# American Institute of Mathematical Sciences

• Previous Article
Supervised distance preserving projection using alternating direction method of multipliers
• JIMO Home
• This Issue
• Next Article
The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications
doi: 10.3934/jimo.2019078

## Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter

 1 School of Management Science and Engineering, Dongbei University of Finance and Economics, Dalian 116025, China 2 School of Business Administration and Collaborative Innovation Center of Financial Security, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author

Received  August 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by NNSFC grant # 71701035 and # 71831003.

Nowadays, the effective credit scoring becomes a very crucial factor for gaining competitive advantages in credit market for both customers and corporations. In this paper, we propose a credit scoring method which combines the non-kernel fuzzy 2-norm quadratic surface SVM model, T-test feature weighting strategy and fuzzy within-class scatter together. It is worth pointing out that this new method not only saves computational time by avoiding choosing a kernel and corresponding parameters in the classical SVM models, but also addresses the "curse of dimensionality" issue and improves the robustness. Besides, we develop an efficient way to calculate the fuzzy membership of each training point by solving a linear programming problem. Finally, we conduct several numerical tests on two benchmark data sets of personal credit and one real-world data set of corporation credit. The numerical results strongly demonstrate that the proposed method outperforms eight state-of-the-art and commonly-used credit scoring methods in terms of accuracy and robustness.

Citation: 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, doi: 10.3934/jimo.2019078
##### References:
 [1] 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.  Google Scholar [2] K. Bache and M. Lichman, Uci machine learning repository, http://archive.ics.uci.edu/ml, 2013. Google Scholar [3] Y. Bai, X. Han, T. Chen and H. Yu, Quadratic kernel-free least squares support vector machine for target diseases classification, Journal of Combinatorial Optimization, 30 (2015), 850-870.  doi: 10.1007/s10878-015-9848-z.  Google Scholar [4] G. Baudat and F. Anouar, Generalized discriminant analysis using a kernel approach, Neural Computation, 12 (2000), 2385-2404.  doi: 10.1162/089976600300014980.  Google Scholar [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar [6] 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.  Google Scholar [7] R. 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.  Google Scholar [8] T. Gestel, B. Baesens and J. Garcia, A support vector machine approach to credit scoring, Journal of Bank and Finance, 2 (2003), 73-82.   Google Scholar [9] J. Han and M. Kamber, Data Mining: Concepts and Techniques, 2nd edition, Morgan Kaufmann, San Francisco, CA, 2006. Google Scholar [10] L. Han and H. Zhao, Orthogonal support vector machine for credit scoring, Engineering Applications of Artificial Intelligence, 26 (2013), 848-862.  doi: 10.1016/j.engappai.2012.10.005.  Google Scholar [11] T. Harris, Credit scoring using the clustered support vector machine, Expert Systems with Applications, 42 (2015), 741-750.  doi: 10.1016/j.eswa.2014.08.029.  Google Scholar [12] C. Huang, M. Chen and C. Wang, Credit scoring with a data mining approach based on support vector machines, Expert Systems with Applications, 33 (2007), 847-856.  doi: 10.1016/j.eswa.2006.07.007.  Google Scholar [13] X. Jiang, Y. Zhang and J. Lv, Fuzzy svm with a new fuzzy membership function, Neural Computing and Applications, 15 (2006), 268-276.  doi: 10.1007/s00521-006-0028-z.  Google Scholar [14] C. Lin and S. Wang, Fuzzy support vector machines, IEEE Transactions on Neural Networks, 13 (2002), 464-471.   Google Scholar [15] F. Liu and X. Xue, Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels, Journal of Industrial and Management Optimization, 12 (2016), 285-301.  doi: 10.3934/jimo.2016.12.285.  Google Scholar [16] J. Luo, S.-C. Fang, Y. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373.  doi: 10.3934/jimo.2016.12.357.  Google Scholar [17] J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic surface support vector machine for binary classification, Asia-Pacific Journal Of Operational Research, 33 (2016), 1650046. doi: 10.1142/S0217595916500469.  Google Scholar [18] A. Marques, V. Garcia and J. Sanchez, On the suitability of resampling techniques for the class imbalance problem in credit scoring, Journal of the Operational Research Society, 64 (2013), 1060-1070.  doi: 10.1057/jors.2012.120.  Google Scholar [19] D. Martin, Early warning of bank failure: a logistic regression approach, Journal of Banking and Finance, 1 (1977), 249-276.   Google Scholar [20] B. Schölkopf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, Cambridge, MA, 2002.  doi: 10.1016/B978-044451378-6/50001-6.  Google Scholar [21] Y. Tian, M. Sun, Z. Deng, J. Luo and Y. Li, A new fuzzy set and non-kernel svm approach for mislabeled binary classification with applications, IEEE Transactions on Fuzzy Systems, 25 (2017), 1536-1545.   Google Scholar [22] W. Tunga, C. Queka and P. Cheng, Genso-ews: A novel neural-fuzzy based early warning system for predicting bank failures, Neural Networks, 17 (2004), 567-587.  doi: 10.1016/j.neunet.2003.11.006.  Google Scholar [23] J. Wiginton, A note on the comparison of logic and discriminant models of customer credit behavior, Journal of Financial and Quantitative Analysis, 15 (1980), 757-770.   Google Scholar [24] X. Yan, Y. Bai, S.-C. Fang and J. Luo, A kernel-free quadratic surface support vector machine for semi-supervised learning, Journal of the Operational Research Society, 67 (2016), 1001-1011.  doi: 10.1007/s10957-015-0843-4.  Google Scholar [25] 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.  Google Scholar [26] H. Zhong, C. Miao, Z. Shen and Y. Feng, Comparing the learning effectiveness of BP, ELM, I-ELM, and SVM for corporate credit ratings, Neurocomputing, 128 (2014), 285-295.  doi: 10.1016/j.neucom.2013.02.054.  Google Scholar [27] L. Zhou, K. Lai and J. Yen, Credit scoring models with auc maximization based on weighted svm, International Journal of Information Technology and Decision Making, 4 (2009), 677-696.  doi: 10.1142/S0219622009003582.  Google Scholar

show all references

##### References:
 [1] 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.  Google Scholar [2] K. Bache and M. Lichman, Uci machine learning repository, http://archive.ics.uci.edu/ml, 2013. Google Scholar [3] Y. Bai, X. Han, T. Chen and H. Yu, Quadratic kernel-free least squares support vector machine for target diseases classification, Journal of Combinatorial Optimization, 30 (2015), 850-870.  doi: 10.1007/s10878-015-9848-z.  Google Scholar [4] G. Baudat and F. Anouar, Generalized discriminant analysis using a kernel approach, Neural Computation, 12 (2000), 2385-2404.  doi: 10.1162/089976600300014980.  Google Scholar [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar [6] 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.  Google Scholar [7] R. 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.  Google Scholar [8] T. Gestel, B. Baesens and J. Garcia, A support vector machine approach to credit scoring, Journal of Bank and Finance, 2 (2003), 73-82.   Google Scholar [9] J. Han and M. Kamber, Data Mining: Concepts and Techniques, 2nd edition, Morgan Kaufmann, San Francisco, CA, 2006. Google Scholar [10] L. Han and H. Zhao, Orthogonal support vector machine for credit scoring, Engineering Applications of Artificial Intelligence, 26 (2013), 848-862.  doi: 10.1016/j.engappai.2012.10.005.  Google Scholar [11] T. Harris, Credit scoring using the clustered support vector machine, Expert Systems with Applications, 42 (2015), 741-750.  doi: 10.1016/j.eswa.2014.08.029.  Google Scholar [12] C. Huang, M. Chen and C. Wang, Credit scoring with a data mining approach based on support vector machines, Expert Systems with Applications, 33 (2007), 847-856.  doi: 10.1016/j.eswa.2006.07.007.  Google Scholar [13] X. Jiang, Y. Zhang and J. Lv, Fuzzy svm with a new fuzzy membership function, Neural Computing and Applications, 15 (2006), 268-276.  doi: 10.1007/s00521-006-0028-z.  Google Scholar [14] C. Lin and S. Wang, Fuzzy support vector machines, IEEE Transactions on Neural Networks, 13 (2002), 464-471.   Google Scholar [15] F. Liu and X. Xue, Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels, Journal of Industrial and Management Optimization, 12 (2016), 285-301.  doi: 10.3934/jimo.2016.12.285.  Google Scholar [16] J. Luo, S.-C. Fang, Y. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373.  doi: 10.3934/jimo.2016.12.357.  Google Scholar [17] J. Luo, S.-C. Fang, Z. Deng and X. Guo, Quadratic surface support vector machine for binary classification, Asia-Pacific Journal Of Operational Research, 33 (2016), 1650046. doi: 10.1142/S0217595916500469.  Google Scholar [18] A. Marques, V. Garcia and J. Sanchez, On the suitability of resampling techniques for the class imbalance problem in credit scoring, Journal of the Operational Research Society, 64 (2013), 1060-1070.  doi: 10.1057/jors.2012.120.  Google Scholar [19] D. Martin, Early warning of bank failure: a logistic regression approach, Journal of Banking and Finance, 1 (1977), 249-276.   Google Scholar [20] B. Schölkopf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, Cambridge, MA, 2002.  doi: 10.1016/B978-044451378-6/50001-6.  Google Scholar [21] Y. Tian, M. Sun, Z. Deng, J. Luo and Y. Li, A new fuzzy set and non-kernel svm approach for mislabeled binary classification with applications, IEEE Transactions on Fuzzy Systems, 25 (2017), 1536-1545.   Google Scholar [22] W. Tunga, C. Queka and P. Cheng, Genso-ews: A novel neural-fuzzy based early warning system for predicting bank failures, Neural Networks, 17 (2004), 567-587.  doi: 10.1016/j.neunet.2003.11.006.  Google Scholar [23] J. Wiginton, A note on the comparison of logic and discriminant models of customer credit behavior, Journal of Financial and Quantitative Analysis, 15 (1980), 757-770.   Google Scholar [24] X. Yan, Y. Bai, S.-C. Fang and J. Luo, A kernel-free quadratic surface support vector machine for semi-supervised learning, Journal of the Operational Research Society, 67 (2016), 1001-1011.  doi: 10.1007/s10957-015-0843-4.  Google Scholar [25] 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.  Google Scholar [26] H. Zhong, C. Miao, Z. Shen and Y. Feng, Comparing the learning effectiveness of BP, ELM, I-ELM, and SVM for corporate credit ratings, Neurocomputing, 128 (2014), 285-295.  doi: 10.1016/j.neucom.2013.02.054.  Google Scholar [27] L. Zhou, K. Lai and J. Yen, Credit scoring models with auc maximization based on weighted svm, International Journal of Information Technology and Decision Making, 4 (2009), 677-696.  doi: 10.1142/S0219622009003582.  Google Scholar
Credit Data Sets
 data set # of features Class $C_1$ Class $C_2$ name # of points name # of points German 20 Creditworthy 700 Non-creditworthy 300 Australian 14 Non-default 383 Default 307 Chinese 7 Good credit 58 Bad credit 48
 data set # of features Class $C_1$ Class $C_2$ name # of points name # of points German 20 Creditworthy 700 Non-creditworthy 300 Australian 14 Non-default 383 Default 307 Chinese 7 Good credit 58 Bad credit 48
German Credit Data Test
 model misclassification rate (%) CPU time (s) mean std LOG_REG 23.04 0.35 0.14 FFBP_NN 24.30 0.57 3.83 SVM_GausKer 24.31 0.71 3.30 W2NSVM_GausKer 23.85 0.56 5.72 W2NSVM_QuadKer 23.92 0.81 5.36 FSVMWCS_GausKer 23.42 1.84 6.87 Clu_SVM 24.49 0.71 0.25 Dagher's QSVM 24.26 0.62 4.63 SQSSVM 23.86 0.59 2.82 FNKSVM-FWS 21.36 0.51 4.23
 model misclassification rate (%) CPU time (s) mean std LOG_REG 23.04 0.35 0.14 FFBP_NN 24.30 0.57 3.83 SVM_GausKer 24.31 0.71 3.30 W2NSVM_GausKer 23.85 0.56 5.72 W2NSVM_QuadKer 23.92 0.81 5.36 FSVMWCS_GausKer 23.42 1.84 6.87 Clu_SVM 24.49 0.71 0.25 Dagher's QSVM 24.26 0.62 4.63 SQSSVM 23.86 0.59 2.82 FNKSVM-FWS 21.36 0.51 4.23
Australian Credit Data Test
 model misclassification rate (%) CPU time (s) mean std LOG_REG 13.56 0.27 0.12 FFBP_NN 14.42 1.16 2.72 SVM_GausKer 15.00 1.06 1.30 W2NSVM_GausKer 14.87 0.53 2.73 W2NSVM_QuadKer 14.59 0.46 3.01 FSVMWCS_GausKer 14.63 3.68 3.75 Clu_SVM 14.34 0.53 0.16 Dagher's QSVM 26.42 1.23 1.63 SQSSVM 14.57 0.57 0.80 FNKSVM-FWS 11.96 0.43 1.56
 model misclassification rate (%) CPU time (s) mean std LOG_REG 13.56 0.27 0.12 FFBP_NN 14.42 1.16 2.72 SVM_GausKer 15.00 1.06 1.30 W2NSVM_GausKer 14.87 0.53 2.73 W2NSVM_QuadKer 14.59 0.46 3.01 FSVMWCS_GausKer 14.63 3.68 3.75 Clu_SVM 14.34 0.53 0.16 Dagher's QSVM 26.42 1.23 1.63 SQSSVM 14.57 0.57 0.80 FNKSVM-FWS 11.96 0.43 1.56
Chinese Credit Data Test
 model misclassification rate (%) CPU time (s) mean std LOG_REG 7.56 0.57 0.235 FFBP_NN 24.01 2.25 4.412 SVM_GausKer 13.75 0.90 0.034 W2NSVM_GausKer 12.13 1.89 0.053 W2NSVM_QuadKer 12.07 2.01 0.062 FSVMWCS_GausKer 21.18 2.88 0.063 Clu_SVM 10.96 0.55 0.048 Dagher's QSVM 11.24 2.33 0.087 SQSSVM 10.87 1.96 0.056 FNKSVM-FWS 8.50 0.51 0.083
 model misclassification rate (%) CPU time (s) mean std LOG_REG 7.56 0.57 0.235 FFBP_NN 24.01 2.25 4.412 SVM_GausKer 13.75 0.90 0.034 W2NSVM_GausKer 12.13 1.89 0.053 W2NSVM_QuadKer 12.07 2.01 0.062 FSVMWCS_GausKer 21.18 2.88 0.063 Clu_SVM 10.96 0.55 0.048 Dagher's QSVM 11.24 2.33 0.087 SQSSVM 10.87 1.96 0.056 FNKSVM-FWS 8.50 0.51 0.083
Robustness of Models on Australian Credit Data
 model mean of misclassification rates (%) without outliers with outliers LOG_REG 13.56 17.87 FFBP_NN 14.42 15.94 SVM_GausKer 15.00 15.80 W2NSVM_GausKer 14.87 15.65 W2NSVM_QuadKer 14.59 15.36 FSVMWCS_GausKer 14.63 18.43 Clu_SVM 14.34 17.84 Dagher's QSVM 26.42 53.21 SQSSVM 14.57 15.58 FNKSVM-FWS 11.96 12.61
 model mean of misclassification rates (%) without outliers with outliers LOG_REG 13.56 17.87 FFBP_NN 14.42 15.94 SVM_GausKer 15.00 15.80 W2NSVM_GausKer 14.87 15.65 W2NSVM_QuadKer 14.59 15.36 FSVMWCS_GausKer 14.63 18.43 Clu_SVM 14.34 17.84 Dagher's QSVM 26.42 53.21 SQSSVM 14.57 15.58 FNKSVM-FWS 11.96 12.61
 [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] Zixue Guo, Fengxuan Song, Yumeng Zheng, Zefang He. An improved fuzzy linear weighting method of multi-objective programming problems and its application. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020175 [3] Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457 [4] Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020046 [5] Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial & Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581 [6] Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082 [7] Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699 [8] Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118 [9] Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619 [10] Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008 [11] Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117 [12] Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611 [13] Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099 [14] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [15] Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007 [16] Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157 [17] Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158 [18] George A. Anastassiou. Fractional Ostrowski-Sugeno Fuzzy univariate inequalities. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020111 [19] Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2017, 2 (5) : 1-20. doi: 10.3934/bdia.2017018 [20] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

2018 Impact Factor: 1.025

## Tools

Article outline

Figures and Tables

[Back to Top]