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November  2020, 16(6): 2743-2756. doi: 10.3934/jimo.2019078

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

Jian Luo 1, , Xueqi Yang 1, , Ye Tian 2,, and  Wenwen Yu 1,

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.

Keywords: Credit scoring, non-kernel SVM, fuzzy membership function, feature weighting, fuzzy within-class scatter.
Mathematics Subject Classification: Primary: 62H30; Secondary: 90C20.
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, 2020, 16 (6) : 2743-2756. 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

Table 1.  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
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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
Table 2.  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
Table Options

Download as excel

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
Table 3.  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
Table Options

Download as excel

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
Table 4.  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
Table Options

Download as excel

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
Table 5.  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
Table Options

Download as excel

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
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