基于Choquet积分的非线性回归模型研究
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摘要
回归模型的建立是回归分析的首要问题。由于Choquet积分中所采用的非可加集函数能够描述属性间的交互作用,所以基于Choquet积分的非线性回归模型在应用于一些实际问题时,取得了很好的效果。在现有的非线性回归模型中,非线性积分所采用的非可加集函数一般是定义在有限论域上的实值集函数。但在一些实际问题中,集函数取区间数或者模糊数更加合理。本文设计了一种基于关于区间值带符号模糊测度的Choquet积分的非线性回归模型。其中,关于区间值带符号模糊测度的Choquet积分(CIIM)的值可以采用线性规划的方法计算。另外,模型中的参数通过一种进化算法—混合蛙跳算法优化确定,实验结果验证了算法的可行性与有效性。
Establishment of regression model is an important part of regression analysis. The nonlinear multiregression models based on Choquet integrals have shown successful applications in practice since the adopted nonadditive set functions in Choquet integrals can effectively describe interaction among the attributes invovled. In the existing nonlinear multiregressin models, the nonlinear integrals are with respect to real-valued set functions. However, in some real problems, it seems more reasonable that the set functions are interval or fuzzy numbers. In this paper, a nonlinear multiregression model is designed based on Choquet integral with respect to an interval-valued signed fuzzy measure. The generalized Choquet integral in the model can be calculated by Linear Programming. Furthermore, we determine the regression coefficients optimally using a population evolutionary algorithm. Experimental results demonstrate the feasibility and effectiveness of the algorithm.
Establishment of regression model is an important part of regression analysis. The nonlinear multiregression models based on Choquet integrals have shown successful applications in practice since the adopted nonadditive set functions in Choquet integrals can effectively describe interaction among the attributes invovled. In the existing nonlinear multiregressin models, the nonlinear integrals are with respect to real-valued set functions. However, in some real problems, it seems more reasonable that the set functions are interval or fuzzy numbers. In this paper, a nonlinear multiregression model is designed based on Choquet integral with respect to an interval-valued signed fuzzy measure. The generalized Choquet integral in the model can be calculated by Linear Programming. Furthermore, we determine the regression coefficients optimally using a population evolutionary algorithm. Experimental results demonstrate the feasibility and effectiveness of the algorithm.
引文
[1]P. R. Halmos. Measure Theory. Van Nostrand, NewYork,1967.
[2]Z. Wang, G. J. Klir. Fuzzy Measure Theory. Plenum, NewYork,1992.
[3]M. Sugeno. Theory of fuzzy integral and its applications. Ph.D. dissertation, Tokyo Institute of Technology,1974.
[4]G. Choquet. Theory of capacities. Annales de l'Institut Fourier 5,1954:131-295.
[5]Z. Wang. Pan integral and Choquet integral. Proc. IFSA World Congress 1993,1993:316-317.
[6]D. Denneberg. Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht, Boston, London,1994.
[7]Z. Wang, K. S. Leung, M. L. Wong, J. Fang, K. Xu. Nonlinear nonnegative multiregressions based on Choquet integrals. International Journal of Approximate Reasoning,2000, (25):71-87.
[8]Z. Wang. A new model of nonlinear multiregressions by projection pursuit based on generalized Choquet integrals. Proc. IEEE International Conference on Fuzzy Systems 2002,2002:1240-1244.
[9]Z. Wang, H. Guo. A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals. Proc. IEEE International Conference on Fuzzy Systems 2003,2003:819-821.
[10]E. Emad, H. Tarek, G. Donald. Comparison among five evolutionary-based optimization algorithms. Advanced Engineering Informatics,2005, (19):43-53.
[11]J. Hui, Z. Wang. Nonlinear multiregressions based on Choquet integral for data with both numerical and categorical attributes. Proc. IFSA World Congress 2005,2005:445-449.
[12]M. Spilde, Z. Wang. Solving nonlinear optimization problems based on Choquet integrals by using a soft computing technique. Proc. IFSA World Congress 2005,2005:450-454.
[13]Z. Wang, K. S. Leung, G. K. Klir. Applying fuzzy measures and nonlinear integral in data mining. Fuzzy Sets and Systems,2005, (156):371-380.
[14]K. Xu, Z. Wang, M. L. Wong, K. S. Leung. Discover dependency pattern among attributes by using a new type of nonlinear multiregression. International Journal of Intelligent system,2001, (16): 949-962.
[15]L. S. Bock, Z. Wang. Using 2-additive measures in nonlinear multiregressions. Proc.IEEE International Conference on Granular Computing 2006,2006:639-642.
[16]P. Mahasukhon, H. Sharif, Z. Wang. Using pseudo gradient search for solving nonlinear multiregression based on 2-additive measures. Proc. IEEE International Conference on Information Reuse and Integration 2006,2006:410-413.
[17]M. Grabisch. A new algorithm for identifying fuzzy measures and its application to pattern recognition. Proc. IEEE International Conference on Fuzzy Systems and 2nd International Fuzzy Engineering Symposium, Yokohama, Japan, March 1995:145-150.
[18]Z. Wang, K. S. Leung, M. L. Wong, J. Fang. A new type of nonlinear integrals and the computational algorithm. Fuzzy Set and System.2000, (112):223-231.
[19]K. Xu, Z. Wang, K.S. Leung. Using a new type of nonlinear integral for multi-regression:an application of evolutionary algorithms in data mining, Proc. IEEE SMC,1998:2326-2331.
[20]Z. Wang, R. Yang, K. S. Leung. On the Choquet integral with fuzzy-valued integrand. Proc. IFSA 2005,2005:433-437.
[21]R. Yang, Z. Wang, P. A. Heng, K. S. Leung. Fuzzy numbers and fuzzification of Choquet integrals. Fuzzy Sets and Systems 153,2005:95-113.
[22]R. Yang, Z. Wang, P. A. Heng, K. S. Leung. Fuzzified Choquet integral with fuzzy-valued integrand and its application on temperature prediction. IEEE T. SMCB 38, No.2,2008:367-380.
[23]Z. Wang, K. H. Lee, K. S. Leung. The Choquet integral with respect to fuzzy-valued signed efficiency measures. WCCI 2008,2008:2143-2148.
[2]Z. Wang, G. J. Klir. Fuzzy Measure Theory. Plenum, NewYork,1992.
[3]M. Sugeno. Theory of fuzzy integral and its applications. Ph.D. dissertation, Tokyo Institute of Technology,1974.
[4]G. Choquet. Theory of capacities. Annales de l'Institut Fourier 5,1954:131-295.
[5]Z. Wang. Pan integral and Choquet integral. Proc. IFSA World Congress 1993,1993:316-317.
[6]D. Denneberg. Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht, Boston, London,1994.
[7]Z. Wang, K. S. Leung, M. L. Wong, J. Fang, K. Xu. Nonlinear nonnegative multiregressions based on Choquet integrals. International Journal of Approximate Reasoning,2000, (25):71-87.
[8]Z. Wang. A new model of nonlinear multiregressions by projection pursuit based on generalized Choquet integrals. Proc. IEEE International Conference on Fuzzy Systems 2002,2002:1240-1244.
[9]Z. Wang, H. Guo. A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals. Proc. IEEE International Conference on Fuzzy Systems 2003,2003:819-821.
[10]E. Emad, H. Tarek, G. Donald. Comparison among five evolutionary-based optimization algorithms. Advanced Engineering Informatics,2005, (19):43-53.
[11]J. Hui, Z. Wang. Nonlinear multiregressions based on Choquet integral for data with both numerical and categorical attributes. Proc. IFSA World Congress 2005,2005:445-449.
[12]M. Spilde, Z. Wang. Solving nonlinear optimization problems based on Choquet integrals by using a soft computing technique. Proc. IFSA World Congress 2005,2005:450-454.
[13]Z. Wang, K. S. Leung, G. K. Klir. Applying fuzzy measures and nonlinear integral in data mining. Fuzzy Sets and Systems,2005, (156):371-380.
[14]K. Xu, Z. Wang, M. L. Wong, K. S. Leung. Discover dependency pattern among attributes by using a new type of nonlinear multiregression. International Journal of Intelligent system,2001, (16): 949-962.
[15]L. S. Bock, Z. Wang. Using 2-additive measures in nonlinear multiregressions. Proc.IEEE International Conference on Granular Computing 2006,2006:639-642.
[16]P. Mahasukhon, H. Sharif, Z. Wang. Using pseudo gradient search for solving nonlinear multiregression based on 2-additive measures. Proc. IEEE International Conference on Information Reuse and Integration 2006,2006:410-413.
[17]M. Grabisch. A new algorithm for identifying fuzzy measures and its application to pattern recognition. Proc. IEEE International Conference on Fuzzy Systems and 2nd International Fuzzy Engineering Symposium, Yokohama, Japan, March 1995:145-150.
[18]Z. Wang, K. S. Leung, M. L. Wong, J. Fang. A new type of nonlinear integrals and the computational algorithm. Fuzzy Set and System.2000, (112):223-231.
[19]K. Xu, Z. Wang, K.S. Leung. Using a new type of nonlinear integral for multi-regression:an application of evolutionary algorithms in data mining, Proc. IEEE SMC,1998:2326-2331.
[20]Z. Wang, R. Yang, K. S. Leung. On the Choquet integral with fuzzy-valued integrand. Proc. IFSA 2005,2005:433-437.
[21]R. Yang, Z. Wang, P. A. Heng, K. S. Leung. Fuzzy numbers and fuzzification of Choquet integrals. Fuzzy Sets and Systems 153,2005:95-113.
[22]R. Yang, Z. Wang, P. A. Heng, K. S. Leung. Fuzzified Choquet integral with fuzzy-valued integrand and its application on temperature prediction. IEEE T. SMCB 38, No.2,2008:367-380.
[23]Z. Wang, K. H. Lee, K. S. Leung. The Choquet integral with respect to fuzzy-valued signed efficiency measures. WCCI 2008,2008:2143-2148.