不确定分布特征函数及拟合有效性检验
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摘要
不确定理论主要用来处理基于专家信度的非决定现象,为解决不确定理论在实际应用中缺少分布拟合检验过程的问题,研究了不确定分布的特征函数,并基于此提出了分布拟合的有效性检验方法.首先,定义了不确定分布的特征函数,基于不确定变量期望计算法则,给出了特征函数的计算方法,并在此基础上,推导了几种典型不确定分布的特征函数;其次,分析了不确定分布特征函数具有的性质,为后续相关理论的证明及计算奠定了基础;再次,根据特征函数的定义及性质,提出了不确定分布拟合有效性的定义及其判定定理,定理分别从分布函数的数字特征及形状相似度两个方面进行判定;最后,通过算例验证了所提方法的可行性和有效性.研究结果表明:不确定理论与随机理论在运算法则上有所不同,难以将随机理论中关于分布拟合检验的方法推广到不确定理论中;不确定分布特征函数与不确定分布函数呈一一对应的关系,不确定分布特征函数可以作为研究不确定分布拟合检验的途径之一.
Uncertainty theory is mainly used to deal with indeterminacy problems based on belief degree. In order to solve the problem of the lack of testing process after fitting uncertain distributions in practical application, characteristic functions with uncertain distributions were studied and effectiveness-of-fit test method was proposed. Firstly, the characteristic function of uncertain distribution was defined, and its calculation formula was given based on the uncertain expectation value. On the basis of the formula, the characteristic functions of several typical uncertain distributions were derived. Secondly, the properties of the characteristic functions were analyzed, which laid foundation for subsequent demonstration of theories and calculation. Thirdly, according to the definition and properties of the characteristic functions, the definition of effectiveness of uncertainty distribution fitting and its judgment theorem were proposed based on the digital characteristics and shape of distribution function. Finally, two examples were given to verify the feasibility and effectiveness of the proposed method. Simulation results show that uncertainty theory and stochastic theory are different in product axiom, and it is difficult to extend the method of distribution fitting test in stochastic theory to the uncertainty theory. In addition, there is a one-to-one correspondence between the uncertainty characteristic function and the uncertainty distribution function, and the uncertainty characteristic function can be used as one of the approaches to study the uncertainty distribution fitting test.
Uncertainty theory is mainly used to deal with indeterminacy problems based on belief degree. In order to solve the problem of the lack of testing process after fitting uncertain distributions in practical application, characteristic functions with uncertain distributions were studied and effectiveness-of-fit test method was proposed. Firstly, the characteristic function of uncertain distribution was defined, and its calculation formula was given based on the uncertain expectation value. On the basis of the formula, the characteristic functions of several typical uncertain distributions were derived. Secondly, the properties of the characteristic functions were analyzed, which laid foundation for subsequent demonstration of theories and calculation. Thirdly, according to the definition and properties of the characteristic functions, the definition of effectiveness of uncertainty distribution fitting and its judgment theorem were proposed based on the digital characteristics and shape of distribution function. Finally, two examples were given to verify the feasibility and effectiveness of the proposed method. Simulation results show that uncertainty theory and stochastic theory are different in product axiom, and it is difficult to extend the method of distribution fitting test in stochastic theory to the uncertainty theory. In addition, there is a one-to-one correspondence between the uncertainty characteristic function and the uncertainty distribution function, and the uncertainty characteristic function can be used as one of the approaches to study the uncertainty distribution fitting test.
引文
[1]LIU Baoding. Uncertainty theory[M]. 5th ed. Berlin: Springer-Verlag, 2016
[2]Peng Zixiong, IWAMURA K. A sufficient and necessary condition of uncertainty distribution[J]. Journal of Interdisciplinary Mathematics, 2010, 13(3): 277. DOI: 10.1080/09720502.2010.10700701
[3]LIU Baoding. Toward uncertain finance theory[J]. Journal of Uncertainty Analysis and Applications, 2013, 1: 1. DOI: 10.1186/2195-5468-1-1
[4]LIU Baoding. Membership functions and operational law of uncertain sets[J]. Fuzzy Optimization and Decision Making, 2012, 11(4): 387. DOI: 10.1007/s10700- 012-9128-7
[5]CHEN Xiaowei. Uncertain calculus with finite variation processes[J]. Soft Computing, 2015, 19(10): 2905. DOI: 10.1007/s00500-014-1452-0
[6]GE Xintong. Neutral uncertain delay differential equations[J]. Wit Transactions on Information & Communication Technologies, 2014, 54:1005. DOI: 10.2495/GCN 131342
[7]LIU Baoding, CHEN Xiaowei. Uncertain multi-objective programming and uncertain goal programming[J]. Journal of Uncertainty Analysis and Applications, 2015, 3: 10. DOI: 10.1186/s40467-015-0036-6
[8]LIU Baoding, YAO Kai. Uncertain multilevel programming: Algorithm and applications[J]. Computers & Industrial Engineering, 2015, 89: 235. DOI: 10.1016/j.cie. 2014.09.029
[9]PENG Jin. Risk metrics of loss function for uncertain system[J]. Fuzzy Optimization and Decision Making, 2013, 12(1): 53. DOI: 10.1007/s10700-012-9146-5
[10]LIU Baoding. Extreme value theorems of uncertain process with application to insurance risk model[J]. Soft Computing, 2013, 17:549. DOI: 10.1007/s00500-012-0930-5
[11]JI Xiaoyu, ZHOU Jian. Option pricing for an uncertain stock model with jumps[J]. Soft Computing, 2015, 19(11): 3323. DOI: 10.1007/s00500-015-1635-3
[12]LIU Baoding. Uncertainty theory: a Branch of mathematics for modeling Human uncertainty[M]. Berlin: Springer-Verlag, 2010. DOI:10.1007/978-3-642-13959-8
[13]GUO Haiying, WANG Xiaosheng, WANG Lili, et al. Delphi method for estimating membership function of uncertain set[J]. Journal of Uncertainty Analysis & Applications, 2016, 4:1. DOI: 10.1186/s40467-016-0044-1
[14]WANG Xiaosheng, PENG Zixiong. Method of moments for estimating uncertainty distributions[J]. Journal of Uncertainty Analysis & Applications, 2014, 2:5. DOI: 10.1186/2195-5468-2-5
[15]LI Runyu, LIU Gang. An uncertain goal programming model for machine scheduling problem[J]. Journal of Intelligent Manufacturing, 2017, 28(3):689. DOI: 10.1007/s10845-014-0982-8
[16]NING Yufu, CHEN Xiumei, WANG Zhiyong, et al. An uncertain multi-objective programming model for machine scheduling problem[J]. International Journal of Machine Learning and Cybernetics, 2017, 8(5):1493. DOI: 10.1007/s13042-016-0522-2
[17]GUO Jiansheng, WANG Zutong, ZHENG Mingfa, et al. Uncertain multi-objective redundancy allocation problem of repairable systems based on artificial bee colony algorithm[J]. Chinese Journal of Aeronautics, 2014, 27(6):1477. DOI: 10.1016/j.cja.2014.10.014
[18]LIU Liying, ZHANG Bo, MA Weimin. Uncertain programming models for fixed charge multi-item solid transportation problem[J]. Soft Computing, 2017(1):1. DOI: 10.1007/s00500-017-2718-0
[19]DALMAN H. Uncertain programming model for multi-item solid transportation problem[J]. International Journal of Machine Learning and Cybernetics, 2018, 9(4):559. DOI: 10.1007/s13042-016-0538-7
[20]王族统. 基于不确定理论的多目标规划方法及其应用研究[D].西安:空军工程大学, 2015WANG Zutong. Research on multi-objective programming and apply based on uncertainty theory[D]. Xi’an: Air Force Engineering University, 2015
[2]Peng Zixiong, IWAMURA K. A sufficient and necessary condition of uncertainty distribution[J]. Journal of Interdisciplinary Mathematics, 2010, 13(3): 277. DOI: 10.1080/09720502.2010.10700701
[3]LIU Baoding. Toward uncertain finance theory[J]. Journal of Uncertainty Analysis and Applications, 2013, 1: 1. DOI: 10.1186/2195-5468-1-1
[4]LIU Baoding. Membership functions and operational law of uncertain sets[J]. Fuzzy Optimization and Decision Making, 2012, 11(4): 387. DOI: 10.1007/s10700- 012-9128-7
[5]CHEN Xiaowei. Uncertain calculus with finite variation processes[J]. Soft Computing, 2015, 19(10): 2905. DOI: 10.1007/s00500-014-1452-0
[6]GE Xintong. Neutral uncertain delay differential equations[J]. Wit Transactions on Information & Communication Technologies, 2014, 54:1005. DOI: 10.2495/GCN 131342
[7]LIU Baoding, CHEN Xiaowei. Uncertain multi-objective programming and uncertain goal programming[J]. Journal of Uncertainty Analysis and Applications, 2015, 3: 10. DOI: 10.1186/s40467-015-0036-6
[8]LIU Baoding, YAO Kai. Uncertain multilevel programming: Algorithm and applications[J]. Computers & Industrial Engineering, 2015, 89: 235. DOI: 10.1016/j.cie. 2014.09.029
[9]PENG Jin. Risk metrics of loss function for uncertain system[J]. Fuzzy Optimization and Decision Making, 2013, 12(1): 53. DOI: 10.1007/s10700-012-9146-5
[10]LIU Baoding. Extreme value theorems of uncertain process with application to insurance risk model[J]. Soft Computing, 2013, 17:549. DOI: 10.1007/s00500-012-0930-5
[11]JI Xiaoyu, ZHOU Jian. Option pricing for an uncertain stock model with jumps[J]. Soft Computing, 2015, 19(11): 3323. DOI: 10.1007/s00500-015-1635-3
[12]LIU Baoding. Uncertainty theory: a Branch of mathematics for modeling Human uncertainty[M]. Berlin: Springer-Verlag, 2010. DOI:10.1007/978-3-642-13959-8
[13]GUO Haiying, WANG Xiaosheng, WANG Lili, et al. Delphi method for estimating membership function of uncertain set[J]. Journal of Uncertainty Analysis & Applications, 2016, 4:1. DOI: 10.1186/s40467-016-0044-1
[14]WANG Xiaosheng, PENG Zixiong. Method of moments for estimating uncertainty distributions[J]. Journal of Uncertainty Analysis & Applications, 2014, 2:5. DOI: 10.1186/2195-5468-2-5
[15]LI Runyu, LIU Gang. An uncertain goal programming model for machine scheduling problem[J]. Journal of Intelligent Manufacturing, 2017, 28(3):689. DOI: 10.1007/s10845-014-0982-8
[16]NING Yufu, CHEN Xiumei, WANG Zhiyong, et al. An uncertain multi-objective programming model for machine scheduling problem[J]. International Journal of Machine Learning and Cybernetics, 2017, 8(5):1493. DOI: 10.1007/s13042-016-0522-2
[17]GUO Jiansheng, WANG Zutong, ZHENG Mingfa, et al. Uncertain multi-objective redundancy allocation problem of repairable systems based on artificial bee colony algorithm[J]. Chinese Journal of Aeronautics, 2014, 27(6):1477. DOI: 10.1016/j.cja.2014.10.014
[18]LIU Liying, ZHANG Bo, MA Weimin. Uncertain programming models for fixed charge multi-item solid transportation problem[J]. Soft Computing, 2017(1):1. DOI: 10.1007/s00500-017-2718-0
[19]DALMAN H. Uncertain programming model for multi-item solid transportation problem[J]. International Journal of Machine Learning and Cybernetics, 2018, 9(4):559. DOI: 10.1007/s13042-016-0538-7
[20]王族统. 基于不确定理论的多目标规划方法及其应用研究[D].西安:空军工程大学, 2015WANG Zutong. Research on multi-objective programming and apply based on uncertainty theory[D]. Xi’an: Air Force Engineering University, 2015