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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.

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 [

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 [

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 [

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 [

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.

Some unclassifiable regions may exist if a data point belongs to more than one class or to none.

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.

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) [

The important researches applying AI techniques in bond-rating prediction are listed in Table

Prior bond rating prediction using AI techniques.

Research | Number of categories | AI methods applied | Data source | Samples size |
---|---|---|---|---|

[ |
2 | BP | U.S | 30/17 |

[ |
2 | BP | U.S | 126 |

[ |
3 | BP | U.S (S&P) | 797 |

[ |
6 | BP, RPS | U.S (S&P) | 110/60 |

[ |
6 | BP | U.S (S&P) | N/A |

[ |
6 | BP | U.S (Moody’s) | 299 |

[ |
5 | BP with OPP | Korea | 126 |

[ |
6 | BP, RBF | U.S (S&P) | 60/60 |

[ |
5 | CBR, GA | Korea | 3886 |

[ |
5 | SVM | U.S (S&P) | N/A |

[ |
5 | BP, SVM | Taiwan, U.S | N/A |

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 [

Support Vector Domain Description (SVDD), proposed by Tax and Duin in 1999 [

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 [

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 [

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 [

Multiclassification Based on SVDD.

Framework of the Proposed Methodology.

The hypersphere, by which SVDD models data points, is specified by its center

Geometry of the SVDD in two dimensions.

In the following, another closely related measure can be obtianed in (

To obtain an exact and compact representation of the data points, the minimization of both the hypersphere radius and

To solve the problem, the Lagrange Function is introduced, where

Setting (

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

Therefore, the dual form of the objective function can be obtained as follows:

And the problem can be formulated as follows:

The center of the hypersphere is a linear combination of data points with weighting factors

Similarly to the methodology based on kernel function proposed by Vapnik [

For example, Kernel function of RBF can be introduced to SVDD algorithm, just as shown as follows:

And it can be determined whether a testing data point

Based on fuzzy clustering algorithm [

Suppose

Let

Let

Fuzzy

Attribute Means Clustering (AMC) is an iterative algorithm by introducing the stable function [

Hence, the relationship of objective function

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 (

Moreover, procedure of minimizing (

And the following equations can be obtained by minimizing

To gain a high-dimensional discriminant, FAMC can be extended to Kernel-based Fuzzy Attribute

Since

And parameters in KFAMC can be estimated by

Moreover, the objective function of KFAMC can be obtained by substituting (

Based on theorem proved in [

Set

For

If

For step

where

In terms of SVDD, only support vectors are necessary to specify hyperspheres. But in the original algorithms [

Just as illustrated in Figure

Thoughts of proposed methodology.

So, the new methodology applied in SVDD is proposed as follows.

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

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.

Data points selection using FAMC.

Fuzzy SVDD establishment.

Training data points obtained by preprocessing

Hypersphere specification after data points preprocessing

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

Hence, the process of Fuzzy SVDD specification can be summarized as follows.

Set a threshold

Survey the membership of each

With hypersphere specified for each class using the updated training set obtained in Step

To accomplish multiclassification for testing data points using hyperspheres specified in Section

distances from the data point to centers of the hyperspheres;

density of the data points belonging to the class implied with values of radius of each hypersphere.

Classification of testing data point.

Just as shown in Figure

So, classification rules can be concluded as follows.

Let

Data point belongs to the class represented by the hypersphere.

Calculate the index of membership of the data point to each hypersphere using (

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

For the purpose of this study, two bond-rating data sets from Korea and China market, which have been used in [

Methods including independent-samples

In terms of Korea data set, 14 variables, which are listed in Table

Table of selected variables.

No. | Description | Definition |
---|---|---|

X1 | Shareholders’ equity | A firm’s total assets minus its total liabilities |

X2 | Sales | Sales |

X3 | Total debt | Total debt |

X4 | Sales per employee | Sales/the number of employees |

X5 | Net income per share | Net income/the number of issued shares |

X6^{*} |
Years after foundation | Years after foundation |

X7 | Gross earning to total asset | Gross earning/total Asset |

X8 | Borrowings-dependency ratio | Interest cost/sales |

X9 | Financing cost to total cost | Financing cost/total cost |

X10 | Fixed ratio | Fixed assets/(total assets-debts) |

X11^{*} |
Inventory assets to current assets | Inventory assets/current assets |

X12 | Short-term borrowings to total borrowings | Short-term borrowings/total borrowings |

X13 | Cash flow to total assets | Cash flow/total assets |

X14 | Cash flow from operating activity | Cash flow from operating activity |

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 [

Experimental results of the proposed method.

Data set | Korea data set | China data set | ||
---|---|---|---|---|

No. | Train (%) | Valid (%) | Train (%) | Valid (%) |

1 | 68.26 | 67.14 | 67.29 | 66.17 |

2 | 80.01^{*} |
71.23 | 68.35 | 67.13 |

3 | 73.21 | 70.62 | 71.56 | 71.01 |

4 | 75.89 | 72.37 | 75.24 | 72.36 |

5 | 76.17 | 74.23 | 84.17^{*} |
83.91^{*} |

6 | 75.28 | 75.01 | 80.02 | 79.86 |

7 | 78.29 | 76.23^{*} |
76.64 | 74.39 |

8 | 77.29 | 74.17 | 72.17 | 71.89 |

9 | 75.23 | 71.88 | 83.27 | 80.09 |

10 | 70.16 | 68.34 | 72.16 | 70.16 |

| ||||

Avg. | 74.98 | 72.12 | 75.09 | 73.70 |

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 [

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

Table of experiment results.

Type | Technique | Korea data set | China data set | ||
---|---|---|---|---|---|

Valid (%) | Time (Second) | Valid (%) | Time (Second) | ||

Prior AI approach | ANN | 62.78 | 1.67 | 67.19 | 1.52 |

| |||||

Conventional MSVM | One-against-all | 70.23 | 2.68 | 71.26 | 2.60 |

One-against-one | 71.76 | 2.70 | 72.13 | 2.37 | |

DAGSVM [ |
69.21 | 2.69 | 71.13 | 2.61 | |

Grammer & Singer [ |
70.07 | 2.62 | 70.91 | 2.50 | |

OMSVM [ |
71.61 | 2.67 | 72.08 | 2.59 | |

| |||||

The sphere-based classifier | Standard SVDD | 72.09 | 1.70 | 72.98 | 1.04 |

proposed method ( |
72.12 | 1.20 | 73.70 | 0.86 |

As shown in Table

Furthermore, as one of modified models based on standard SVDD, the proposed method accomplishes data preprocessing using KFAMC. Since the fuzzy area is determined by threshold

In the following experiment, the proposed method with various threshold values is tested based on different data sets, just as shown in Figure

Experiment results of generalization ability on data sets. (AUC represents hit-ratio of testing samples).

Experiments based on korea data set

Experiments based on china data set

The results illustrated in Figure

Moreover, training time of proposed method can be also compared with standard SVDD, just as illustrated in Figure

Experiment results of training time on data sets.

Experiments based on korea data set

Experiments based on china data set

Just as shown in Figure

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 bond-rating 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

Bezdek-type FCM is an inner-product-induced distance-based least-squared error criterion nonlinear optimization algorithm with constrains,

The above formula is also called as Mahalanobis distance, where

The paper was sponsored by 985-3 project of Xi’an Jiaotong University.