复模糊积分及其应用
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摘要
多分类器融合是解决复杂模式识别问题的有效办法,而提高多分类器融合系统的分类精确率和改善系统的稳健性,通常使用模糊积分作为一种融合工具。在基于模糊积分的多分类器融合系统中,模糊密度决定着模糊积分,从而对分类结果产生很大影响。基于这样的考虑,本文提出了基于多分类器融合应用的复模糊积分,并从以下几个方面对多分类器融合进行了研究:复模糊积分定义的给出以及讨论了相关性质定理、复模糊测度的确定、对相关样例进行了检验。首先,本文给出了复模糊积分的概念,并将经典积分中的一些性质和定理推广到复模糊积分上。然后,本文给出了复模糊测度的确定方法,并给出复模糊积分在多分类器融合应用中的融合过程。最后,本文对几个样例进行了详细讨论,并证明了该方法具有一定的可行性。
Multi-classifiers fusion is a powerful solution to the difficult pattern recognition problem,but who can improve the accuracy of classification and the robustness of systems ,we often use used as an integration tool for fuzzy integral .In the model of multi-classifiers fusion based on fuzzy integrals ,the fuzzy results are heavily dependent on fuzzy densities which represent the important of individual classifier to the fusion results. In this thesis, we proposed complex fuzzy integral which based on the application of multiple classifier fusion, and from the following aspects of the fusion of multiple classifiers were studied: the concepts of complex fuzzy integral,and discuss their properties and some converge theorems,methods to confirm the complex fuzzy measure, tested on the relevant samples. Firstly, the concept of complex fuzzy integral are given, and promoted some theorems of the classical integral to the complex fuzzy integral. Secondly, methods to confirm the complex fuzzy measure are given, and the fusion process of Complex Fuzzy integral in multi-classifier fusion applications are proposed. Finally, several examples were discussed in detail and the methods are verified to be practical.
Multi-classifiers fusion is a powerful solution to the difficult pattern recognition problem,but who can improve the accuracy of classification and the robustness of systems ,we often use used as an integration tool for fuzzy integral .In the model of multi-classifiers fusion based on fuzzy integrals ,the fuzzy results are heavily dependent on fuzzy densities which represent the important of individual classifier to the fusion results. In this thesis, we proposed complex fuzzy integral which based on the application of multiple classifier fusion, and from the following aspects of the fusion of multiple classifiers were studied: the concepts of complex fuzzy integral,and discuss their properties and some converge theorems,methods to confirm the complex fuzzy measure, tested on the relevant samples. Firstly, the concept of complex fuzzy integral are given, and promoted some theorems of the classical integral to the complex fuzzy integral. Secondly, methods to confirm the complex fuzzy measure are given, and the fusion process of Complex Fuzzy integral in multi-classifier fusion applications are proposed. Finally, several examples were discussed in detail and the methods are verified to be practical.
引文
[1].J.J.Buckley.Fuzzy complex numbers.Fuzzy Sets and Systems[J]. 1989,33:333-345.
[2].J.J.Buckley.Fuzzy complex analysisⅠ:Differentiation[J].Fuzzy Sets and Systems.1991,41:268-284.
[3].J.J.Buckley.Fuzzy complex analysisⅡ:Integration[J]. Fuzzy Sets and Systems. 1992,49:171-179.
[4].张跃,王光远.模糊随机动力系统理论[M].北京.科学出版社.1993.
[5].马生全.模糊复数及其几个运算性质[J].兰州大学学报(自然科学版)Vol.32.1996.643-645
[6].马生全.极模型模糊复数概念及简性[J].西北民族学院学报(自然科学版)Vol.18.No.1.1997:8-9.
[7].马生全等.矩形模糊复数的几个运算性质[J].西北民族学院学报(自然科学版).Vol.19.No.1.1998:62-63.
[8].马生全,曹纯.模糊复分析[M].北京.民族出版社.2001.
[9].马生全,曹纯.复模糊函数的微分法[C].98’中国模糊数学与模糊系统第九届年会论文选集.保定.河北大学出版社.1998:162-166.
[10].马生全.模糊复围道积分[J].模糊系统与数学(专辑).Vo1.16.2002.:199-202.
[11]王震源,李法朝.Fuzzy积分在综合评判过程中的应用[J].模糊数学,1985,5:109-114.
[12]. Leszczynski K, Penczek P, Grochuiski W. Sugeno’s fuzzy measure and fuzzy clustering[J]. Fuzzy Sets and Systems,1985,15:147-158.
[13]. Onisava T. Fuzzy measure analysis of public attitude towards the use of nuclear energy[J]. Fuzzy Sets and Systems,1986,20:259-289.
[14].马生全.复Fuzzy级数及其收敛性的进一步讨论[J].模糊系统与数学.Vo1.16.No.2.2002:92-96.
[15].马生全等.模糊复数项级数及其收敛性[J].西北民族学院学报.Vol.23.No.3.2002:10-13.
[16].Ma shengquan and Gong zengtai.The Series of Complex Fuzzy Number and Complex Fuzzy-valued Functions and Its Convergence[J]. International Journal of Pure and Applied Mathematics .Vol.7.No.1.2003.49-62.
[17].Ma Shengquan.The Series of Complex fuzzy Valued and its Convergence[J]. The Journal of Fuzzy Mathematics,Vol.13,No.2(2005)(USA)467-475.
[18].R.Battiti M.Colla.Democracy in neural nets:Voting schemes for classification[J].Neural Networks,1994,7(4):691-707.
[19].G.Rogova.Combining the results of several neural network classifiers[J].Neural Networks,1994,7(5):777-781.
[20].刘汝杰,袁保宗.一种考虑分类器不确定性的多通道D-S融合方法[J].铁道学报,2000,22(4):42-45.
[21].边肇祺,张学工.模式识别[M].清华大学出版社,北京,2002
[22].Jia Wang,Zhenyuan Wang.Using Neural Networks to Determine Sugeno Measures by Statistics[J]. Neural Networks,1997,10(1):183-195.
[23].Kuncheva L.I.,Bezdek J.C.,Duin R.P.W..Decision templates for multiple classifier fusion:An experimental comparion[J].pattern recognition,2001,34(2):299-314.
[24].Tuan D.Pham,Hong Yan.Information Fusion by Fuzzy Integral[J]. Proc.1996 Australian New Zealand Conf.On Intelligent Information Systems,1996:18-20.
[25].李玉榕,乔斌,蒋静坪.基于模糊积分和遗传算法的分类器组合算法[J].计算机工程与应用,2002,12:119-121.
[26].Zhenyuan Wang,Kwong-Sak Leng.A genetic algorithm for determining no additive set functions in information fusion[J].Fuzzy set and systems ,1999,102:463-469.
[27].Michel Grabisch.Fuzzy Integral for Classification and Feature Extraction[J].Fuzzy measures and integral:Theory and Application.Physical-Verlag,2000:415-434.
[28].Daniel Yeung,XiZhao Wang.Handing interaction in fuzzy production rule reasoning[J].IEEE Transaction on systems and cybernetics,2004, 34(5):1-9.
[29].Verikas,A.etal.Fusion neural networks through fuzzyintegration[M]. In:Bunke,H.Kandel,A.(eds.),Hybird methods in pattern recognition,World Scientific,2001.
[30].Zhenyuan Wang,George J.Klir.Fuzzy Measure Theory[J].Plenum Press, 1992:46-47,62-63
[31].张文修.模糊数学基础[M],西安交通大学出版社,1984.
[32].哈明虎,吴从火斤,模糊测度与模糊积分理论[M].科学出版社,1998
[33].Sansanee Auephanwiriyakul,James M.Keller,Paul D.Gader.Generalized Choquet Fuzzy Integral Fusion[J].Information Fusion,2002,3:69-85.
[34].罗志增,蒋静坪.机器人感觉与多信息融合[M].机械工业出版社,2002, 113-114,116-117
[35].Ludmila I.Kuncheva.Fuzzy Classifier Design[M].Physica-Verlag Heidelberg,2000.
[36].Ludmila I.Kuncheva.Using measures of similarity and inclusion for multiple classifier fusion by decision templates[J]. Fuzzy set and systems,2000,122:401-407.
[37].Sung-Bae Cho,Jin H.Kim.Combining Multiple Networks by Fuzzy Integral for Robust Classification[J].IEEE Transactions onSystems,Man,and Cybernetics,1995,25(2):380-384.
[38].Paul D.Gader,Magdi A.Mohamed,James M.Keller.Fusion of Handwritten Word Classifiers[J].Pattern Recognition Letters,2001,17:577-584.
[39].姚明海,何通能.一种基于模糊积分的多分类器联合方法[J].浙江工业大学学报,2002,30(2):156-159.
[40].姚明海,李澎林.基于遗传算法和模糊积分的多分类器集成[J].计算机应用与软件,2003,20(8):66-68.
[41].王晓君.基于模糊积分的多分类器融合研究[M].河北大学.2005,6
[42].谢颖.信息融合算法研究及其应用[D].重庆:重庆大学软件学院,2008.4
[43].罗承忠.模糊集引论[M].北京师范大学出版社,2007.
[44].仇计清,李法朝,苏连青.复Fuzzy测度与复Fuzzy积分[J].河北轻化工学院学报,1997,1: 1-4.
[45].M.Sugeno.Theory of Fuzzy Integrals and Applications[M]. Ph.D.Disertation.okyo Institute of Technology,1974.
[46].张文修,马计丰.模糊经济系统模型[J].模糊系统与数学,1987,1:1-12
[47].赵汝怀.模糊积分[M].数学研究与评论,1981,10
[48].张文修等.模糊数学引论[M].西安交通大学出版社,1991.
[49].吴从炘.模糊分析学基础[M].北京,国防工业出版社,1991.
[50].王熙照.模糊测度和模糊积分及在分类技术中的应用[M].科学出版社,2008.
[51].马鸿,张捷,王玉静.模糊测度与模糊积分的应用[J].沈阳大学学报,2003, 6:82-83.
[52].刘永祥,黎湘,庄钊文.基于Choquet模糊积分的决策层信息融合目标识别[J].电子信息学报,2003,25(5):695-699.
[53].孔志周,蔡自兴.分类器融合中模糊积分理论研究进展[J].小型微型计算机,2008,29(6):1093-1098.
[54].段宝彬,孙梅兰.基于多分类器多模糊积分的信息融合方法[J].重庆科技学院学报:自然科学版,2008,10(3):87-89
[55].李素文.模糊积分在多分类器融合中的应用[M].河北大学.2006,6
[56].赵志伟.基于模糊积分的多分类器融合方法研究[M].长沙理工大学.2009,3
[57].冯慧敏.多分类器融合中的模糊测度研究[M].河北大学.2005,6
[58].张德利,郭彩梅,吴从火斤,模糊积分论进展[J],模糊系统与数学,2003, Vol.17,No.4:1-10.
[59].汪培庄.模糊集合论及其应用[M].上海科学技术出版社,1983.
[2].J.J.Buckley.Fuzzy complex analysisⅠ:Differentiation[J].Fuzzy Sets and Systems.1991,41:268-284.
[3].J.J.Buckley.Fuzzy complex analysisⅡ:Integration[J]. Fuzzy Sets and Systems. 1992,49:171-179.
[4].张跃,王光远.模糊随机动力系统理论[M].北京.科学出版社.1993.
[5].马生全.模糊复数及其几个运算性质[J].兰州大学学报(自然科学版)Vol.32.1996.643-645
[6].马生全.极模型模糊复数概念及简性[J].西北民族学院学报(自然科学版)Vol.18.No.1.1997:8-9.
[7].马生全等.矩形模糊复数的几个运算性质[J].西北民族学院学报(自然科学版).Vol.19.No.1.1998:62-63.
[8].马生全,曹纯.模糊复分析[M].北京.民族出版社.2001.
[9].马生全,曹纯.复模糊函数的微分法[C].98’中国模糊数学与模糊系统第九届年会论文选集.保定.河北大学出版社.1998:162-166.
[10].马生全.模糊复围道积分[J].模糊系统与数学(专辑).Vo1.16.2002.:199-202.
[11]王震源,李法朝.Fuzzy积分在综合评判过程中的应用[J].模糊数学,1985,5:109-114.
[12]. Leszczynski K, Penczek P, Grochuiski W. Sugeno’s fuzzy measure and fuzzy clustering[J]. Fuzzy Sets and Systems,1985,15:147-158.
[13]. Onisava T. Fuzzy measure analysis of public attitude towards the use of nuclear energy[J]. Fuzzy Sets and Systems,1986,20:259-289.
[14].马生全.复Fuzzy级数及其收敛性的进一步讨论[J].模糊系统与数学.Vo1.16.No.2.2002:92-96.
[15].马生全等.模糊复数项级数及其收敛性[J].西北民族学院学报.Vol.23.No.3.2002:10-13.
[16].Ma shengquan and Gong zengtai.The Series of Complex Fuzzy Number and Complex Fuzzy-valued Functions and Its Convergence[J]. International Journal of Pure and Applied Mathematics .Vol.7.No.1.2003.49-62.
[17].Ma Shengquan.The Series of Complex fuzzy Valued and its Convergence[J]. The Journal of Fuzzy Mathematics,Vol.13,No.2(2005)(USA)467-475.
[18].R.Battiti M.Colla.Democracy in neural nets:Voting schemes for classification[J].Neural Networks,1994,7(4):691-707.
[19].G.Rogova.Combining the results of several neural network classifiers[J].Neural Networks,1994,7(5):777-781.
[20].刘汝杰,袁保宗.一种考虑分类器不确定性的多通道D-S融合方法[J].铁道学报,2000,22(4):42-45.
[21].边肇祺,张学工.模式识别[M].清华大学出版社,北京,2002
[22].Jia Wang,Zhenyuan Wang.Using Neural Networks to Determine Sugeno Measures by Statistics[J]. Neural Networks,1997,10(1):183-195.
[23].Kuncheva L.I.,Bezdek J.C.,Duin R.P.W..Decision templates for multiple classifier fusion:An experimental comparion[J].pattern recognition,2001,34(2):299-314.
[24].Tuan D.Pham,Hong Yan.Information Fusion by Fuzzy Integral[J]. Proc.1996 Australian New Zealand Conf.On Intelligent Information Systems,1996:18-20.
[25].李玉榕,乔斌,蒋静坪.基于模糊积分和遗传算法的分类器组合算法[J].计算机工程与应用,2002,12:119-121.
[26].Zhenyuan Wang,Kwong-Sak Leng.A genetic algorithm for determining no additive set functions in information fusion[J].Fuzzy set and systems ,1999,102:463-469.
[27].Michel Grabisch.Fuzzy Integral for Classification and Feature Extraction[J].Fuzzy measures and integral:Theory and Application.Physical-Verlag,2000:415-434.
[28].Daniel Yeung,XiZhao Wang.Handing interaction in fuzzy production rule reasoning[J].IEEE Transaction on systems and cybernetics,2004, 34(5):1-9.
[29].Verikas,A.etal.Fusion neural networks through fuzzyintegration[M]. In:Bunke,H.Kandel,A.(eds.),Hybird methods in pattern recognition,World Scientific,2001.
[30].Zhenyuan Wang,George J.Klir.Fuzzy Measure Theory[J].Plenum Press, 1992:46-47,62-63
[31].张文修.模糊数学基础[M],西安交通大学出版社,1984.
[32].哈明虎,吴从火斤,模糊测度与模糊积分理论[M].科学出版社,1998
[33].Sansanee Auephanwiriyakul,James M.Keller,Paul D.Gader.Generalized Choquet Fuzzy Integral Fusion[J].Information Fusion,2002,3:69-85.
[34].罗志增,蒋静坪.机器人感觉与多信息融合[M].机械工业出版社,2002, 113-114,116-117
[35].Ludmila I.Kuncheva.Fuzzy Classifier Design[M].Physica-Verlag Heidelberg,2000.
[36].Ludmila I.Kuncheva.Using measures of similarity and inclusion for multiple classifier fusion by decision templates[J]. Fuzzy set and systems,2000,122:401-407.
[37].Sung-Bae Cho,Jin H.Kim.Combining Multiple Networks by Fuzzy Integral for Robust Classification[J].IEEE Transactions onSystems,Man,and Cybernetics,1995,25(2):380-384.
[38].Paul D.Gader,Magdi A.Mohamed,James M.Keller.Fusion of Handwritten Word Classifiers[J].Pattern Recognition Letters,2001,17:577-584.
[39].姚明海,何通能.一种基于模糊积分的多分类器联合方法[J].浙江工业大学学报,2002,30(2):156-159.
[40].姚明海,李澎林.基于遗传算法和模糊积分的多分类器集成[J].计算机应用与软件,2003,20(8):66-68.
[41].王晓君.基于模糊积分的多分类器融合研究[M].河北大学.2005,6
[42].谢颖.信息融合算法研究及其应用[D].重庆:重庆大学软件学院,2008.4
[43].罗承忠.模糊集引论[M].北京师范大学出版社,2007.
[44].仇计清,李法朝,苏连青.复Fuzzy测度与复Fuzzy积分[J].河北轻化工学院学报,1997,1: 1-4.
[45].M.Sugeno.Theory of Fuzzy Integrals and Applications[M]. Ph.D.Disertation.okyo Institute of Technology,1974.
[46].张文修,马计丰.模糊经济系统模型[J].模糊系统与数学,1987,1:1-12
[47].赵汝怀.模糊积分[M].数学研究与评论,1981,10
[48].张文修等.模糊数学引论[M].西安交通大学出版社,1991.
[49].吴从炘.模糊分析学基础[M].北京,国防工业出版社,1991.
[50].王熙照.模糊测度和模糊积分及在分类技术中的应用[M].科学出版社,2008.
[51].马鸿,张捷,王玉静.模糊测度与模糊积分的应用[J].沈阳大学学报,2003, 6:82-83.
[52].刘永祥,黎湘,庄钊文.基于Choquet模糊积分的决策层信息融合目标识别[J].电子信息学报,2003,25(5):695-699.
[53].孔志周,蔡自兴.分类器融合中模糊积分理论研究进展[J].小型微型计算机,2008,29(6):1093-1098.
[54].段宝彬,孙梅兰.基于多分类器多模糊积分的信息融合方法[J].重庆科技学院学报:自然科学版,2008,10(3):87-89
[55].李素文.模糊积分在多分类器融合中的应用[M].河北大学.2006,6
[56].赵志伟.基于模糊积分的多分类器融合方法研究[M].长沙理工大学.2009,3
[57].冯慧敏.多分类器融合中的模糊测度研究[M].河北大学.2005,6
[58].张德利,郭彩梅,吴从火斤,模糊积分论进展[J],模糊系统与数学,2003, Vol.17,No.4:1-10.
[59].汪培庄.模糊集合论及其应用[M].上海科学技术出版社,1983.