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January  2016, 12(1): 357-373. doi: 10.3934/jimo.2016.12.357

## Fuzzy quadratic surface support vector machine based on fisher discriminant analysis

 1 School of Management Science and Engineering, Dongbei University of Finance and Economics, Dalian 116025, China 2 Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906 3 Department of Mathematics, Shanghai University, Shanghai 200444 4 School of Management, University of Chinese Academy of Sciences, Beijing, 100190

Received  August 2014 Revised  January 2015 Published  April 2015

In this paper, using the concept of Fisher discriminant analysis and a new fuzzy membership function, a kernel-free fuzzy quadratic surface support vector machine model is proposed for binary classification. The membership function is specially designed to consider not only the quadratic-margin distance'' between a training point and its related quadratic center surface'' but also the affinity among training points. A decomposition algorithm is designed to solve the proposed model. Computational results on artificial and four real-world classifying data sets indicate that the proposed model outperforms fuzzy support vector machine models with Gaussian or Quadratic kernel and soft quadratic surface support vector machine model, especially, when the data sets contain a large amount of outliers and noises.
Citation: Jian Luo, Shu-Cherng Fang, Yanqin Bai, Zhibin Deng. Fuzzy quadratic surface support vector machine based on fisher discriminant analysis. Journal of Industrial and Management Optimization, 2016, 12 (1) : 357-373. doi: 10.3934/jimo.2016.12.357
##### References:
 [1] L. T. H. An and P. D. Tao, A continuous approach for the concave cost supply problem via DC programming and DCA, Discrete Applied Mathematics, 156 (2008), 325-338. doi: 10.1016/j.dam.2007.03.024. [2] W. An and M. Liang, Fuzzy support vector machine based on within-class scatter for classification problems with outliers or noises, Neurocomputing, 110 (2013), 101-110. doi: 10.1016/j.neucom.2012.11.023. [3] K. Bache and M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science, 2013. Available from: http://archive.ics.uci.edu/ml. [4] M. Bicego and M. A. Figueiredo, Soft clustering using weighted one-class support vector machines, Pattern Recognition, 42 (2009), 27-32. doi: 10.1016/j.patcog.2008.07.004. [5] J. P. Brooks, Support vector machines with the ramp loss and the hard margin loss, Operations Research, 59 (2011), 467-479. doi: 10.1287/opre.1100.0854. [6] H.-G. Chew and C.-C. Lim, On regularisation parameter transformation of support vector machines, Journal of Industrial and Management Optimization, 5 (2009), 403-415. doi: 10.3934/jimo.2009.5.403. [7] I. Dagher, Quadratic kernel-free non-linear support vector machine, Journal of Global Optimization, 41 (2008), 15-30. doi: 10.1007/s10898-007-9162-0. [8] R. A. Fisher, The use of multiple measurements in taxonomic problems, Annals of Human Genetics, 7 (1936), 179-188. doi: 10.1111/j.1469-1809.1936.tb02137.x. [9] X. Jiang, Y. Zhang and J. C. Lv, Fuzzy SVM with a new fuzzy membership function, Neural Computing and Applications, 15 (2006), 268-276. doi: 10.1007/s00521-006-0028-z. [10] T. Joachims, Text categorization with support vector machines: learning with many relevant features, Machine Learning: ECML-98, Lecture Notes in Computer Science, 1398 (1998), 137-142. doi: 10.1007/BFb0026683. [11] S. B. Kazmi, Q. Ain and M. A. Jaffar, Wavelets-based facial expression recognition using a bank of support vector machines, Soft Computing, 16 (2012), 369-379. doi: 10.1007/s00500-011-0721-4. [12] C. F. Lin and S. D. Wang, Fuzzy support vector machines, IEEE Transactions on Neural Networks, 13 (2002), 464-471. [13] Y. Liu and M. Yuan, Reinforced multicategory support vector machines, Journal of Computational and Graphical Statistics, 20 (2011), 901-919. doi: 10.1198/jcgs.2010.09206. [14] J. Luo, Z. Deng, D. Bulatov, J. E. Lavery and S.-C. Fang, Comparison of an $l_1$-regression-based and a RANSAC-based planar segmentation procedure for urban terrain data with many outliers, Image and Signal Processing for Remote Sensing XIX, 8892 (2013), p889209. doi: 10.1117/12.2028627. [15] J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic Surface Support Vector Machine for Binary Classification, Submitted to Neurocomputing, 2014. [16] K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476. doi: 10.3934/jimo.2005.1.465. [17] F. E. H. Tay and L. Cao, Application of support vector machines in financial time series forecasting, Omega, 29 (2001), 309-317. doi: 10.1016/S0305-0483(01)00026-3. [18] V. N. Vapnik, The Nature of Statistical Learning Theory, $2^{nd}$ edition, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4757-3264-1. [19] C. Wu, C. Li and Q. Long, A DC programming approach for sensor network localization with uncertainties in archor positions, Journal of Industrial and Management Optimization, 10 (2014), 817-826. doi: 10.3934/jimo.2014.10.817. [20] Y. Wu and Y. Liu, Robust truncated hinge loss support vector machines, Journal of the American Statistical Association, 102 (2007), 974-983. doi: 10.1198/016214507000000617. [21] X. Zhang, X. Xiao and G. Xu, Fuzzy support vector machine based on affinity among samples, Journal of Software, 17 (2006), 951-958. doi: 10.1360/jos170951. [22] G. Zhang, S. Wang, Y. Wang and W. Liu, LS-SVM approximate solution for affine nonlinear systems with partially unknown systems, Journal of Industrial and Management Optimization, 10 (2014), 621-636. doi: 10.3934/jimo.2014.10.621.

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##### References:
 [1] L. T. H. An and P. D. Tao, A continuous approach for the concave cost supply problem via DC programming and DCA, Discrete Applied Mathematics, 156 (2008), 325-338. doi: 10.1016/j.dam.2007.03.024. [2] W. An and M. Liang, Fuzzy support vector machine based on within-class scatter for classification problems with outliers or noises, Neurocomputing, 110 (2013), 101-110. doi: 10.1016/j.neucom.2012.11.023. [3] K. Bache and M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science, 2013. Available from: http://archive.ics.uci.edu/ml. [4] M. Bicego and M. A. Figueiredo, Soft clustering using weighted one-class support vector machines, Pattern Recognition, 42 (2009), 27-32. doi: 10.1016/j.patcog.2008.07.004. [5] J. P. Brooks, Support vector machines with the ramp loss and the hard margin loss, Operations Research, 59 (2011), 467-479. doi: 10.1287/opre.1100.0854. [6] H.-G. Chew and C.-C. Lim, On regularisation parameter transformation of support vector machines, Journal of Industrial and Management Optimization, 5 (2009), 403-415. doi: 10.3934/jimo.2009.5.403. [7] I. Dagher, Quadratic kernel-free non-linear support vector machine, Journal of Global Optimization, 41 (2008), 15-30. doi: 10.1007/s10898-007-9162-0. [8] R. A. Fisher, The use of multiple measurements in taxonomic problems, Annals of Human Genetics, 7 (1936), 179-188. doi: 10.1111/j.1469-1809.1936.tb02137.x. [9] X. Jiang, Y. Zhang and J. C. Lv, Fuzzy SVM with a new fuzzy membership function, Neural Computing and Applications, 15 (2006), 268-276. doi: 10.1007/s00521-006-0028-z. [10] T. Joachims, Text categorization with support vector machines: learning with many relevant features, Machine Learning: ECML-98, Lecture Notes in Computer Science, 1398 (1998), 137-142. doi: 10.1007/BFb0026683. [11] S. B. Kazmi, Q. Ain and M. A. Jaffar, Wavelets-based facial expression recognition using a bank of support vector machines, Soft Computing, 16 (2012), 369-379. doi: 10.1007/s00500-011-0721-4. [12] C. F. Lin and S. D. Wang, Fuzzy support vector machines, IEEE Transactions on Neural Networks, 13 (2002), 464-471. [13] Y. Liu and M. Yuan, Reinforced multicategory support vector machines, Journal of Computational and Graphical Statistics, 20 (2011), 901-919. doi: 10.1198/jcgs.2010.09206. [14] J. Luo, Z. Deng, D. Bulatov, J. E. Lavery and S.-C. Fang, Comparison of an $l_1$-regression-based and a RANSAC-based planar segmentation procedure for urban terrain data with many outliers, Image and Signal Processing for Remote Sensing XIX, 8892 (2013), p889209. doi: 10.1117/12.2028627. [15] J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic Surface Support Vector Machine for Binary Classification, Submitted to Neurocomputing, 2014. [16] K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476. doi: 10.3934/jimo.2005.1.465. [17] F. E. H. Tay and L. Cao, Application of support vector machines in financial time series forecasting, Omega, 29 (2001), 309-317. doi: 10.1016/S0305-0483(01)00026-3. [18] V. N. Vapnik, The Nature of Statistical Learning Theory, $2^{nd}$ edition, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4757-3264-1. [19] C. Wu, C. Li and Q. Long, A DC programming approach for sensor network localization with uncertainties in archor positions, Journal of Industrial and Management Optimization, 10 (2014), 817-826. doi: 10.3934/jimo.2014.10.817. [20] Y. Wu and Y. Liu, Robust truncated hinge loss support vector machines, Journal of the American Statistical Association, 102 (2007), 974-983. doi: 10.1198/016214507000000617. [21] X. Zhang, X. Xiao and G. Xu, Fuzzy support vector machine based on affinity among samples, Journal of Software, 17 (2006), 951-958. doi: 10.1360/jos170951. [22] G. Zhang, S. Wang, Y. Wang and W. Liu, LS-SVM approximate solution for affine nonlinear systems with partially unknown systems, Journal of Industrial and Management Optimization, 10 (2014), 621-636. doi: 10.3934/jimo.2014.10.621.
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