American Institute of Mathematical Sciences

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

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.  Google Scholar

[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.  Google Scholar

[3]

K. Bache and M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science,, 2013. Available from: , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. F. Lin and S. D. Wang, Fuzzy support vector machines,, IEEE Transactions on Neural Networks, 13 (2002), 464.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic Surface Support Vector Machine for Binary Classification,, Submitted to Neurocomputing, (2014).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. N. Vapnik, The Nature of Statistical Learning Theory,, $2^{nd}$ edition, (2000).  doi: 10.1007/978-1-4757-3264-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

K. Bache and M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science,, 2013. Available from: , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. F. Lin and S. D. Wang, Fuzzy support vector machines,, IEEE Transactions on Neural Networks, 13 (2002), 464.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic Surface Support Vector Machine for Binary Classification,, Submitted to Neurocomputing, (2014).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. N. Vapnik, The Nature of Statistical Learning Theory,, $2^{nd}$ edition, (2000).  doi: 10.1007/978-1-4757-3264-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]

Huiqin Zhang, JinChun Wang, Meng Wang, Xudong Chen. Integration of cuckoo search and fuzzy support vector machine for intelligent diagnosis of production process quality. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020150
[4]

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
[5]

Jian Luo, Xueqi Yang, Ye Tian, Wenwen Yu. Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2743-2756. doi: 10.3934/jimo.2019078
[6]

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
[7]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2020018
[8]

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
[9]

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
[10]

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
[11]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020024
[12]

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
[13]

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
[14]

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
[15]

Florian Dumpert. Quantitative robustness of localized support vector machines. Communications on Pure & Applied Analysis, 2020, 19 (8) : 3947-3956. doi: 10.3934/cpaa.2020174
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences