Fuzzy quadratic surface support vector machine based on fisher discriminant analysis

Previous Article
A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain
JIMO Home
This Issue
Next Article
A criterion for an approximation global optimal solution based on the filled functions

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

Fuzzy quadratic surface support vector machine based on fisher discriminant analysis

Jian Luo ^1, , Shu-Cherng Fang ^2, , Yanqin Bai ^3, and Zhibin Deng ^4,

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

Abstract
Related Papers

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.

Keywords: kernel-free SVM, Support vector machine, quadratic surface., Fisher discriminant analysis, fuzzy membership function.

Mathematics Subject Classification: Primary: 62H30; Secondary: 90C2.

Citation: Jian Luo, Shu-Cherng Fang, Yanqin Bai, Zhibin Deng. Fuzzy quadratic surface support vector machine based on fisher discriminant analysis. Journal of Industrial & 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. 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. 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: , ().
[4]	M. Bicego and M. A. Figueiredo, Soft clustering using weighted one-class support vector machines,, Pattern Recognition, 42 (2009), 27. 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. 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. 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. 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. 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. 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, 1398* (1998), 137. 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. 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.
[13]	Y. Liu and M. Yuan, Reinforced multicategory support vector machines,, Journal of Computational and Graphical Statistics, 20 (2011), 901. 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). 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. 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. doi: 10.1016/S0305-0483(01)00026-3.
[18]	V. N. Vapnik, The Nature of Statistical Learning Theory,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.3934/jimo.2014.10.621.

show all references

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. 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. 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: , ().
[4]	M. Bicego and M. A. Figueiredo, Soft clustering using weighted one-class support vector machines,, Pattern Recognition, 42 (2009), 27. 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. 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. 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. 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. 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. 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, 1398* (1998), 137. 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. 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.
[13]	Y. Liu and M. Yuan, Reinforced multicategory support vector machines,, Journal of Computational and Graphical Statistics, 20 (2011), 901. 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). 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. 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. doi: 10.1016/S0305-0483(01)00026-3.
[18]	V. N. Vapnik, The Nature of Statistical Learning Theory,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.3934/jimo.2014.10.621.

[1]	Ye Tian, Wei Yang, Gene Lai, Menghan Zhao. Predicting non-life insurer's insolvency using non-kernel fuzzy quadratic surface support vector machines. Journal of Industrial & Management Optimization, 2019, 15 (2) : 985-999. doi: 10.3934/jimo.2018081
[2]	Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial & Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529
[3]	Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial & Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611
[4]	Wenjuan Jia, Yingjie Deng, Chenyang Xin, Xiaodong Liu, Witold Pedrycz. A classification algorithm with Linear Discriminant Analysis and Axiomatic Fuzzy Sets. Mathematical Foundations of Computing, 2019, 2 (1) : 73-81. doi: 10.3934/mfc.2019006
[5]	Xin Li, Ziguan Cui, Linhui Sun, Guanming Lu, Debnath Narayan. Research on iterative repair algorithm of Hyperchaotic image based on support vector machine. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1199-1218. doi: 10.3934/dcdss.2019083
[6]	Ning Lu, Ying Liu. Application of support vector machine model in wind power prediction based on particle swarm optimization. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1267-1276. doi: 10.3934/dcdss.2015.8.1267
[7]	Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[8]	Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
[9]	Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[10]	Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
[11]	K. Schittkowski. Optimal parameter selection in support vector machines. Journal of Industrial & Management Optimization, 2005, 1 (4) : 465-476. doi: 10.3934/jimo.2005.1.465
[12]	Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial & Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031
[13]	Hong-Gunn Chew, Cheng-Chew Lim. On regularisation parameter transformation of support vector machines. Journal of Industrial & Management Optimization, 2009, 5 (2) : 403-415. doi: 10.3934/jimo.2009.5.403
[14]	Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure & Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379
[15]	Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[16]	Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013
[17]	Keiji Tatsumi, Masashi Akao, Ryo Kawachi, Tetsuzo Tanino. Performance evaluation of multiobjective multiclass support vector machines maximizing geometric margins. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 151-169. doi: 10.3934/naco.2011.1.151
[18]	M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.
[19]	Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[20]	Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018157

2017 Impact Factor: 0.994

American Institute of Mathematical Sciences