模糊积分及其在多分类问题中的应用
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摘要
近年来,在模式识别、机器学习等领域,信息融合技术得到了迅速发展和广泛应用。信息融合包括三个阶段:数据融合、特征融合和决策融合,大量的分类器融合方法都是决策融合。考虑到分类器之间存在着交互影响,本文使用Choquet模糊积分这个融合算子,将已训练好的神经网络作为分类器进行融合,并将模糊积分融合系统的分类精度与单个分类器的分类精度进行比较。由于Choquet模糊积分的计算可以转化成模糊测度的线性组合,关于模糊测度是可微的,可以使用标准的优化技术来确定模糊测度。本文使用线性规划和二次规划来确定模糊测度,并在pima数据库上进行测试。实验发现,基于二次规划确定的模糊测度的系统融合精度要高于线性规划确定的模糊测度的系统融合精度,但是二者的时间复杂度却相差不大。
Recently, in the fields of pattern recognition and machine learning ,the technology of information fusion develops very fast and expands its wide application. There are three stages in information fusion: data fusion; character fusion and decision fusion. A lot of approaches to classifier fusion are decision fusion. In many realistic cases, multiple classifiers are not independent but have interaction. In this thesis, we use the choquet fuzzy integral operator to fuse the different neural network classifiers which have been trained in advance. We compare the fusion system's classification accuracy and an individual classifier's classification accuracy. The calculation of the choquet integral can be transferred into the linear combination about the fuzzy measure, which is differentiable. This quality allows us to use the standard optimization technology to determine the fuzzy measure. In this thesis, we use the linear programming and quadratic programming to determine the fuzzy measure, these methods are tested in pima data set. The experiments shows that: the fusion accuracy based on the fuzzy measure which is determined by the quadratic programming is higher than the accuracy based on the fuzzy measure which is determined by the linear programming. There are almost no different in time and space complexity.
Recently, in the fields of pattern recognition and machine learning ,the technology of information fusion develops very fast and expands its wide application. There are three stages in information fusion: data fusion; character fusion and decision fusion. A lot of approaches to classifier fusion are decision fusion. In many realistic cases, multiple classifiers are not independent but have interaction. In this thesis, we use the choquet fuzzy integral operator to fuse the different neural network classifiers which have been trained in advance. We compare the fusion system's classification accuracy and an individual classifier's classification accuracy. The calculation of the choquet integral can be transferred into the linear combination about the fuzzy measure, which is differentiable. This quality allows us to use the standard optimization technology to determine the fuzzy measure. In this thesis, we use the linear programming and quadratic programming to determine the fuzzy measure, these methods are tested in pima data set. The experiments shows that: the fusion accuracy based on the fuzzy measure which is determined by the quadratic programming is higher than the accuracy based on the fuzzy measure which is determined by the linear programming. There are almost no different in time and space complexity.
引文
[1] Tahani H. , Keller J. M. Information fusion in computer vision using fuzzy integral. IEEE Transactions on Systems, Man, and Cybernetics, 1990, 20: 733-741.
[2] Woods K. , Kegelmeyer W. P. , Bowyer K. Combination of multiple classifiers using local accuracy estimates. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19: 405-410.
[3] Ludmila I. Kuncheva. Switching Between Selection and Fusion in Combining classifiers: An experiment. IEEE Transactions on systems, man and cybernetics, 2002, 32 (2) : 146-156.
[4] Michel Grabisch, Jean-Marie Nicolas. Classification by fuzzy integral: performance and tests. Fuzzy set and systems, 1994, 65: 255-271.
[5] R. Battiti M. Colla. Democracy in neural nets: Voting schemes for classification. Neural Networks, 1994, 7 (4) : 691-707.
[6] G. Rogova. Combining the results of several neural network classifiers. Neural Networks, 1994, 7 (5) : 777-781.
[7] 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.铁道学报,2000,22(4):42-45.
[8] 边肇祺,张学工.模式识别.清华大学出版社,北京,2002.
[9] Jia Wang, Zhenyuan Wang. Using Neural Networks to Determine Sugeno Measures by Statistics. Neural Networks, 1997, 10 (1) : 183-195.
[10] Kuncheva L. I. , Bezdek J. C. , Duin R. P. W. . Decision templates for multiple classifier fusion: An experimental comparion, pattern recognition, 2001, 34 (2) : 299-314.
[11] Tuan D. Pham, Hong Yan. Information Fusion by Fuzzy Integral. Proc. 1996 Australian New Zealand Conf. On Intelligent Information Systems, 1996: 18-20.
[12] 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.计算机工程与应用,2002,12:119-121.
[13] Zhenyuan Wang, Kwong-Sak Leng. A genetic algorithm for determining no additive set functions in information fusion. Fuzzy set and systems, 1999, 102: 463-469.
[14] Michel Grabisch. Fuzzy Integral for Classification and Feature Extraction, Fuzzy measures and integrals: Theory and Applications. Physical-Verlag, 2000: 415-434.
[15] Daniel Yeung, XiZhao Wang. Handing interaction in fuzzy production rule reasoning. IEEE Transaction on systems and cybernetics, 2004, 34 (5) : 1-9.
[16] Verikas, A. et al. Fusion neural networks through fuzzy integration. In: Bunke, H. Kandel, A. (eds.), Hybird methods in pattern recgnition, World Scientific, 2001.
[17] 王振源,测度论.
[18] 程其襄,张奠宙等.《实变函数与泛函分析基础》.高等教育出版社,1997.
[19] Zhenyuan Wang, George J. Klir. Fuzzy Measure Theory. New York, 1992 Plenum Press.
[20] M. Sugeno. Theory of fuzzy integrals and its applications, Doct. Thesis. Tokyo Institute of technology, 1974.
[21] T. Murofushi, M. Sugeno. An interpretation of fuzzy measure and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Set and Systems, 1989, 29: 201-227.
[22] Zhenyuan Wang, Kwong-sak Leung, Man-Leung Wong. A new type of nonlinear integrals and the computational algorithm. Fuzzy Set and Systems, 2000, 112: 223-231.
[23] 袁曾任.人工神经元网络及其应用.清华大学出版社,北京,1999.
[24] 神经网络简介:http://www.2nsoft.com/netmodel/neuralNetwork.jsp.
[25] UCI Repository of machine learning databases and domain theories. FTP address: ftp://ftp.ics.uci.edu/pub/machine-leaming-databases.
[2] Woods K. , Kegelmeyer W. P. , Bowyer K. Combination of multiple classifiers using local accuracy estimates. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19: 405-410.
[3] Ludmila I. Kuncheva. Switching Between Selection and Fusion in Combining classifiers: An experiment. IEEE Transactions on systems, man and cybernetics, 2002, 32 (2) : 146-156.
[4] Michel Grabisch, Jean-Marie Nicolas. Classification by fuzzy integral: performance and tests. Fuzzy set and systems, 1994, 65: 255-271.
[5] R. Battiti M. Colla. Democracy in neural nets: Voting schemes for classification. Neural Networks, 1994, 7 (4) : 691-707.
[6] G. Rogova. Combining the results of several neural network classifiers. Neural Networks, 1994, 7 (5) : 777-781.
[7] 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.铁道学报,2000,22(4):42-45.
[8] 边肇祺,张学工.模式识别.清华大学出版社,北京,2002.
[9] Jia Wang, Zhenyuan Wang. Using Neural Networks to Determine Sugeno Measures by Statistics. Neural Networks, 1997, 10 (1) : 183-195.
[10] Kuncheva L. I. , Bezdek J. C. , Duin R. P. W. . Decision templates for multiple classifier fusion: An experimental comparion, pattern recognition, 2001, 34 (2) : 299-314.
[11] Tuan D. Pham, Hong Yan. Information Fusion by Fuzzy Integral. Proc. 1996 Australian New Zealand Conf. On Intelligent Information Systems, 1996: 18-20.
[12] 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.计算机工程与应用,2002,12:119-121.
[13] Zhenyuan Wang, Kwong-Sak Leng. A genetic algorithm for determining no additive set functions in information fusion. Fuzzy set and systems, 1999, 102: 463-469.
[14] Michel Grabisch. Fuzzy Integral for Classification and Feature Extraction, Fuzzy measures and integrals: Theory and Applications. Physical-Verlag, 2000: 415-434.
[15] Daniel Yeung, XiZhao Wang. Handing interaction in fuzzy production rule reasoning. IEEE Transaction on systems and cybernetics, 2004, 34 (5) : 1-9.
[16] Verikas, A. et al. Fusion neural networks through fuzzy integration. In: Bunke, H. Kandel, A. (eds.), Hybird methods in pattern recgnition, World Scientific, 2001.
[17] 王振源,测度论.
[18] 程其襄,张奠宙等.《实变函数与泛函分析基础》.高等教育出版社,1997.
[19] Zhenyuan Wang, George J. Klir. Fuzzy Measure Theory. New York, 1992 Plenum Press.
[20] M. Sugeno. Theory of fuzzy integrals and its applications, Doct. Thesis. Tokyo Institute of technology, 1974.
[21] T. Murofushi, M. Sugeno. An interpretation of fuzzy measure and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Set and Systems, 1989, 29: 201-227.
[22] Zhenyuan Wang, Kwong-sak Leung, Man-Leung Wong. A new type of nonlinear integrals and the computational algorithm. Fuzzy Set and Systems, 2000, 112: 223-231.
[23] 袁曾任.人工神经元网络及其应用.清华大学出版社,北京,1999.
[24] 神经网络简介:http://www.2nsoft.com/netmodel/neuralNetwork.jsp.
[25] UCI Repository of machine learning databases and domain theories. FTP address: ftp://ftp.ics.uci.edu/pub/machine-leaming-databases.
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