# Developing an Intelligent Model Based on Fuzzy Multilayer Neural Network for Improving Credit Risk Management under Uncertain Conditions

Authors: Zohreh Ghasemi; Mozhdeh Afshar Kermani; Tofigh Allahviranloo
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This study utilizes the advantages of soft calculations, represented as intelligent combined methods and computational intelligence methods, to enhance credit risk management. For this purpose, the proposed method contains the fuzzy regression and artificial neural network (ANN). In this way, the parameters of neural network are fuzzy, encompassing weights and errors to model under uncertain conditions. Then, fuzzy neural networks form the system where the optimal decision is obtained using the highest degree of superiority by fuzzy inferences. Finally, using the credit information of some countries, the efficiency of the proposed combined model in credit scoring analysis has been shown.


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## Body

## 1. Introduction

Regarding the nature of activity, banks encounter several types of risks from the very beginning, such as credit, market, asset, operational, liquidity, and interest rate risks, such that they have tried to identify and manage these risks, even under incoherent circumstances. Nonetheless, concerning the widespread and variety of banks’ activities, as well as entering the diverse range of complex financial and credit services, the risk management problem has been paid attention to as one of the management issues in large and long-term decision-making and also in daily activities.Among the several dimensions of risk management, credit ranking and scoring are one of the fundamental basics. This is because the first role of the bank in the financial markets is to gather deposits and then grant them in the form of bank facilities to applicants in order to obtain income, growth, and economic development in communities [1]. In this regard, credit scoring was first introduced as an accessible tool to commit microcredits such as mortgages, credit cards, installment loans, and microcommercial credit. In recent years, these models have been employed to control and manage default risk in the credit portfolio of a financial institution such as firms, banks, financial institutions, and project financing.Here, it is worth noting that the credit scoring models are not only utilized for credit approval but also in other fields such as pricing, storage, regulatory capital calculation, and securities. Today, considering the widespread success of the application of scoring in credit environment, financial institutions have begun to employ the scoring methods based on a range of other business objectives. The scoring program assesses the borrower’s risk when allocating the credit. When a loan defaults, the scoring system is used to support the scores received and then update the risk parameters. In this regard, many statistical methods have been successfully developed in terms of analysis of credit scoring problems.Some of them are logistic regression [2], linear separation analysis [3], and second degree separation analysis [4]. Artificial intelligence methods such as support vector machines [5], ANNs (ANN) [6], decision trees [7], genetic algorithm [8], and data envelopment analysis method [9] as another group of classification models have a successful performance in the analysis of scoring problems. It is suggested that although statistical methods and artificial intelligence are considered accurate and efficient methods, they are both deterministic models using classical logic. Thus, in a situation where we encountered with complexity and uncertainty, models will not be efficient [10]. Therefore, in such a situation, predicting and fuzzy classification models such as fuzzy linear regression models and fuzzy time series can be useful, even though the performance of such models will not always be satisfactory [11].Many researchers believe that the combination of models can lead to improved efficiency. Based on this, in previous research studies, in order to cope with the gaps of single models as well as to enhance the efficiency of classification models, the use of combined models was considered common. The purpose of combining models is to decrease the risk of improper models as well as to achieve effective results. In other words, the main motivation for using combined models is the result of the fact that each of the models mentioned here alone does not have sufficient ability to diagnose the data production process and recognize the characteristics of the data. Recently, several hybrid classification models have been developed to raise accuracy and efficiency, some of which will be briefly mentioned in the next section.This study develops a novel method by utilizing fuzzy inference and ANN in order to enhance the quality of financial decisions in terms of granting credit points to customers in banks and financial institutions. To do so, the proposed system is first formed using the ANN, and then, the optimal policy is specified using the highest degree of superiority by fuzzy inference. By implementing the proposed method, the achieved result reveals both the high accuracy and the method efficiency to analyze credit scoring problems.The following sections are presented as follows: In Section 2, the subject literature is briefly mentioned. In Section 3, the basic concepts are described. In Section 4, the integrated model of fuzzy regression and ANNs is demonstrated. In Section 5, some explanations are provided about the credit information of Japan and Australia used in this study. Section 6 addresses the use of the integrated model for credit scoring and customer classification into two groups: creditworthy and unworthy, and finally, the conclusion is provided.


## 2. Subject Literature

The literature on the subject of hybrid models is very extensive, and since the previous research studies have been conducted until now, many studies have been done in this field. In this section, a number of hybrid classification models that use fuzzy logic or ANNs are described in the analysis and evaluation of financial markets, especially credit scoring. In a study, Akkoc [12] compared common econometric models, adaptive neural-fuzzy inference systems (ANFISs), and ANNs for credit scoring analysis. He came to the conclusion that the adaptive neural-fuzzy inference system is more efficient than the artificial neural network and other traditional econometric models in order to provide an accurate interpretation of the decision-making process. In their study, Zhao et al. [13] improved the credit scoring models through the multilayer perceptron neural network and a new method called random selection of the mean, with the optimization approach of data distribution. Sohn and his colleagues [14] modeled the problems of credit assessment using explanatory variables and fuzzy logistic regression. Alaraj and Abbod [15] presented a composite credit scoring model in order to enhance its accuracy. In this model, combined methods were used both in the data preprocessing phase and in the customer classification phase. Neto and his colleagues [16] used computational intelligence tools and classical models to create a framework of calculations based on model-oriented approaches for the purpose of preprocessing large amounts of information.


## 3. Basic Concepts

### 3.1. ANNs

ANN is well known as one of the methods to estimate multiple nonlinear problems, which is a flexible computational framework for several nonlinear problems. Here, it should be mentioned that as ANN is a universal approximator, it is one of its obvious benefits over other nonlinear models, which can approximate any type of function via desired accuracy [17]. Moreover, the parallel processing of data and information is an advantage of neural network. Note that such networks do not need the model shape to form the process and contains a data-based model.Here, it is worthwhile to mention that one of the most useful models of neural networks is the leading neural network via a hidden layer to predict time series [17]. Such models consist of three simple layers of connected information processing. In these networks, the relationship between inputs yt−1,…,yt−p and output yt is as follows:(1)yt=fb0+∑j=1qωj.∑j=1pωi,j.yt−i+εt,where ωj and ωi,j are the model parameters, represented as the connection weights. q is the number of hidden nodes, while p is the number of input nodes, denoting the output layer activation and middle layer functions, respectively. Note that linear functions and sigmoid equation (1) are among the functions employed for the hidden and output layers of the cord:(2)Sigx=11+exp−x.It should be mentioned that two models of ANN’s equation (1) will be utilized as a nonlinear mapping from past observations to future values:(3)yt=fyt−1,…,yt−p,ω+εt,where f is a function determined by neural network structure and the connection weights, while ω is the vector of all parameters. As such, the network can be equivalent to a nonlinear autoregression model. Moreover, equation (3) provides that a tron (output) on the output side is exploited to predict a step.


### 3.2. Fuzzy Logic

Definition 1.
The parametric form of a fuzzy number using the ordered pair of functions asa1α,b1α is defined as follows: whereas, 0≤α≤1, in which these conditions are satisfied as follows:(1)
a1α is a bounded ascending function that is continuous from the right of the interval 0,1(2)
b1α is a bounded descending function that is continuous from the left of the interval 0,1.Definition 2.
If A and B denote the fuzzy numbers ofAα=a1α,a2α and Bα=b1α,b2α, where α∈0,1, the fuzzy operations between them can be described as follows [18]:(4)A+Bα=a1α+b1α,a2α+b2α,−Aα=−a2α,−a1α,A−Bα=a1α−b2α,a2α−b1α,λAα=λa1α,λa2α,λ≥0,λAα=−λa2α,−λa1α,λ<0.


### 3.3. Fuzzy Regression

Classical regression includes strong assumptions about the statistical features of regression models. For example, the nonexistence or normality of autocorrelation and the fixed error variance are the hypotheses that violating them would invalidate the results of classical regression. Note that justifying these assumptions is difficult or cannot properly be employed in some cases, such as human observations, and judgments may be influential in the definitions or observations of a system, or the inaccuracy and inadequacy of information in terms of variables. Generally, although classical regression has several applications, it would be misleading as follows:(1)
The number of observations is insufficient(2)
Errors do not have the normal distribution(3)
The relationship between dependent and independent variables is ambiguous(4)
An ambiguity exists about an event(5)
The linearization assumptions are incorrect.Once classical regression methodology and justifying its assumptions are difficult, the use of fuzzy regression can raise the understanding of the system and provide better results, which represent a membership function with a possible distribution for inaccuracy or ambiguity [1]. The basic concept of fuzzy regression is that the error terms are not generated from the residuals between the estimated values and the original values, but are used in the uncertainty of the model parameters and the possibility of distribution in relation to real observations. Of course, many fuzzy regression models have been developed. This paper employs a fuzzy possibilistic regression model to obtain the best regression equation by reducing the degree of fuzzyness. For this purpose, an optimization model should be developed to obtain a good fit.Regarding the fact that the membership functions of fuzzy numbers are triangular, fuzzy regression can be generated in the form of a linear programming problem. To do so, the input and output of observations are nonfuzzy numbers, whereas the computational parameters are fuzzy numbers. Besides, the membership function of the regression model coefficients is symmetric triangular fuzzy numbers:(5)μβ˜iβi=1−αi−βici,αi−ci≤βi≤αi+ci,0,otherwise,where μβ˜iβi is a function of fuzzy set membership of the regression coefficients. αi and ci are the center and width of the membership function, respectively. Thus, A˜i=αi,ci [19]. Accordingly, the regression relationship is as follows:(6)Y˜=β˜0+β˜1X1+⋯+β˜nXn+ε.


## 3.1. ANNs

ANN is well known as one of the methods to estimate multiple nonlinear problems, which is a flexible computational framework for several nonlinear problems. Here, it should be mentioned that as ANN is a universal approximator, it is one of its obvious benefits over other nonlinear models, which can approximate any type of function via desired accuracy [17]. Moreover, the parallel processing of data and information is an advantage of neural network. Note that such networks do not need the model shape to form the process and contains a data-based model.Here, it is worthwhile to mention that one of the most useful models of neural networks is the leading neural network via a hidden layer to predict time series [17]. Such models consist of three simple layers of connected information processing. In these networks, the relationship between inputs yt−1,…,yt−p and output yt is as follows:(1)yt=fb0+∑j=1qωj.∑j=1pωi,j.yt−i+εt,where ωj and ωi,j are the model parameters, represented as the connection weights. q is the number of hidden nodes, while p is the number of input nodes, denoting the output layer activation and middle layer functions, respectively. Note that linear functions and sigmoid equation (1) are among the functions employed for the hidden and output layers of the cord:(2)Sigx=11+exp−x.It should be mentioned that two models of ANN’s equation (1) will be utilized as a nonlinear mapping from past observations to future values:(3)yt=fyt−1,…,yt−p,ω+εt,where f is a function determined by neural network structure and the connection weights, while ω is the vector of all parameters. As such, the network can be equivalent to a nonlinear autoregression model. Moreover, equation (3) provides that a tron (output) on the output side is exploited to predict a step.


## 3.2. Fuzzy Logic

Definition 1.
The parametric form of a fuzzy number using the ordered pair of functions asa1α,b1α is defined as follows: whereas, 0≤α≤1, in which these conditions are satisfied as follows:(1)
a1α is a bounded ascending function that is continuous from the right of the interval 0,1(2)
b1α is a bounded descending function that is continuous from the left of the interval 0,1.Definition 2.
If A and B denote the fuzzy numbers ofAα=a1α,a2α and Bα=b1α,b2α, where α∈0,1, the fuzzy operations between them can be described as follows [18]:(4)A+Bα=a1α+b1α,a2α+b2α,−Aα=−a2α,−a1α,A−Bα=a1α−b2α,a2α−b1α,λAα=λa1α,λa2α,λ≥0,λAα=−λa2α,−λa1α,λ<0.


## 3.3. Fuzzy Regression

Classical regression includes strong assumptions about the statistical features of regression models. For example, the nonexistence or normality of autocorrelation and the fixed error variance are the hypotheses that violating them would invalidate the results of classical regression. Note that justifying these assumptions is difficult or cannot properly be employed in some cases, such as human observations, and judgments may be influential in the definitions or observations of a system, or the inaccuracy and inadequacy of information in terms of variables. Generally, although classical regression has several applications, it would be misleading as follows:(1)
The number of observations is insufficient(2)
Errors do not have the normal distribution(3)
The relationship between dependent and independent variables is ambiguous(4)
An ambiguity exists about an event(5)
The linearization assumptions are incorrect.Once classical regression methodology and justifying its assumptions are difficult, the use of fuzzy regression can raise the understanding of the system and provide better results, which represent a membership function with a possible distribution for inaccuracy or ambiguity [1]. The basic concept of fuzzy regression is that the error terms are not generated from the residuals between the estimated values and the original values, but are used in the uncertainty of the model parameters and the possibility of distribution in relation to real observations. Of course, many fuzzy regression models have been developed. This paper employs a fuzzy possibilistic regression model to obtain the best regression equation by reducing the degree of fuzzyness. For this purpose, an optimization model should be developed to obtain a good fit.Regarding the fact that the membership functions of fuzzy numbers are triangular, fuzzy regression can be generated in the form of a linear programming problem. To do so, the input and output of observations are nonfuzzy numbers, whereas the computational parameters are fuzzy numbers. Besides, the membership function of the regression model coefficients is symmetric triangular fuzzy numbers:(5)μβ˜iβi=1−αi−βici,αi−ci≤βi≤αi+ci,0,otherwise,where μβ˜iβi is a function of fuzzy set membership of the regression coefficients. αi and ci are the center and width of the membership function, respectively. Thus, A˜i=αi,ci [19]. Accordingly, the regression relationship is as follows:(6)Y˜=β˜0+β˜1X1+⋯+β˜nXn+ε.


## 4. The Formulation of the Proposed Model

It should be noted that although ANNs are accurate predictive models, they have restricted past data to achieve accurate results. Recently, due to the lack of environmental uncertainty as well as the rapid development of new technologies, predicting future situations can be necessary using low data and in short time frames. As a result, forecasting methods are required that need less data and are efficient in such conditions. In this regard, although the fuzzy regression model is an appropriate predictive model in terms of the limited available data, it does not always work well.This study exploits the benefits of fuzzy regression in order to predict time series as well as to handle the limitation of needing large data in ANN methods. Its parameters (i.e., weights and biases) are definite values (b0، b0,j، ωj و ωi,j), while in the proposed method, the factors of the final layer are considered to be a triangular fuzzy number ω˜j,b˜0, instead of using above definite values. Moreover, this study employs the method proposed by Ishibuchi and Tanaka [19] for situations where the scope of the prediction is wide. Hence, a hybrid model with fuzzy functions and factors is as follows:(7)y˜t=fb˜0+∑j=1qω˜j.gb0,j+∑j=1pωi,j.yt−i,where y˜t are the observations, and b˜0 and ω˜j are the fuzzy numbers. Now, equation (7) is replaced by ut,j=b0,j+∑j=1pωi,j.yt−i as follows:(8)y˜t=b˜0+∑j=1qω˜j.ut,j.Note that the fuzzy parameter is a triangular fuzzy number according to equation (3). Now, we set y˜t=yαt,yct, in which yαt,yct are defined as follows:(9)yαt=b0αut,0+ω1αut,1+⋯+ωqαut,q+ε1,(10)yct=b0cut,0+ω1cut,1+⋯+ωqcut,q+ε2,forj=0,1,…,q,t=1,2,…,k,where ωj,α and ωj,s are the connection weights between the output neurons and j neurons. So,(11)Ep=Minb0αut,0+ω1αut,1+⋯+ωqαut,q−yαt2+16b0cut,0+ω1cut,1+⋯+ωqcut,q−yct2.In short, the following describes the main steps of the hybrid method:Step 1.
Teaching an ANN using the observations (which are nonfuzzy). Its result is the optimal answer of parameterω∗=ω0∗,ω1∗,…,ωq∗, represented as one of the input datasets in the second step.Step 2.
Determining the minimum ambiguity using equation (11), as well as the weights obtained from the first step ω∗=ω0∗,ω1∗,…,ωq∗. Note that the number of observations is equal to the number of constraints. Then, the hybrid model is as follows:(12)y˜t=α0,c0ut,0+α1,c1ut,1+⋯+αq,cqut,q,where αj,cj are the centers and expansions of fuzzy numbers, respectively.Step 3.
According to Ishibuchi, once the hybrid model is expanded, the upper and lower bounds are removed. It should be mentioned that to generate such a model, the data may be large if the dataset includes out-of-bounds items or specific differences. According to Ishibuchi and Tanaka [20], the data of the upper and lower bounds are removed, and then, the model is formulated again.
Here,x˜t,j=gω˜0,j+∑i=1pω˜i,j.yt−i. The connection weights are triangular fuzzy numbers ω˜i,j=ai,j,bi,j,ci,j and is defined as follows:(13)μω˜i,jωi,j=1bi,j−ai,jωi,j−ai,j,ai,j≤ωi,j≤bi,j,1bi,j−ci,jωi,j−ci,j,bi,j≤ωi,j≤ci,j,0,otherwise,where μω˜i,jωi,j is a member of a fuzzy set specified by ωi,j parameters. Now, using the expansion principle, the membership function x˜t,j=gω˜0,j+∑i=1pω˜i,j.yt−i is defined as follows [21]:(14)μX˜t,jXt,j=Xt,j−g∑i=0pai,jyt,ig∑i=0pbi,jyt,i−g∑i=0pai,jyt,i,ifg∑i=0pai,jyt,i≤Xt,j≤g∑i=0pbi,jyt,i,Xt,j−g∑i=0pci,jyt,ig∑i=0pbi,jyt,i−g∑i=0pci,jyt,i,ifg∑i=0pbi,jyt,i≤Xt,j≤g∑i=0pci,jyt,i,0,otherwise,where yt,i=yt−i, t=1,2,…,k,i=1,2,…,p.(15)yt,i=1,t=1,2,…,k,i=0.
Regarding the triangular fuzzy dataX˜t,j with the membership function based on equation (13) and triangle fuzzy parameters ω˜j, the membership function is as follows:(16)y˜t=fω˜0+∑j=1qω˜j.x˜i,j=f∑i=0qω˜j.x˜i,j,B1=∑j=0qdj.g∑i=0pbi,jyt,i−g∑i=0pai,jyt,i+g∑i=0pai,jyt,i.ej−dj,A2=∑j=0qfj−ej.g∑i=0pci,jyt,i−g∑i=0pbi,jyt,i,B2=∑j=0qfj.g∑i=0pci,jyt,i−g∑i=0pbi,jyt,i+g∑i=0pci,jyt,i.fj−ej,C1=∑j=0qdj.g∑i=0pai,jyt,i,C2=∑j=0qfj.g∑i=0pci,jyt,i,C3=∑j=0qfj.g∑i=0pbi,jyt,i,μY˜yt≅−B12A1+B12A12−C1−f−1ytA11/2,ifC1≤f−1yt≤C3,−B12A1+B12A12−C1−f−1ytA11/2,ifC3≤f−1yt≤C2,0,otherwise,where(17)A1=∑j=0qej−dj.g∑i=0pbi,j.yt,i−g∑i=0pai,j.yt,i.
Now, all the observations of a nonlinear programming model are obtained using the following equation by considering the threshold levelh in terms of the value of membership functions:(18)Min∑t=1k∑j=0qfjg∑i=0pci,jyt,i−djg∑i=0pai,jyt,i,Subject.to−B12A1+B12A12−C1−f−1ytA11/2≤h,ifC1≤f−1yt≤C3,fort=1,2,…,k,B22A2+B22A22−C2−f−1ytA21/2≤h,ifC3≤f−1yt≤C2,fort=1,2,…,k,0,otherwise.
At the end, the model output is fuzzy and continuous, while the output is nonfuzzy and discrete in classification problems. As such, we should enhance and modify the model to apply the proposed model to classification problems. To do so, in the first step, a number value is assigned to each class. Then, the probability of output membership is as follows:(19)PA=1−PB=∫−∞mfxdx∫−∞+∞fxdx=1−∫m+∞fxdx∫−∞+∞fxdx,where PA and PB are the probabilities of membership in each class of A and B, respectively. Ultimately, a sample is assigned to those classes whose output has a higher probability. As previously mentioned, the output is fuzzy. Hence, it is better to consider large numerical values for each class so that the model is more sensitive to changes in input.


## 5. Benchmark Credit Dataset

A set of credit information related to Japan and Australia is used to survey the effectiveness and appropriateness of the proposed model for classifying financial data. An example of each of information is derived in the next section. This set of information has been considered by many researchers in the field of financial management because it has a suitable combination of features, including both small and large numerical values.


### 5.1. Japan Credit Dataset

Overall, this dataset contains 690 views. Among these, 307 samples (about 44.5%) of the studied individuals are among the credit worthy, where 383 samples (about 5.55%) are credit unworthy. This dataset contains a total of 37 missing values. In this study, all 37 of these observations have been removed from the original dataset. Therefore, the new dataset contains 653 observations, in which 357 are credit worthy, whereas 296 are credit unworthy. Besides, this set of information includes 15 features, of which 9 are batch types, while 6 of them are numeric. It should be noted that due to the confidentiality and protection of information, the exact expression of the names of the features and their values have been avoided. Table1 reports a summary of the information. The data were randomly divided into two classes, such as education data and test data. Of these, 50% of the data are related to education, whereas the remaining 50% are related to the model test. Diagrams of two-dimensional distribution of the comparison of both features have been drawn, which can be seen in Figure 1 as an example of the diagrams related to the comparison of A2,A3, A2,A8,A3,A8, and A14,A15 features.Table 1
Japan credit information collection.
AttributesTypeAmountsValues (previous)1Batch0, 1a, b2Numerical80.25–13.7580.25–13.753Numerical0–280–284Batch1, 2, 3u, y, i5Batch1, 2, 3g, p, gg6Batch1, 2, 3, …, 14ff, d, i, k, j, aa, m, c, w, q, r, cc, x7Batch1, 2, 3, …, 9ff, dd, j, bb, v, n, o, h, z8Numerical0–28.50–28.59Batch0.1t, f10Batch0, 1t, f11Numerical0–670–6712Batch0, 1t, f13Batch1, 2, 3s, g, p14Numerical0–20000–200015Numerical0–1000000–100000ClassDiscrete0, 1−, +Figure 1
Two-dimensional distribution of the features of Japanese credit cards.


### 5.2. Australian Credit Dataset

Overall, this dataset of information includes 690 observations and 14 features. 8 of these features have discrete values, while 6 have continuous values. Among them, 307 samples (about 44.5%) of the studied individuals are credit worthy, whereas 383 cases (about 55.5%) are credit unworthy. Table2 lists a summary of the information. The data were randomly divided into two groups: education data and test data. Of these, 50% of the data are related to education, while the remaining 50% are related to the model test. The data were randomly divided into two classes, such as education data and test data. Of these, 50% of the data are related to education, whereas the remaining 50% are related to the model test. Diagrams of two-dimensional distribution of the comparison of both features have been drawn, which can be seen in Figure 2 as an example of the diagrams related to the comparison of A2,A3, A2,A7,A3,A10, and A2,A10 features.Table 2
Australian credit information collection.
AttributesTypeAmountsValues (previous)1Discrete0.1a, b2Continuous80.25–13.7580.25–13.753Continuous0–280–284Discrete1, 2, 3P, g, g, g5Discrete1, 2, 3, …, 14ff, d, i, k, j, aa, m, c, w, e, q, r, cc, x6Discrete1, 2, 3, …, 9ff, dd, j, bb, v, n, o, h, z7Continuous0–28.50–28.58Discrete0, 1t, f9Discrete0, 1t, f10Continuous0–670–6711Discrete0, 1t, f12Discrete1, 2, 3s, g, p13Continuous0–20000–200014Continuous0–1000000–100000ClassDiscrete0, 1−, +Figure 2
Two-dimensional distribution of the features of Australian credit cards.


### 5.3. Employing the Model in Credit Scoring Analysis

Now, the performance of the model is assessed to analyze the credit rating using the credit information of Japan and Australia. At first, in order to achieve the best ANN structure using the concept of multilayer perceptors [16] using MATLAB 7 software, the neural networks were compared under different structures in terms of performance. Afterwards, the ideal credit information network of Japan, networks consisting of 15 input neurons, 16 intermediate neurons, and one output neuron, is represented in the form of N (15-16-1), and the ideal network for credit information in Australia, network including 14 input neurons, 14 middle neurons, and one output neuron, is represented as N (14-14-1).When the optimal structure is specified, the postdiffusion algorithm under 100 replications trains the network. Then, in the second step, the minimum ambiguity in the fuzzy parameters regarding the threshold levelh = 0 is set using equation (4.10) in order to obtain the fuzzy parameters using the minimum width. Moreover, the misclassification rate in each model as well as the improvement percent by taking into account (h = 0) are listed in Tables 3 and 4, respectively, as compared to other classification models.Table 3
Incorrect classification rate in each model.
ModelIncorrect classification rate (%)Japan credit informationAustralian credit informationLinear discriminant analysis (LDA)13.814.0Quadratic discriminant analysis (QDA)15.619.9K-nearest neighbor13.814.2Support vector machines13.822.5ANNs11.412.3Adaptive fuzzy neural inference system10.811.7Proposed hybrid model9.710.9Table 4
Incorrect classification rate in each model.
ModelIncorrect classification rate (%)Japan credit informationAustralian credit informationLinear discriminant analysis (LDA)29.7122.14Quadratic discriminant analysis (QDA)37.8245.23K-nearest neighbor (KNN)29.7123.24Support vector machines (SVMs)29.7151.56ANNs (ANN)14.9111.38Adaptive fuzzy neural inference system (ANFIS)10.186.83According to the obtained results, to perform the credit analysis of the information of Japan, the proposed model shows high performance associated with 9.7% error of misclassification rate and 14.91% improvement in the information evaluation as compared to the multilayer perceptron model. Furthermore, the proposed model obtains 29.71%, 37.82, 29.71, and 10.18% improvements over the linear separation analysis models (c = 0), quadratic separation analysis (c = 0), support vector machines (c = 0), Kth nearest neighbor (K = 5), and fuzzy neural inference systems.Similarly, to analyze the credit information of Australia, the proposed model contains the lowest error rate of 10.9 percent as compared to other models. Besides, this hybrid model would result in 51.56, 23.24, 45.23, 22.14, 11.38, and 6.83 improvements, as compared to the linear separation analysis models (c = 0), quadratic separation analysis (c = 0), Kth nearest neighbor (K = 13), supporting vector machines (c = 1), ANNs, and adaptive fuzzy neural inference systems.


## 5.1. Japan Credit Dataset

Overall, this dataset contains 690 views. Among these, 307 samples (about 44.5%) of the studied individuals are among the credit worthy, where 383 samples (about 5.55%) are credit unworthy. This dataset contains a total of 37 missing values. In this study, all 37 of these observations have been removed from the original dataset. Therefore, the new dataset contains 653 observations, in which 357 are credit worthy, whereas 296 are credit unworthy. Besides, this set of information includes 15 features, of which 9 are batch types, while 6 of them are numeric. It should be noted that due to the confidentiality and protection of information, the exact expression of the names of the features and their values have been avoided. Table1 reports a summary of the information. The data were randomly divided into two classes, such as education data and test data. Of these, 50% of the data are related to education, whereas the remaining 50% are related to the model test. Diagrams of two-dimensional distribution of the comparison of both features have been drawn, which can be seen in Figure 1 as an example of the diagrams related to the comparison of A2,A3, A2,A8,A3,A8, and A14,A15 features.Table 1
Japan credit information collection.
AttributesTypeAmountsValues (previous)1Batch0, 1a, b2Numerical80.25–13.7580.25–13.753Numerical0–280–284Batch1, 2, 3u, y, i5Batch1, 2, 3g, p, gg6Batch1, 2, 3, …, 14ff, d, i, k, j, aa, m, c, w, q, r, cc, x7Batch1, 2, 3, …, 9ff, dd, j, bb, v, n, o, h, z8Numerical0–28.50–28.59Batch0.1t, f10Batch0, 1t, f11Numerical0–670–6712Batch0, 1t, f13Batch1, 2, 3s, g, p14Numerical0–20000–200015Numerical0–1000000–100000ClassDiscrete0, 1−, +Figure 1
Two-dimensional distribution of the features of Japanese credit cards.


## 5.2. Australian Credit Dataset

Overall, this dataset of information includes 690 observations and 14 features. 8 of these features have discrete values, while 6 have continuous values. Among them, 307 samples (about 44.5%) of the studied individuals are credit worthy, whereas 383 cases (about 55.5%) are credit unworthy. Table2 lists a summary of the information. The data were randomly divided into two groups: education data and test data. Of these, 50% of the data are related to education, while the remaining 50% are related to the model test. The data were randomly divided into two classes, such as education data and test data. Of these, 50% of the data are related to education, whereas the remaining 50% are related to the model test. Diagrams of two-dimensional distribution of the comparison of both features have been drawn, which can be seen in Figure 2 as an example of the diagrams related to the comparison of A2,A3, A2,A7,A3,A10, and A2,A10 features.Table 2
Australian credit information collection.
AttributesTypeAmountsValues (previous)1Discrete0.1a, b2Continuous80.25–13.7580.25–13.753Continuous0–280–284Discrete1, 2, 3P, g, g, g5Discrete1, 2, 3, …, 14ff, d, i, k, j, aa, m, c, w, e, q, r, cc, x6Discrete1, 2, 3, …, 9ff, dd, j, bb, v, n, o, h, z7Continuous0–28.50–28.58Discrete0, 1t, f9Discrete0, 1t, f10Continuous0–670–6711Discrete0, 1t, f12Discrete1, 2, 3s, g, p13Continuous0–20000–200014Continuous0–1000000–100000ClassDiscrete0, 1−, +Figure 2
Two-dimensional distribution of the features of Australian credit cards.


## 5.3. Employing the Model in Credit Scoring Analysis

Now, the performance of the model is assessed to analyze the credit rating using the credit information of Japan and Australia. At first, in order to achieve the best ANN structure using the concept of multilayer perceptors [16] using MATLAB 7 software, the neural networks were compared under different structures in terms of performance. Afterwards, the ideal credit information network of Japan, networks consisting of 15 input neurons, 16 intermediate neurons, and one output neuron, is represented in the form of N (15-16-1), and the ideal network for credit information in Australia, network including 14 input neurons, 14 middle neurons, and one output neuron, is represented as N (14-14-1).When the optimal structure is specified, the postdiffusion algorithm under 100 replications trains the network. Then, in the second step, the minimum ambiguity in the fuzzy parameters regarding the threshold levelh = 0 is set using equation (4.10) in order to obtain the fuzzy parameters using the minimum width. Moreover, the misclassification rate in each model as well as the improvement percent by taking into account (h = 0) are listed in Tables 3 and 4, respectively, as compared to other classification models.Table 3
Incorrect classification rate in each model.
ModelIncorrect classification rate (%)Japan credit informationAustralian credit informationLinear discriminant analysis (LDA)13.814.0Quadratic discriminant analysis (QDA)15.619.9K-nearest neighbor13.814.2Support vector machines13.822.5ANNs11.412.3Adaptive fuzzy neural inference system10.811.7Proposed hybrid model9.710.9Table 4
Incorrect classification rate in each model.
ModelIncorrect classification rate (%)Japan credit informationAustralian credit informationLinear discriminant analysis (LDA)29.7122.14Quadratic discriminant analysis (QDA)37.8245.23K-nearest neighbor (KNN)29.7123.24Support vector machines (SVMs)29.7151.56ANNs (ANN)14.9111.38Adaptive fuzzy neural inference system (ANFIS)10.186.83According to the obtained results, to perform the credit analysis of the information of Japan, the proposed model shows high performance associated with 9.7% error of misclassification rate and 14.91% improvement in the information evaluation as compared to the multilayer perceptron model. Furthermore, the proposed model obtains 29.71%, 37.82, 29.71, and 10.18% improvements over the linear separation analysis models (c = 0), quadratic separation analysis (c = 0), support vector machines (c = 0), Kth nearest neighbor (K = 5), and fuzzy neural inference systems.Similarly, to analyze the credit information of Australia, the proposed model contains the lowest error rate of 10.9 percent as compared to other models. Besides, this hybrid model would result in 51.56, 23.24, 45.23, 22.14, 11.38, and 6.83 improvements, as compared to the linear separation analysis models (c = 0), quadratic separation analysis (c = 0), Kth nearest neighbor (K = 13), supporting vector machines (c = 1), ANNs, and adaptive fuzzy neural inference systems.


## 6. Conclusion

Recently, credit risk analysis is known as one of the essential research areas in terms of financial management. In this matter, risk management and credit scoring are the most important and effective methods of the credit risk assessment as well as one of the main applications of classification problems, which is based on separating credit cards of customers and dividing them into two groups of credit worthy and unworthy. In this way, granting credit to customers requires a comprehensive analysis of factors and variables affecting it in order to make the right decision. Nevertheless, in these cases, the ambiguity and complexity between these variables have made decision-making very difficult.Theoretical and practical results indicated that the use of hybrid models was popular to decrease the misclassification rate, particularly when the models had a completely different structure. In this study, a multilayer perceptron model was developed as an alternative model for the correct classification of customers, in which the special advantages of soft intelligent computation and fuzzy logic were employed. In order to better model the financial markets in conditions of uncertainty, the parameters of the multilayer perceptron neural network were fuzzy, such as weights and errors.Afterwards, the system was first formed using ANNs, and then, the policy and optimal decision were specified using the highest superiority by applying fuzzy inference. The proposed model did not have any assumptions, such as the existence of a relationship between independent and dependent variables, in contrast to the quadratic separation analysis and linear models. Meanwhile, it did not require the storage of training data, unlike the K class of the nearest neighbor. It should be mentioned that although support vector machines require determining the penalty parameter and the kernel function, in the proposed model, the final classifier is determined only by relying on the training process without setting these parameters. Eventually, in contrast to multilayer perceptron, it did not require a large amount of information to obtain efficient results. The obtained results revealed the high accuracy and efficiency of the proposed method as compared to other statistical and intelligent classification models to analyze credit scoring problems [22].



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