4
Application of RBF Neural Network to Enterprise Credit Comprehensive Evaluation Wengao Lou Luoping Kuang College of Management, The University of Shanghai For Science and Technology, Shanghai 200093, China Email: [email protected] [email protected] Abstract-Radial basis function (RBF) neural network possesses the characteristics of fast training speed, reaching the global minimum and escaping from local minimum, et al. Under fewer samples, the RBF neural network model (RBFNN-based model) for credit comprehensive evaluation is established according to actual enterprise's credit state. For the 6 non-training enterprises, the comparison of the calculated results of several models shows that the accuracy of the RBFNN-based model established in this paper is the highest. It is effective and suitable to apply RBFNN to enterprise credit comprehensive evaluation. I. INTRODUCTION The enterprise credit comprehensive evaluation is carried out according to credit characteristics of the enterprise's fmancial indexes, and then the enterprise credit risk was calculated and judged. This process is a process of pattern recognition as well as mathematical approximation. Furthermore, the relationship between the enterprise's financial indexes and credit risk levels is obviously non-linear. For instance, one financial index (net income/gloss assets) increases from -0.1 to 0.1, or from 0.4 to 0.6, the same result will be got by using linear method because the index both changes 0.2. As a matter of fact, the increase from -0.1 to 0.1 means that the company turns losses into profits and gains survival and developmental environment. And the increase from 0.4 to 0.6 just means that the company has an increasing profitability. These two situations are completely different from each other. On the other hand, the artificial neural networks (ANNs) are especially effective to non-linear system modeling. In 1990, Odom firstly introduced ANNs to credit risk evaluation. And scholars abroad compared the applicability of various methods, such as discriminate analysis, cluster analysis, classification tree and ANNs, and got good results for a large number of set data. But at home, due to the incomplete market economy system, and the shortage of the joint-stock company's financial data, it is difficult to obtain adequate financial data to set up reliable credit evaluation model. At the same time, the enterprises and banks urgently need reliable enterprise credit evaluation model under fewer samples to escape from investment risk. At present, under fewer samples, there is no effective and reliable model at home and abroad. Radial basis function neural network (RBFNN) with fast approximating ability and global minimum is applied to enterprise credit comprehensive evaluation in this paper. As a case, the credit levels of 6 actual enterprises are calculated using several methods, and compared with each other. The results show that the RBFNN model possesses the highest precision and best generalization ability under fewer samples. H. ENTERPRISE CREDIT COMPREHENSIVE EVALUATION SYSTEM Enterprise credit score is the comprehensive state of the economy space with different levels and different factors. In study, it is necessary to take account of the representation, reliability, and the selective error of the extracted samples. Ref [7] described the enterprise credit evaluation system as four types with 10 variables which are listed in Table I. In order to compare the results of the various methods, the same data used in Ref [7] are also applied in this paper. There are 30 set data, and 10 evaluation indexes as input variables, one enterprise credit score as output variable. Of the 30 set data, 6 set data of the actual enterprise fmancial data used as test set data in Ref [7] are also used as test set data in this paper. III. ESTABLISHMENT OF THE RADAL BASIS FUNCTION NEURAL NETWORK (RBFNN) MODEL 3.1 The Radial Basis Function Neural Network (RBFNN) Artificial neural networks (ANNs) with excellent non-linear approximation ability quickly developed since 1980s are widely used in non-linear fields such as pattern recognition and approximation, et al. BP neural network (BPNN) has several disadvantages such as existence of local minimum, lower coverage, and hard to decide the reasonable number of neurons on hidden layers, etc. The 0-7803-9422-4/05/$20.00 ©2005 IEEE 1340

[IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Application of RBF

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Page 1: [IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Application of RBF

Application ofRBF Neural Network toEnterprise Credit Comprehensive Evaluation

Wengao Lou Luoping KuangCollege of Management, The University ofShanghai For Science and Technology,

Shanghai 200093, ChinaEmail: [email protected]

[email protected]

Abstract-Radial basis function (RBF) neural networkpossesses the characteristics of fast training speed, reaching theglobal minimum and escaping from local minimum, et al.Under fewer samples, the RBF neural network model(RBFNN-based model) for credit comprehensive evaluation isestablished according to actual enterprise's credit state. For the6 non-training enterprises, the comparison of the calculatedresults of several models shows that the accuracy of theRBFNN-based model established in this paper is the highest. Itis effective and suitable to apply RBFNN to enterprise creditcomprehensive evaluation.

I. INTRODUCTION

The enterprise credit comprehensive evaluation is carriedout according to credit characteristics of the enterprise'sfmancial indexes, and then the enterprise credit risk wascalculated and judged. This process is a process of patternrecognition as well as mathematical approximation.Furthermore, the relationship between the enterprise'sfinancial indexes and credit risk levels is obviouslynon-linear. For instance, one financial index (netincome/gloss assets) increases from -0.1 to 0.1, or from 0.4to 0.6, the same result will be got by using linear methodbecause the index both changes 0.2. As a matter of fact, theincrease from -0.1 to 0.1 means that the company turnslosses into profits and gains survival and developmentalenvironment. And the increase from 0.4 to 0.6 just meansthat the company has an increasing profitability. These twosituations are completely different from each other. On theother hand, the artificial neural networks (ANNs) areespecially effective to non-linear system modeling.

In 1990, Odom firstly introduced ANNs to credit riskevaluation. And scholars abroad compared the applicabilityof various methods, such as discriminate analysis, cluster

analysis, classification tree and ANNs, and got goodresults for a large number of set data. But at home, due tothe incomplete market economy system, and the shortage ofthe joint-stock company's financial data, it is difficult toobtain adequate financial data to set up reliable creditevaluation model. At the same time, the enterprises and

banks urgently need reliable enterprise credit evaluationmodel under fewer samples to escape from investment risk.At present, under fewer samples, there is no effective andreliable model at home and abroad. Radial basis functionneural network (RBFNN) with fast approximating abilityand global minimum is applied to enterprise creditcomprehensive evaluation in this paper. As a case, the creditlevels of 6 actual enterprises are calculated using severalmethods, and compared with each other. The results showthat the RBFNN model possesses the highest precision andbest generalization ability under fewer samples.

H. ENTERPRISE CREDIT COMPREHENSIVE EVALUATIONSYSTEM

Enterprise credit score is the comprehensive state of theeconomy space with different levels and different factors. Instudy, it is necessary to take account of the representation,reliability, and the selective error of the extracted samples.Ref [7] described the enterprise credit evaluation system asfour types with 10 variables which are listed in Table I.

In order to compare the results of the various methods,the same data used in Ref [7] are also applied in this paper.There are 30 set data, and 10 evaluation indexes as inputvariables, one enterprise credit score as output variable. Ofthe 30 set data, 6 set data of the actual enterprise fmancialdata used as test set data in Ref [7] are also used as test setdata in this paper.

III. ESTABLISHMENT OF THE RADAL BASIS FUNCTIONNEURALNETWORK (RBFNN) MODEL

3.1 The Radial Basis Function Neural Network(RBFNN)

Artificial neural networks (ANNs) with excellentnon-linear approximation ability quickly developed since1980s are widely used in non-linear fields such as patternrecognition and approximation, et al. BP neural network(BPNN) has several disadvantages such as existence of localminimum, lower coverage, and hard to decide thereasonable number of neurons on hidden layers, etc. The

0-7803-9422-4/05/$20.00 ©2005 IEEE1340

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establishment ofBPNN is thus time-consuming, tedious andtrivial. On the other hand, the RBFNN has the contrastingadvantages and disadvantages of BPNN. RBFNN cantherefore be trained extremely quickly. The training ofRBFNN does not suffer from local minimum. The RBFNNis thus applied to enterprise credit comprehensive evaluationunder fewer samples in this paper.

TABLE IENTERPRISE CREDIT EVALUATION INDEX SYSTEM

VariableMain aim Sub aim (financial Attention

ratios )earning per

share

Profitability net assets return Investorratiototal assets re

enterprise accoutcredit Management receivablecomprehen ability tumover ratiosive inventoryevaluation turnover ratio Creditorsystem Repayment debt-to-total-assets

capability liquidity ratioAcid-test ratio

Developing market share

capability Capital Governmentappreciation ratio

RBFNN, a typical function approximator and an idealtool for continuous variables, with high training speed, hasan input layer, a hidden layer of radial basis neurons and anoutput layer of linear neurons, its topology shown in Fig. I[1-3]. The centers of the basis function on hidden layer areCj, widths as,, the connections between hidden layer andoutput layerEvjk . In training ofRBFNN, the centers stored inthe radial hidden layer are optimized fast, typically usingK-means, widths (or the spread of the data, or radialdeviation) assigned by Isotropic or K-nearest neighboralgorithm, and the linear output layer optimized usingpseudo-inverse technique.

z.

Fig .1 The Topology ofRBFNN

The training of RBFNN is no fewer prone to discoveringsub-optimum combinations, but it can become overtrainedwhen the number of centers is greater than that is requiredby the samples. The key problem for RBFNN modeling isstill to identify how many neurons should be used on hiddenlayer. The software of STATISTICA Neural Networks isapplied in this study. For this study, the RBFNN model hasone input layer with 10 neurons, one output layer with oneneuron describing enterprise credit score.

Firstly, 30 samples totally as training set data are used totrain the REFNN whose number of hidden neurons is 2, 5, 6,7, 8, 9, 10, 13, 15, 17, 20, 25 and 29, respectively. TheRMSEs of these RBFNN model are 0.126, 0.1159, 0.1132,0.1117, 0.1070, 0.1037, 0.1035, 0.1044, 0.1002, 0.0934,0.0856. 0.0802, 0.0332 and2 x 10-', respectively. It is thusknown that, the RMSEs are almost the same when theneurons are 6, 7, 8, 9, and 10, respectively. The RMSE isnearly zero when the number ofhidden neurons is 29, that is,the number of hidden neurons equals to the number ofsamples -1. The RBFNN model accurately approximates thetraining set data at this time. Taking the complexity of thetopology and the RMSE of the model into considerationcomprehensively, the topology of RBFNN model isreasonable 10-7-1 in this study.

In order to compare the calculated results with that of Ref[7], the same 6 actual enterprise data are selected as test data,and the other 24 samples are used as training samples. TheRBFNN model is retrained according to the foregoingprocedure. The number of neurons on hidden layer is 2, 5, 7,10, 13 and 15 respectively and the RMSEs of training setdata are 0.1476, 0.1285, 0.1257, 0.1140, 0.1054, and 0.0935,respectively. The topology for the RBFNN model isreasonably 10-7-1 because the RMSE of test data reaches theminimum. The RMSE of training set data and test set dataare 0.1257. and 0.0639, respectively.

It is obvious that the RMSE of test set data is muchsmaller than that of the training set data. It is discovered byanalyzing the training sample errors that the -error of sample15 is the greatest, and reaches 0.4883, exceeds 30 (oa is thestandard deviation of the data), the sample 15 is thusregarded as abnormal and eliminated. Retraining the RBFNN,the RMSEs of training set data and test set data are 0.07323and 0.04819, respectively, the average absolute errors (AAEs)are 0.0553 and 0.0440, respectively. These performancesshow that the trained RBFNN model has the similarapproximating ability for training set data as well as test setdata, that is, the RBFNN model has good generalizationability and can be applied to evaluate the enterprise creditscore. In this way, the RBFNN model for actual enterprisecredit comprehensive evaluation is thereby established.

3.2 Modeling the RBFNNComprehensive Credit Evaluation

for EnterpriseIV. VERIFICATION AND ANSLYSIS OF RBFNN MODEL

GENERALIZATION

4.1 The Output ofRBFNNModel

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The value of 10 evaluation variables of the 6 actualenterprises as test set data is inputted into the establishedRBFNN model, and these 6 enterprise credit scores (outputsof the model) are shown in Table 2. The AAE and theMAPE of the 6 enterprise credit scores are 0.0440 and5.51% respectively, compared with the expert evaluationscore, which indicate that the established RBFNN modelexpresses the essence and principle of the enterprise creditcomprehensive evaluation system perfectly.

In enterprise credit comprehensive evaluation, theenterprise credit absolute score as well as the relative levelshould both be studied synchronously. The relative levelandthe order of 6 actual enterprises are calculated, analyzedand also listed in Table II. The order calculated in this paperis totally the same as the order of expert evaluated score.

In addition, if two sections [0.60, 0.80] and [0.80, 1.0] aredivided into and regarded as two credit grades, that is, thethreshold value is 0.80. As to the RBFNN model, the ratioof 6 enterprises' credit level correctly recognized is high as83.3%.

4.2 Comparison and Analysis ofthe Results of VariousModelsThe GA-NN model using genetic algorithm in Ref [7]

improves the traditional BP neural network and promotesthe generalization. The results calculated in this paper iscompared with that of Ref [7], listed in the Table II. In Ref[7], for the 6 enterprises, credit scores of 3 enterprisesexceed 1.0, much higher than expert score and greater thanthe highest possible value, credit scores of 2 enterprisesfewer than 0.60, much lower than the expert score. But iftaking two sections [0.60, 0.80] and [0.80, 1.0] as two creditgrades, and the ratio of 6 enterprises' credit level correctlyrecognized is thus as low as 16.7% in Ref [7]. The AAE andMAPE of the 6 enterprises is 0.2957 and 35.98% forGA-NN model. The order of one enterprise is seriously inwrong position as to that of expert score.Under large number of samples, it was drawn that

Logistic model can gain satisfied results for enterprise creditcomprehensive evaluation. Logistic regression technique isalso used to set up model of the 24 training samples in thispaper. It is also discovered that the 15th sample is anabnormal data and eliminated. And then the Logistic modelfor enterprise credit comprehensive evaluation is establishedbased on the other 23 samples. The credit scores of 6 testenterprises of Logistic model and their order are also listedin Table II. The credit score of one enterprise is fewer than0.60, and the order of 6 enterprises using Logistic model isin good agreement with that of expert score.The credit scores of 6 enterprises and their relative order

of various models are all listed in Table II. In order tofurther verify and study the modeling ability of variousmethodology, for 6 test enterprises, the expert scores and thecalculated scores of various models are linear-regressed andtheir interception A, slope K, correlation coefficient R, AAE,MAPE, maximum absolute error E,,. and ratio of

recognized correct to mistaken are all listed in Table Ill. Theschematic diagram of regression line ofRBFNN model, andGA-NN model and Logistic model are shown in Fig. II.

TABLE IITHE CREDIT SCORESAND THEIR RELATIVE ORDER OFVARIOUS MODELS FOR 6 ACTUAL ENTERPRISES

Model Enterpriseand

credit CA CB CC CD CE CForderExpert 0.69 0.98 0.9 0.8 0.78 0.66score___

Expertscore 5 1 2 3 4 6orderoutputof 0.7364 0.946 0.9 0.8 0.72 0.6402

RBFNNorBFN 4 1 2 3 5 6orderOutputof 0.4947 1.47 1.2 1 1.27 0.4965

GA-NN___GA-N 6 1 3 4 2 5

N order

Outputof 0.6614 0.966 0.9 0.8 0.71 0.5478

Logistic __4_6Logistic 5 1 2 3 4 6order _____ ____

V. DISCUSSIONS AND CONCLUSIONS

5.1 Characteristics ofRBFNNModelingThe RBFNN has a number of advantages over BPNN.

First, using a single hidden layer, RBFNN can model anynon-linear function. Second, the simple linear transformationon the output layer can be optimized fully using traditionallinear modeling techniques, which are fast and do no sufferfrom problems such as local minimum. RBFNN cantherefore be trained extremely quickly. So in the trainingprocess there is not over-fitting problem, and the verificationset data are unnecessary to monitor the training process. Italso has high training speed, but with a large number of inputvariables, the RBFNN model would face severe problems.

5.2 Factors Affecting the RBFNN Model Performanceand its Generalization

Modeling RBFNN, it is necessary to determine thereasonable neurons on hidden layer and to optimize thecenters Cj and widths cj of basis function on hidden layerand the connections between hidden layer and output layer.There are many algorithms to optimize above parameters,and the calculated results are slightly different from eachother. Presently, random selection and self-organizationselection are the main methods for determining thecenters C1 of basis function on hidden layer. Among these

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methods, K-nearest neighbor was widely used. Modelingshows that the widths of basis function on hidden layerdirectly affect the generalization of RBFFNN. The usualcalculation methods are Isotropic and K-nearest optimization.Modeling experiments indicate that K-nearest optimizationhas better generalization than that of Isotropic. Furthermore,the number of neurons on hidden layer greatly influencesRBFNN generalization. The construction algorithm isapplied to determine the reasonable number of neurons onhidden layer in this paper.

TABLE III

THE COMPARISON OF PERFORMANCES OF VARIOUSMODELS

1.6

1.4

1.2

1.0

0. a

0.4

0.2

0.0

the principle and essence of the credit evaluation system atall. Furthermore, changing the training samples, theestablished RBFNN model is changed too. For modelingRBFNN, it is very important to use enough representativeand typical samples, otherwise, the established model mayhave no actual practice meaning. Modeling RBFNN, noverification required, can reach the global minimumabsolutely, but the characteristics of the training sample, thenumber of neurons on hidden layer, and the algorithm tooptimize the basis function centers and widths all haveinfluence on the generalization. The generalization ofRBFNN model can only be appraised and judged bynon-training samples, such as test set data.

5.3 Validations ofthe RBFNN ModelUnder fewer samples, there is no reliable algorithm for

establishing ANN model at home and abroad. The modelingRBFNN of 24 training samples and the testing of 6 actualenterprises show that RBFNN-base model can successfullybe applied to evaluate enterprise credit score under fewersamples and has good generalization too. The rightrecognition ratio and relative credit order are superior to thatofGA-NN model as well as Logistic model.

REFERENCES

[1] Pingfan Yan and Shuichang Zhang. "Aritificial neural network and

_ simulated evolution processs," Beijing: Tsinghua University Press,2000.

[2] Haykin,S. "Principle ofneural network," Beijing: China Machine Press,srE 2004.

[3] Statsoft. "STATISTICA neural networks," Tulsa: Statsoft, Inc., 1999Theoretical [4] Odom, M D and Sharda, R A. "A neural network modelfor bankruptcy

prediction," Proceedings of the IEEE National Joint Conference on

A GA-NE Neural Networks, 1990, 2:163-168[5] West, D. "Neural network credit scoring models," Computers &

X Logistic Operations Research, 2000 (27):1131-1152[6] Piramuthu, S. "Financial credit-risk evaluation with neural and

lN neurofuzzy systems,"European Journal ofOperational Research, 1999,1 12(2):3 10-321

Theorticl [7] Desheng Wu and Liang Liang. "An application ofpattern recognitionon scoring Chinese corporations financial conditions based on

--Theoretical backpropagation neural network," Management Science ofChina,2004,12 (1): 68-76[8] Jingmei Wu. "Capital credit evaluation," Beijing: China Audit Press,

2001.0.0 0.2 0.4 0. 6 0. 1.0 1.2 1.4 1.6

Fig. II Schematic diagram of linear-regression lines ofRBFNN model,GA-NN model and Logistic model

Modeling also shows that the number of neurons onhidden layer is equal to N-I (N is the number of trainingsamples), the RMSE of the model may be as small as zero,but the model may have very poor generalization. For thestudied case, the RMSEs of training set data and the test setdata are 6xio-15 and 2.036. The credit scores of the 6actual enterprises are 1.226, 4.218, 3.672, 2.489, 2.064 and2.065, which are impossible and too greater than the expertscores. The RBFNN model is nonsense and do not describe

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Model ______ Ability index

A K R AAE MAPE Em.RBFNN 0.1017 0.8384 0.9371 0.0440 5.51% 0.0757

GA-NN -1.453 3.021 0.8871 0.2958 35.98% 0.4932

Logistic -0.2874 1.3202 0.9625 0.0504 6.73% 0.1122

Theoretic O 1OKIal ___I _I