois D, Prade H. Possibility Theory: An Approach to Computerized Processing of Un-certainty. New York: Plenum, 1988.
[19] Heilpern E. The expected value of a fuzzy number. Fuzzy Sets and Systems, 1992, 47:81–86.
[20] Kaufmann A, Gupta M. Introduction to Fuzzy Arithmetic: Theory and Applications. NewYork: Van Nostrand Reinhold, 1985.
[21] Klir G, Folger T. Fuzzy Sets, Uncertainty and Information. Englewood Cliffs, NJ: PrenticeHall, 1988.
[22] Klir G, Yuan B. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Englewood Cliffs,NJ: Prentice Hall, 1995.
[23] De Cooman G. Possibility theory I-III. International Journal of General Systems, 1997,25:291–371.
[24] Liu B, Liu Y. Expected value of fuzzy variable and fuzzy expected value models. IEEETransactions on Fuzzy Systems, 2002, 10(4):445–450.
[25] Liu B. Theory and Practice of Uncertain Programming. Heidelberg: Physica-Verlag, 2002.
[26] Gao X. Some properties of covariance of fuzzy variables. Proceedings of Proceedings ofthe Third International Conference on Information and Management Sciences, Dunhuang,China, 2004. 304–307.
[27] Li X, Liu B. Moment estimation theorems for various types of uncertain variable. Technicalreport, 2008.
[28] Li P, Liu B. Entropy of credibility distributions for fuzzy variables. IEEE Transactions onFuzzy Systems, 2008, 16(1):123–129.
[29] You C, Wen M. The entropy of fuzzy vectors. Computers and Mathematics with Applica-tions, 2008, 56:1626–1633.
[30] Li X, Liu B. Maximum entropy principle for fuzzy variables. International Journal ofUncertainty, Fuzziness & Knowledge-Based Systems, 2007, 15:43–52.
[31] Gao X, You C. Maximum entropy membership functions for discrete fuzzy variables. In-formation Sciences. doi:10.1016/j.ins.2009.03.010.
[32] Liu B. Uncertainty Theory. Berlin: Springer-Verlag, 2004.
[33] Liu B. A survey of credibility theory. Fuzzy Optimization and Decision Making, 2006,5(4):387–408.
[34] Liu B. Uncertainty Theory, 2nd ed. Berlin: Springer-Verlag, 2007.
[35]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(一)运行风险的提出与发展.电力系统自动化, 2006, 30(1):17–23.
[36]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(二)理论基础.电力系统自动化, 2006, 30(2):11–21.
[37]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(三)应用与工程实践.电力系统自动化, 2006, 30(3):11–16.
[38] Yang L, Li K, Gao Z. Train timetable problem on a single-line railway with fuzzy passengerdemand. IEEE Transactions on Fuzzy Systems. In press.
[39] Yang L, Liu L. Fuzzy fixed charge solid transportation problem and algorithm. Applied SoftComputing, 2007, 7(3):879–889.
[40] Zhou J, Liu B. Modeling capacitated location-allocation problem with fuzzy demands. Com-puters & Industrial Engineering, 2007, 53:454–468.
[41] Wen M, Iwamura K. Fuzzy facility location-allocation problem under the Hurwicz criterion.European Journal of Operational Research, 2008, 184:627–635.
[42] Ke H, Liu B. Project scheduling problem with mixed uncertainty of randomness and fuzzi-ness. European Journal of Operational Research, 2007, 183:135–147.
[43] Zheng Y, Liu B. Fuzzy vehicle routing model with credibility measure and its hybrid intel-ligent algorithm. Applied Mathematics and Computation, 2006, 176:673–683.
[44] Liu L, Gao X. Fuzzy weighted equilibrium multi-job assignment problem and genetic algo-rithm. Applied Mathematical Modelling. doi:10.1016/j.apm.2009.01.014.
[45] Gao X, Kar S. Information sharing model in supply chain management in fuzzy environ-ment. Far East Journal of Experimental and Theoretical Artificial Intelligence. In press.
[46] Ji X, Zhao X, Zhou D. A fuzzy programming approach for supply chain network design.International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, 2007, 15:75–87.
[47] Huang X. Portfolio selection with fuzzy returns. Journal of Intelligent & Fuzzy Systems,2007, 18:383–390.
[48] Huang X. Mean-semivariance models for fuzzy portfolio selection. Journal of Computa-tional and Applied Mathematics, 2008, 217:1–8.
[49] Huang X. Mean-entropy models for fuzzy portfolio selection. IEEE transactions on FuzzySystems, 2008, 16(4):1096–1101.
[50] Qin Z, Li X, Ji X. Portfolio selection based on fuzzy cross-entropy. Journal of Computa-tional and Applied Mathematics, 2009, 228:139–149.
[51] Liu B. Some research problems in uncertainty theory. Journal of Uncertain Systems, 2009,3(1):3–10.
[52] Choquet G. Theory of capacities. Annales de l’Institut Fourier, 1954, 5:131–295.
[53] Dempster A. Upper and lower probabilities induced by a multivalued mapping. Annals ofMathematical Statistics, 1967, 38:325–339.
[54] Shafer G. A Mathematical Theory of Evidence. Princeton (NJ): Princeton University Press,1976.
[55] Sugeno M. Theory of Fuzzy Integrals and Its Applications[D]. Tokoy Institute of Technol-ogy, 1974.
[56] Zadeh L. A theory of approximate reasoning. Proceedings of Mathematical Frontiers of theSocial and Policy Sciences, Boulders, Cororado, 1979.
[57] Liu B. Uncertainty Theory, 3rd ed. http://orsc.edu.cn/liu/ut.pdf.
[58]刘郁涵.如何生成不确定测度.第十届中国青年信息与管理学者大会论文集,洛阳,中国, 2008. 23–26.
[59] Peng Z. Some properties of product uncertain measure. Technical report. http://orsc.edu.cn/online/081228.pdf.
[60] You C. Some convergence theorems of uncertain sequences. Mathematical and ComputerModelling, 2009, 49(3-4):482–487.
[61] Liu B. Theory and Practice of Uncertain Programming, 3rd ed. http://orsc.edu.cn/liu/up.pdf.
[62] Liu B. Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems,2008, 2(1):3–16.
[63] Chen X, Liu B. Existence and uniqueness theorem for uncertain differential equations.Technical report. http://orsc.edu.cn/online/081230.pdf.
[64] Li X, Liu B. Hybrid logic and uncertain logic. Journal of Uncertain Systems, 2009, 3(2):83–94.
[65] Chen X, Dai W. Linearity property of expected value operator for uncertain variables. Tech-nical report. http://orsc.edu.cn/online/090311.pdf.
[66] Gao X. Some properties of continuous uncertain measure. International Journal of Uncer-tainty, Fuzziness & Knowledge-Based Systems. In press.
[67] Gao X, Chen X. Maximum entropy principle for uncertain variables. Technical report, 2009.
[68] Gao X. Cross-entropy of uncertain variables and its properties. Technical report, 2008.
[69] Li X, Liu B. Cross-entropy for fuzzy variables. Technical report, 2007.
[70] Jang J. ANFIS: Adaptive-network-based fuzzy inference system. IEEE transactions onsystems, man, and cybernetics, 1993, 23:665–685.
[71]吴望名.模糊推理的原理和方法.贵阳:贵州科技出版社, 1994.
[72]王国俊.模糊推理的全蕴涵三I算法.中国科学, E辑, 1999, 29(1):43–53.
[73]张文修,梁怡,徐萍.基于包含度的不确定推理.北京:清华大学出版社, 2007.
[74]王国俊.非经典数理逻辑与近似推理,第2版.北京:科学出版社, 2008.
[75] Li X, Liu B. Foundation of credibilistic logic. Fuzzy Optimization and Decision Making,2009, 8(1):91–102.
[76] Rescher N. Many-valued logic. New York: McGraw-Hill, 1969.
[77] Zadeh L. Outline of a new approach to the analysis of complex systems and decision pro-cesses. IEEE Transactions on Systems, Man, and Cybernetics, 1973, (1):28–44.
[78] Mamdani E, Assilian S. An experiment in linguistic synthesis with a fuzzy logic controller.International Journal of Man-Machine Studies, 1975, 7(1):1–13.
[79] Gao X. On Liu’s inference rule for uncertain systems. Technical report. http://orsc.edu.cn/online/090121.pdf.
[2] Laha R, Rohatgi V. Probability Theory. New York: Wiley, 1979.
[3] Loe`ve M. Probability Theory I. 4th Edition. New York: Springer-Verlag, 1977.
[4] Kall P. Stochastic Linear Programming. Berlin: Springer-Verlag, 1976.
[5] Kall P, Wallace S. Stochastic Programming. Chichester: John Wiley and Sons, 1994.
[6] Shannon C. A mathematic theory of communication. Bell System Techical Journal, 1948,27:373–423, 623–656.
[7] Zadeh L. Fuzzy sets. Information and Control, 1965, 8:338–353.
[8] Zadeh L. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1978,1:3–28.
[9] Kaufmann A. Introduction to the Theory of Fuzzy Subsets. New York: Academic Press,1975.
[10] Nahmias S. Fuzzy variables. Fuzzy Sets and Systems, 1978, 1:97–110.
[11] Zadeh L. The concept of a linguistic variable and its application to approximate reasoning.Information Sciences, 1975, 8:199–251.
[12] Buckley J. Fuzzy complex numbers. Fuzzy Sets and Systems, 1989, 33:333–345.
[13] Dubois D, Prade H. Operations on fuzzy numbers. International Journal of System Science,1978, 9:613–626.
[14] Dubois D, Prade H. Operations in a fuzzy-valued logic. Information and Control, 1979,43:224–240.
[15] Dubois D, Prade H. Fuzzy Sets and Systems, Theory and Applications. New York: Aca-demic press, 1980.
[16] Yager R. A foundation for a theory of possibility. Journal of Cybernetics, 1980, 10:177–204.
[17] Dubois D, Prade H. The mean value of a fuzzy number. Fuzzy Sets and Systems, 1987,24:279–300.
[18] Dubois D, Prade H. Possibility Theory: An Approach to Computerized Processing of Un-certainty. New York: Plenum, 1988.
[19] Heilpern E. The expected value of a fuzzy number. Fuzzy Sets and Systems, 1992, 47:81–86.
[20] Kaufmann A, Gupta M. Introduction to Fuzzy Arithmetic: Theory and Applications. NewYork: Van Nostrand Reinhold, 1985.
[21] Klir G, Folger T. Fuzzy Sets, Uncertainty and Information. Englewood Cliffs, NJ: PrenticeHall, 1988.
[22] Klir G, Yuan B. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Englewood Cliffs,NJ: Prentice Hall, 1995.
[23] De Cooman G. Possibility theory I-III. International Journal of General Systems, 1997,25:291–371.
[24] Liu B, Liu Y. Expected value of fuzzy variable and fuzzy expected value models. IEEETransactions on Fuzzy Systems, 2002, 10(4):445–450.
[25] Liu B. Theory and Practice of Uncertain Programming. Heidelberg: Physica-Verlag, 2002.
[26] Gao X. Some properties of covariance of fuzzy variables. Proceedings of Proceedings ofthe Third International Conference on Information and Management Sciences, Dunhuang,China, 2004. 304–307.
[27] Li X, Liu B. Moment estimation theorems for various types of uncertain variable. Technicalreport, 2008.
[28] Li P, Liu B. Entropy of credibility distributions for fuzzy variables. IEEE Transactions onFuzzy Systems, 2008, 16(1):123–129.
[29] You C, Wen M. The entropy of fuzzy vectors. Computers and Mathematics with Applica-tions, 2008, 56:1626–1633.
[30] Li X, Liu B. Maximum entropy principle for fuzzy variables. International Journal ofUncertainty, Fuzziness & Knowledge-Based Systems, 2007, 15:43–52.
[31] Gao X, You C. Maximum entropy membership functions for discrete fuzzy variables. In-formation Sciences. doi:10.1016/j.ins.2009.03.010.
[32] Liu B. Uncertainty Theory. Berlin: Springer-Verlag, 2004.
[33] Liu B. A survey of credibility theory. Fuzzy Optimization and Decision Making, 2006,5(4):387–408.
[34] Liu B. Uncertainty Theory, 2nd ed. Berlin: Springer-Verlag, 2007.
[35]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(一)运行风险的提出与发展.电力系统自动化, 2006, 30(1):17–23.
[36]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(二)理论基础.电力系统自动化, 2006, 30(2):11–21.
[37]冯永青,张伯明,吴文传,等.基于可信性理论的电力系统运行风险评估(三)应用与工程实践.电力系统自动化, 2006, 30(3):11–16.
[38] Yang L, Li K, Gao Z. Train timetable problem on a single-line railway with fuzzy passengerdemand. IEEE Transactions on Fuzzy Systems. In press.
[39] Yang L, Liu L. Fuzzy fixed charge solid transportation problem and algorithm. Applied SoftComputing, 2007, 7(3):879–889.
[40] Zhou J, Liu B. Modeling capacitated location-allocation problem with fuzzy demands. Com-puters & Industrial Engineering, 2007, 53:454–468.
[41] Wen M, Iwamura K. Fuzzy facility location-allocation problem under the Hurwicz criterion.European Journal of Operational Research, 2008, 184:627–635.
[42] Ke H, Liu B. Project scheduling problem with mixed uncertainty of randomness and fuzzi-ness. European Journal of Operational Research, 2007, 183:135–147.
[43] Zheng Y, Liu B. Fuzzy vehicle routing model with credibility measure and its hybrid intel-ligent algorithm. Applied Mathematics and Computation, 2006, 176:673–683.
[44] Liu L, Gao X. Fuzzy weighted equilibrium multi-job assignment problem and genetic algo-rithm. Applied Mathematical Modelling. doi:10.1016/j.apm.2009.01.014.
[45] Gao X, Kar S. Information sharing model in supply chain management in fuzzy environ-ment. Far East Journal of Experimental and Theoretical Artificial Intelligence. In press.
[46] Ji X, Zhao X, Zhou D. A fuzzy programming approach for supply chain network design.International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, 2007, 15:75–87.
[47] Huang X. Portfolio selection with fuzzy returns. Journal of Intelligent & Fuzzy Systems,2007, 18:383–390.
[48] Huang X. Mean-semivariance models for fuzzy portfolio selection. Journal of Computa-tional and Applied Mathematics, 2008, 217:1–8.
[49] Huang X. Mean-entropy models for fuzzy portfolio selection. IEEE transactions on FuzzySystems, 2008, 16(4):1096–1101.
[50] Qin Z, Li X, Ji X. Portfolio selection based on fuzzy cross-entropy. Journal of Computa-tional and Applied Mathematics, 2009, 228:139–149.
[51] Liu B. Some research problems in uncertainty theory. Journal of Uncertain Systems, 2009,3(1):3–10.
[52] Choquet G. Theory of capacities. Annales de l’Institut Fourier, 1954, 5:131–295.
[53] Dempster A. Upper and lower probabilities induced by a multivalued mapping. Annals ofMathematical Statistics, 1967, 38:325–339.
[54] Shafer G. A Mathematical Theory of Evidence. Princeton (NJ): Princeton University Press,1976.
[55] Sugeno M. Theory of Fuzzy Integrals and Its Applications[D]. Tokoy Institute of Technol-ogy, 1974.
[56] Zadeh L. A theory of approximate reasoning. Proceedings of Mathematical Frontiers of theSocial and Policy Sciences, Boulders, Cororado, 1979.
[57] Liu B. Uncertainty Theory, 3rd ed. http://orsc.edu.cn/liu/ut.pdf.
[58]刘郁涵.如何生成不确定测度.第十届中国青年信息与管理学者大会论文集,洛阳,中国, 2008. 23–26.
[59] Peng Z. Some properties of product uncertain measure. Technical report. http://orsc.edu.cn/online/081228.pdf.
[60] You C. Some convergence theorems of uncertain sequences. Mathematical and ComputerModelling, 2009, 49(3-4):482–487.
[61] Liu B. Theory and Practice of Uncertain Programming, 3rd ed. http://orsc.edu.cn/liu/up.pdf.
[62] Liu B. Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems,2008, 2(1):3–16.
[63] Chen X, Liu B. Existence and uniqueness theorem for uncertain differential equations.Technical report. http://orsc.edu.cn/online/081230.pdf.
[64] Li X, Liu B. Hybrid logic and uncertain logic. Journal of Uncertain Systems, 2009, 3(2):83–94.
[65] Chen X, Dai W. Linearity property of expected value operator for uncertain variables. Tech-nical report. http://orsc.edu.cn/online/090311.pdf.
[66] Gao X. Some properties of continuous uncertain measure. International Journal of Uncer-tainty, Fuzziness & Knowledge-Based Systems. In press.
[67] Gao X, Chen X. Maximum entropy principle for uncertain variables. Technical report, 2009.
[68] Gao X. Cross-entropy of uncertain variables and its properties. Technical report, 2008.
[69] Li X, Liu B. Cross-entropy for fuzzy variables. Technical report, 2007.
[70] Jang J. ANFIS: Adaptive-network-based fuzzy inference system. IEEE transactions onsystems, man, and cybernetics, 1993, 23:665–685.
[71]吴望名.模糊推理的原理和方法.贵阳:贵州科技出版社, 1994.
[72]王国俊.模糊推理的全蕴涵三I算法.中国科学, E辑, 1999, 29(1):43–53.
[73]张文修,梁怡,徐萍.基于包含度的不确定推理.北京:清华大学出版社, 2007.
[74]王国俊.非经典数理逻辑与近似推理,第2版.北京:科学出版社, 2008.
[75] Li X, Liu B. Foundation of credibilistic logic. Fuzzy Optimization and Decision Making,2009, 8(1):91–102.
[76] Rescher N. Many-valued logic. New York: McGraw-Hill, 1969.
[77] Zadeh L. Outline of a new approach to the analysis of complex systems and decision pro-cesses. IEEE Transactions on Systems, Man, and Cybernetics, 1973, (1):28–44.
[78] Mamdani E, Assilian S. An experiment in linguistic synthesis with a fuzzy logic controller.International Journal of Man-Machine Studies, 1975, 7(1):1–13.
[79] Gao X. On Liu’s inference rule for uncertain systems. Technical report. http://orsc.edu.cn/online/090121.pdf.