A Corporate Credit Rating Model Using Support Vector Domain Combined with Fuzzy Clustering Algorithm

Corporate credit-rating prediction using statistical and artificial intelligence techniques has received considerable attentions in the literature. Different from the thoughts of various techniques for adopting support vector machines as binary classifiers originally, a new method, based on support vector domain combined with fuzzy clustering algorithm for multiclassification, is proposed in the paper to accomplish corporate credit rating. By data preprocessing using fuzzy clustering algorithm, only the boundary data points are selected as training samples to accomplish support vector domain specification to reduce computational cost and also achieve better performance. To validate the proposed methodology, real-world cases are used for experiments, with results compared with conventional multiclassification support vector machine approaches and other artificial intelligence techniques. The results show that the proposed model improves the performance of corporate credit-rating with less computational consumption.


Introduction
Techniques of credit ratings have been applied by bond investors, debt issuers, and governmental officials as one of the most efficient measures of risk management.However, company credit ratings are too costly to obtain, because agencies including Standard and Poor's S&P , and Moody's are required to invest lots of time and human resources to accomplish critical analysis based on various aspects ranging from strategic competitiveness to operational level in detail 1-3 .Moreover, from a technical perspective, credit rating constitutes a typical multiclassification problem, because the agencies generally have much more than two categories of ratings.For example, ratings from S&P range from AAA for the highest-quality bonds to D for the lowest-quality ones.
The final objective of credit rating prediction is to develop the models, by which knowledge of credit risk evaluation can be extracted from experiences of experts and to be applied in much broader scope.Besides prediction, the studies can also help users capture fundamental characteristics of different financial markets by analyzing the information applied by experts.
Although rating agencies take emphasis on experts' subjective judgment in obtaining ratings, many promising results on credit rating prediction based on different statistical and Artificial Intelligence AI methods have been proposed, with a grand assumption that financial variables extracted from general statements, such as financial ratios, contain lots of information about company's credit risk, embedded in their valuable experiences 4, 5 .
Among the technologies based on AI applied in credit rating prediction, the Artificial Neural Networks ANNs have been applied in the domain of finance because of the ability to learn from training samples.Moreover, in terms of defects of ANN such as overfitting, Support Vector Machine SVM has been regarded as one of the popular alternative solutions to the problems, because of its much better performance than traditional approaches such as ANN 6-11 .That is, an SVM's solution can be globally optimal because the models seek to minimize the structural risk 12 .Conversely, the solutions found by ANN tend to fall into local optimum because of seeking to minimize the empirical risk.
However, SVM, which was originally developed for binary classification, is not naturally modified for multiclassification of many problems including credit ratings.Thus, researchers have tried to extend original SVM to multiclassification problems 13 , with some techniques of multiclassification SVM MSVM proposed, which include approaches that construct and combine several binary classifiers as well as the ones that directly consider all the data in a single optimization formulation.
In terms of multiclassification in the domain of credit rating containing lots of data, current approaches applied in MSVM still have some drawbacks in integration of multiple binary classifiers as follows.
1 Some unclassifiable regions may exist if a data point belongs to more than one class or to none.
2 Training binary classifiers based on two-class SVM multiple times for the same data set often result in a highly intensive time complexity for large-scale problems including credit ratings prediction to improve computational consumption.
To overcome the drawbacks associated with current MSVM in credit rating prediction, a novel model based on support vector domain combined with kernel-based fuzzy clustering is proposed in the paper to accomplish multiclassification involved in credit ratings prediction.

Credit Rating Using Data Mining Techniques
Major researches applying data mining techniques for bond rating prediction can be found in the literature.
Early investigations of credit rating techniques mainly focused on the applicability of statistical techniques including multiple discriminant analysis MDA 14, 15 and logistic regression analysis LRA 16 , and so forth, while typical techniques of AI including ANN  1.In summary, the most prior ones accomplish prediction using ANN with comparison to other statistical methods, with general conclusions that neural networks outperformed conventional statistical methods in the domain of bond rating prediction.
On the other hand, to overcome the limitations such as overfitting of ANN, techniques based on MSVM are applied in credit rating in recent years.Among the models based on MSVM in credit rating, method of Grammar and Singer was early proposed by Huang et al., with experiments based on different parameters so as to find the optimal model 29 .Moreover, methodologies based on One-Against-All, One-Against-One, and DAGSVM are also proposed to accomplish S&P's bond ratings prediction, with kernel function of Gaussian RBF applied and the optimal parameters derived form a grid-search strategy 28 .Another automatic-classification model for credit rating prediction based on One-Against-One approach was also applied 30 .And Lee applied MSVM in corporate credit rating prediction 31 , with experiments showing that model based on MSVM outperformed other AI techniques such as ANN, MDA, and CBR.

Multiclassification by Support Vector Domain Description
Support Vector Domain Description SVDD , proposed by Tax and Duin in 1999 32 and extended in 2004 33 , is a method for classification with the aim to accomplish accurate estimation of a set of data points originally.The methods based on SVDD differ from two or multiclass classification in that a single object type is interested rather than to be separated from other classes.The SVDD is a nonparametric method in the sense that it does not assume any particular form of distribution of the data points.The support of unknown distribution of data points is modeled by a boundary function.And the boundary is "soft" in the sense that atypical points are allowed outside it.
The boundary function of SVDD is modeled by a hypersphere rather than a hyperplane applied in standard SVM, which can be made with less constrains by mapping the data points to a high-dimensional space using methodology known as kernel trick, where the classification is performed.SVDD has been applied in a wide range as a basis for new methodologies in statistical and machine learning, whose application in anomaly detection showed that the model based on it can improve accuracy and reduce computational complexity 34 .Moreover, ideas of improving the original SVDD through weighting each data point by an estimate of its corresponding density were also proposed 35 and applied in area of breast cancer, leukemia, and hepatitis, and so forth.Other applications including pump failure detection 36 , face recognition 37 , speaker recognition 38 , and image retrieval 39 are argued by researchers.
The capability of SVDD in modeling makes it one of the alternative to large-margin classifiers such as SVM.And some novel methods applied in multiclass classification were proposed based on SVDD 40 combined with other algorithms such as fuzzy theories 41, 42 and Bayesian decision 36 .

The Proposed Methodology
In terms of SVDD, which is a boundary-based method for data description, it needs more boundary samples to construct a closely fit boundary.Unfortunately, more boundary ones usually imply that more target objects have to be rejected with the overfitting problem arising and computational consumption increased.To accomplish multiclassification in corporate credit rating, a method using Fuzzy SVDD combined with fuzzy clustering algorithm is proposed in the paper.By mapping data points to a high-dimensional space by Kernel Trick, the hypersphere applied to every category is specified by training samples selected as boundary ones, which are more likely to be candidates of support vectors.After preprocessing using fuzzy clustering algorithm, rather than by original ones directly in standard SVDD 32, 33 , one can improve accuracy and reduce computational consumption.Thus, testing samples are classified by the classification rules based on hyperspheres specified for every class.And the thoughts and framework of the proposed methodology can be illustrated in Figures 1 and 2, respectively.

Introduction to Hypersphere Specification Algorithm
The hypersphere, by which SVDD models data points, is specified by its center a and radius R. Let X x 1 , x 2 , x 3 , . . .denote the data matrix with n data points and p variables, which implies that a is p-dimensional while R is scalar.The geometry of one solution to SVDD in two dimensions is illustrated in Figure 3, where ω i represents the perpendicular distance from the boundary to an exterior points x i .In terms of interior points, and the ones positioned on the boundary, ω i is to be assigned as 0. Hence, ω i can be calculated using the following equation: 3.1 In the following, another closely related measure can be obtianed in 3.2 in terms of exterior points To obtain an exact and compact representation of the data points, the minimization of both the hypersphere radius and ξ i to any exterior point is required.Moreover, inspired by fuzzy set theory, matrix X can be extended to X x 1 , s 1 , x 2 , s 2 , x 3 , s 3 , . . .with coefficients s i representing fuzzy membership associated with x i introduced.So, the data domain description can be formulated as 3.3 , where nonnegative slack variables ξ i are a measure of error in SVDD, and the term s i ξ i is the one with different weights based on fuzzy set theory min

3.3
To solve the problem, the Lagrange Function is introduced, where α i , β i ≥ 0 are Lagrange Multipliers shown as follows: Setting 3.4 to 0, the partial derivates of L leads to the following equations:

3.5
That is,

3.6
The Karush-Kuhn-Tucker complementarities conditions result in the following equations:

3.7
Therefore, the dual form of the objective function can be obtained as follows: And the problem can be formulated as follows: 3.9 The center of the hypersphere is a linear combination of data points with weighting factors α i obtained by optimizing 3.9 .And the coefficients α i , which are nonzero, are thus selected as support vectors, only by which the hypersphere is specified and described.Hence, to judge whether a data point is within a hypersphere, the distance to the center should be calculated with 3.10 in order to judge whether it is smaller than the radius R.And the decision function shown as 3.12 can be concluded from

Introduction to Fuzzy SVDD Based on Kernel Trick
Similarly to the methodology based on kernel function proposed by Vapnik 12 , the Fuzzy SVDD can also be generalized to high-dimensional space by replacing its inner products by kernel functions K •, • Φ • • Φ • .For example, Kernel function of RBF can be introduced to SVDD algorithm, just as shown as follows:

3.13
And it can be determined whether a testing data point x is within the hypersphere with 3.14 by introducing kernel function based on 3.12 l i 1 3.14

3.15
Hence, the relationship of objective function ρ t and its weight function is described by sable function, which was introduced to propose AMC.
According to current researches, some alternative functions including squared stable function, Cauchy stable function, and Exponential stable function are recommended.
Based on previous researches, AMC and FCM are extended to FAMC, which is also an iterative algorithm to minimize the following objective function shown as 3.16 , where m > 1, which is a coefficient of FCM introduced in Appendix Moreover, procedure of minimizing 3.16 can be converted to an iterative objective function shown as 3.17 43 3.17 And the following equations can be obtained by minimizing Q i U i , p , Q i U, p i 1 , respectively, which can be seen in 43, 45 in detail

Introduction to Kernel-Based Fuzzy Clustering
To gain a high-dimensional discriminant, FAMC can be extended to Kernel-based Fuzzy Attribute C-means Clustering KFAMC .That is, the training samples can be first mapped into high-dimensional space by the mapping Φ using kernel function methods addressed in Section 3.1.2. Since Mathematical Problems in Engineering when Kernel function of RBF is introduced, 3.19 can be given as follows

Algorithms of Kernel-Based Fuzzy Attribute C-Means Clustering
Based on theorem proved in 45 , the updating procedure of KFAMC can be summarized in the following iterative scheme.
Step 1. Set c, m, ε and t max , and initialize U 0 , W 0 .
Step 2. For i 1, calculate fuzzy cluster centers P i , U i .and W i .
Step 4. For step i i 1, update P i 1 , U i 1 , and W i , turn to Step 3, where i denotes iterate step, t max represents the maximum iteration times, and W i denotes the weighting matrix, respectively, which can be seen in 45 in detail.

Classifier Establishment
In terms of SVDD, only support vectors are necessary to specify hyperspheres.But in the original algorithms 32, 33, 41 , all the training samples are analyzed and thus computational cost is high consumption.Hence, if the data points, which are more likely to be candidates of support vectors, can be selected as training samples, the hypersphere will be specified with much less computational consumption.Just as illustrated in Figure 4, only the data points, such as M, N positioned in fuzzy areas, which are more likely to be candidates of support vectors, are necessary to be classified with SVDD, while the ones in deterministic areas can be regarded as data points belonging to certain class.
So, the new methodology applied in SVDD is proposed as follows.
1 Preprocess data points using FAMC to reduce amount of training samples.That is, if fuzzy membership of a data point to a class is great enough, the data point can be ranked to the class directly.Just as shown in Figure 5, the data points positioned in deterministic area shadow area A are to be regarded as samples belonging to the class, while the other ones are selected as training samples.
2 Accomplish SVDD specification with training samples positioned in fuzzy areas, which has been selected using KFAMC.That is, among the whole data points, only the ones in fuzzy area, rather than all the data points, are treated as candidates of support vectors.And the classifier applied in multiclassification can be developed based on Fuzzy SVDD by specifying hypersphere according to every class.
Hence, the main thoughts of Fuzzy SVDD establishment combined with KFAMC can be illustrated in Figure 6.
The process of methods proposed in the paper can be depicted as follows.
In high-dimensional space, the training samples are selected according to their fuzzy memberships to clustering centers.Based on preprocessing with KFAMC, a set of training samples is given, which is represented by and μ m l ∈ 0, 1 denote the number of training data, input pattern, and membership to class m, respectively.
Hence, the process of Fuzzy SVDD specification can be summarized as follows.
Step 1. Set a threshold θ > 0, and apply KFAMC to calculate the membership of each x i , i 1, 2, . . ., l, to each class.If μ m i ≥ θ, μ m i is to be set as 1 and μ t i , t / m, is to be set as 0.
Step 2. Survey the membership of each x i , i 1, 2, . . ., l.If μ m i 1, x i is to be ranked to class m directly and removed from the training set.And an updated training set can be obtained.Step 3.With hypersphere specified for each class using the updated training set obtained in Step 2, classifier for credit rating can be established using the algorithm of Fuzzy SVDD, just as illustrated in Figure 6.

Classification Rules for Testing Data Points
To accomplish multiclassification for testing data points using hyperspheres specified in Section 3.3.1, the following two factors should be taken into consideration, just as illustrated in Figure 7: Case II d 0 or d > 1 .Calculate the index of membership of the data point to each hypersphere using 3.24 , where R c denotes the radius of hypersphere c, D x i , c denotes the distance from data point x i to the center of hypersphere c

3.24
And the testing data points can be classified according to the following rules represented with

Data Sets
For the purpose of this study, two bond-rating data sets from Korea and China market, which have been used in 46, 47 , are applied, in order to validate the proposed methodology.The data are divided into the following four classes: A1, A2, A3, and A4.

Variables Selection
Methods including independent-samples t-test and F-value are applied in variable selection.
In terms of Korea data set, 14 variables, which are listed in Table 2, are selected from original ones, which were known to affect bond rating.For better comparison, similar methods were also used in China data set, with 12 variables among them being selected.

Experiment Results and Discussions
Based on the two data sets, some models based on AI are introduced for experiments.To evaluate the prediction performance, 10-fold cross validation, which has shown good performance in model selection 48 , is followed.In the research, all features, which are represented with variables listed in Table 2, of data points range from 0 to 1 after Min-max transformation.To validate the methodology oriented multiclassification problem in credit rating, ten percent of the data points for each class are selected as testing samples.And the results of experiments on proposed method, with 0.9 being chosen as the value of threshold intuitively, are shown in Table 3.
To compare with other methods, the proposed model is compared with some other MSVM techniques, namely, ANN, One-Against-All, One-Against-One, DAGSVM, Grammer & Singer, OMSVM 46 , and standard SVDD.The results concluded in the paper are all shown as average values obtained following 10-fold cross validation based on platform of Matlab 7.0.
To compare the performance of each algorithm, hit-ratio, which is defined according to the samples classified correctly, is applied.And the experiment results are listed in Table 4.
As shown in Table 4, the proposed method based on thoughts of hypersphere achieves better performance than conventional SVM models based on thoughts of hyperplane.Moreover, as one of modified models, some results obtained imply that the proposed method has better generalization ability and less computational complexity, which can be partially measured with training time labeled with "Time," than standard SVDD.Moreover, training time of proposed method can be also compared with standard SVDD, just as illustrated in Figure 9.
Just as shown in Figure 9, results of experiments based on different data sets are similar.That is, with decline of threshold, more samples were eliminated from training set through preprocessing based on KFAMC to reduce training time.Hence, smaller values of threshold will lead to less computational consumption partly indicated as training time, while classification accuracy may be decreased due to lack of necessary training samples.Overall, threshold selection, which involves complex tradeoffs between computational consumption and classification accuracy, is essential to the proposed method.

Conclusions and Directions for Future Research
In the study, a novel algorithm based on Fuzzy SVDD combined with Fuzzy Clustering for credit rating is proposed.The underlying assumption of the proposed method is that sufficient boundary points could support a close boundary around the target data but too many ones might cause overfitting and poor generalization ability.In contrast to prior researches, which just applied conventional MSVM algorithms in credit ratings, the algorithm based on sphere-based classifier is introduced with samples preprocessed using fuzzy clustering algorithm.
As a result, through appropriate threshold setting, generalization performance measured by hit-ratio of the proposed method is better than that of standard SVDD, which outperformed many kinds of conventional MSVM algorithms argued in prior literatures.Moreover, as a modified sphere-based classifier, proposed method has much less computational consumption than standard SVDD.
One of the future directions is to accomplish survey studies comparing different bondrating processes, with deeper market structure analysis also achieved.Moreover, as one of the MSVM algorithms, the proposed method can be applied in other areas besides credit ratings.And some more experiments on data sets such as UCI repository 49 are to be accomplished in the future.A.3

Figure 7 :
Figure 7: Classification of testing data point.

bFigure 8 : 1 b
Figure 8: Experiment results of generalization ability on data sets.AUC represents hit-ratio of testing samples .

Figure 9 :
Figure 9: Experiment results of training time on data sets.

18 Mathematical
Problems in Engineeringadopt Euclidean distance in the rest of the paper.So, the parameters of FCM are estimated by updating min J m U, P according to the formulas:

Table 1 :
Prior bond rating prediction using AI techniques.
17, 18 and case-based reasoning CBR 19 , and so forth are applied in the second phase of research.The important researches applying AI techniques in bond-rating prediction are listed in Table {C 1 , C 2 , . .., C c }, where c is the cluster number.For ∀x ∈ χ, let μ x C k denote the attribute measure of x, with c k 1 μ Let μ kn denote the attribute measure of the nth sample belonging to the kth cluster.That is, µ kn μ n p k , U µ kn , p p 1 , p 2 , . .., p k .The task of fuzzy clustering is to calculate the attribute measure μ kn , and determine the cluster which x n belongs to according to the maximum cluster index.Fuzzy C-means FCM is an inner-product-induced distance based on the leastsquared error criterion.A brief review of FCM can be found in Appendix based on coefficients definitions mentioned above.
Based on fuzzy clustering algorithm 42 , Fuzzy Attribute C-means Clustering FAMC 43 was proposed as extension of Attribute Means Clustering AMC and Fuzzy C-means FCM .Suppose χ ⊂ R d denote any finite sample set, where χ {x 1 , x 2 , . . ., x n }, and each sample is defined as x n x 1n , x 2n , . . ., x dn 1 ≤ n ≤ N .The category of attribute space is F x C k 1.Let p k p k1 , p k2 , . . ., p kd denote the kth prototype of cluster C k , where 1 ≤ k ≤ c.

Table 2 :
Table of selected variables.
* Indicates variables excluded in China data set.