A real-valued genetic algorithm to optimize the parameters of
support vector machine for predicting bankruptcy
Chih-Hung Wu
a, Gwo-Hshiung Tzeng
b,c,*, Yeong-Jia Goo
d, Wen-Chang Fang
daDepartment of Business Administration, Takming College, No. 56, Sec.1, Huanshan Rd., Neihu District, Taipei City 114, Taiwan bDepartment of Business Administration, Kainan University, No. 1, Kainan Rd., Luchn, Taoyuan 338, Taiwan
c
Institute of Management of Technology, National Chiao Tung University, 100 Ta-Hsueh Rd., Hsinchu 300, Taiwan
d
Department of Business Administration, National Taipei University, Taipei, Taiwan
Abstract
Two parameters, C and r, must be carefully predetermined in establishing an efficient support vector machine (SVM) model. There-fore, the purpose of this study is to develop a genetic-based SVM (GA-SVM) model that can automatically determine the optimal param-eters, C and r, of SVM with the highest predictive accuracy and generalization ability simultaneously. This paper pioneered on employing a real-valued genetic algorithm (GA) to optimize the parameters of SVM for predicting bankruptcy. Additionally, the pro-posed GA-SVM model was tested on the prediction of financial crisis in Taiwan to compare the accuracy of the propro-posed GA-SVM model with that of other models in multivariate statistics (DA, logit, and probit) and artificial intelligence (NN and SVM). Experimental results show that the GA-SVM model performs the best predictive accuracy, implying that integrating the RGA with traditional SVM model is very successful.
Ó 2005 Elsevier Ltd. All rights reserved.
Keywords: Support vector machine (SVM); Real-valued; Genetic algorithm (GM); Financial distress; Prediction; Bootstrap simulation
1. Introduction
Predicting corporate failure has been an important research topic in accounting and finance for the last three decades (Lee, Han, & Kwon, 1996; Salcedo-Sanz, Fernan-dez-Villacanas, Segovia-Vargas, & Bousono-Calzon, 2005). The financial crisis in East Asia provoked particularly extensive studies of the financial distress of institutions with various financial and ownership structures that arose across countries with very diverse institutional setups in East Asia in 1997 and 1998 (Claessens, Djankov, & Klap-per, 2003). Classical studies on ratio analysis and the clas-sification of bankruptcy was performed by Beaver’s dichotomous classification test in 1967.Altman (1968)
pro-posed the Z-score model, which applied multivariate dis-criminant analysis (MDA) and employed financial ratios as input variables to predict financial distress. Subsequent studies have developed more precise model to predict bank-ruptcy.Deakin (1972)revised Altman and Beaver’s studies, using a quadratic function to construct a more precise clas-sification model of financial distress and thus increase the accuracy for predicting financial distress. After that, logit regression (Ohlson, 1980; Platt & Platt, 1990; Tseng & Lin, 2005; Zavgren, 1985) or probit regression (Zmijewski, 1984) have widely adopted in subsequent work. Neverthe-less, empirical results have shown that most of financial ratios violate the assumptions of the multivariate statistical model used in these previous studies. In recently studies, several revised financial distress models such as the revised the Z score and ZETA models and the hybrid system (Lee et al., 1996; Tam & Kiang, 1992) have been demonstrated the results of highly adaptable and outperformed in predicting bankruptcy. In addition, statistical learning 0957-4174/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2005.12.008
*
Corresponding author. Fax: +886 3 3412430.
E-mail addresses:ericwu@mail.takming.edu.tw(C.-H. Wu),ghtzeng@ cc.nctu.edu.tw, ghtzeng@mail.knu.edu.tw (G.-H. Tzeng), goo@ntpu. edu.tw(Y.-J. Goo),fang@ntpu.edu.tw(W.-C. Fang).
www.elsevier.com/locate/eswa Expert Systems with Applications 32 (2007) 397–408
approaches such as neural networks or support vector machines (SVM) has been successfully applied to this kind of problems (Min & Lee, 2005; Salcedo-Sanz et al., 2005; Tam, 1991; Tam & Kiang, 1992; Wu, 2004).
Previous application of neural networks in finance and accounting, notably in bankruptcy prediction, are limited to back-propagation neural networks (Yang, Platt, & Platt, 1999). Recently, new algorithms in machine learning, Sup-port vector machines (SVMs), were developed by Boster,
Guyon, and Vapnik (1992) to provide better solutions to
decision boundary than could be obtained using the tradi-tional neural network. Since the new model was proposed (Boster et al., 1992; Cortes & Vapnik, 1995), SVM has been successfully applied to numerous applications, including the handwriting recognition, particle identification (e.g., muons), digital images identification (e.g., face identifica-tion), text categorization, bioinformatics (e.g., gene expres-sion), function approximation and regression, time series forecasting (Cao, 2003; Kim, 2003; Mukherjee, Osuna, & Girosi, 1997; Mu¨ller, Smola, Ra¨tsch, & Scho¨lkopf, 1999; Tay & Cao, 2001, 2002), chaotic system (Mattera & Hay-kin, 1999) and bankruptcy prediction (Min & Lee, 2005; Salcedo-Sanz et al., 2005). This study examines the possi-bility of enhancing the accuracy of predicting bankruptcy by adopting the SVM model.
Min and Lee (2005) stated that the optimal parameter
search on SVM plays a crucial role to build a bankruptcy prediction model with high prediction accuracy and stabil-ity. To make an efficient SVM model, two extra parame-ters: C and r2 (sigma squared) have to be carefully predetermined. The first parameter, C, determines the trade-offs between the minimization of the fitting error and the minimization of the model complexity. The second parameter, r2, is the bandwidth of the radial basis function (RBF) kernel. Consequently, the purpose of this study is to propose a model that can determine the optimal parame-ters (C and r2) of SVMs to yield the highest predictive accuracy and generalization ability for predicting bank-ruptcy. The model was tested on the prediction of financial crisis of Taiwan to compare its accuracy with that of other models that based on multivariate statistics and AI approaches.
The remainder of this paper is organized as follows. The basic ideas of methods for bankruptcy prediction is reviewed and discussed in Section 2. Research design for modeling genetic based SVM is proposed in Section 3 to describe its ideas and procedures. An example of empirical analysis for predicting bankruptcy is used to demonstrate the proposed method in Section 4. Discussions are pre-sented in Section5 and conclusions are in the last section. 2. Basic ideas of methods for bankruptcy prediction
In this section, the basic ideas of methods for bank-ruptcy prediction from the perspective of the non-linear SVM are provides. Then, the real-valued genetic algorithm is briefly introduced. Parameters optimization approaches
are discussed in the following section. Finally, statistical approaches for predicting bankruptcy are overviewed in the final section.
2.1. The non-linear support vector machine
The basic idea in designing a non-linear SVM model is to map the input vector x2 Rn into vectors z of a higher-dimensional feature space F (z = u(x), where u denotes the mapping Rn ! RfÞ, and to solve a linear clas-sification problem in this feature space
x2 Rn ! zðxÞ ¼ a
1u1ðxÞ; a2u2ðxÞ; . . . ; anunðxÞ
½ T2 Rf:
ð1Þ Namely, the basic idea in non-linear SVM is to map the data x into a high-dimensional feature space via a mapping function u(x) (also called kernel function), which is selected by the user in advance. By replacing the inner product for non-linear pattern problem, the kernel function can per-form a non-linear mapping to a high-dimensional feature space (Vapnik, 1995). Kernel functions perform the non-linear mapping between input space and a feature space.
The approximating feature map for the Mercer kernel is K(x, y) = u(x)Tu(y), which performs the non-linear map-ping. Currently, popular kernel functions in machine learn-ing theories are as follows (Campbell, 2002; Kecman, 2001).
GaussianðRBFÞ kernel : Kðxi; xjÞ ¼ exp
kxi xjk 2 2r2 ! ; ð2Þ Polynomial kernel: Kðxi; xjÞ ¼ ð1 þ xTixjÞ d ; ð3Þ Linear kernel: Kðxi; xjÞ ¼ xTixj; ð4Þ
Multilayer perceptron: Kðxi; xjÞ ¼ tanh½ðxTxiÞ þ b. ð5Þ
In Eq.(2), r2denotes the variance of the Gaussian ker-nel. A certain value of b is used only in the multilayer perceptron.
The learning algorithm for a non-linear classifier SVM follows the design of an optimal separating hyperplane in a feature space. The procedure the same as associated with hard and soft margin classifier SVMs in x-space. Accord-ingly, the dual Lagrangian in z-space is
LdðaÞ ¼ Xl i¼1 ai 1 2 Xl i;j¼1 yiyjaiajzTizj; ð6Þ
and using the chosen kernels, the Lagrangian is maximized as follows. Maximize: LdðaÞ ¼ Xl i¼1 ai 1 2 Xl i;j¼1 yiyjaiajKðxixjÞ ð7Þ Subject to aiP0; i¼ 1; . . . ; l; ð8Þ Xl i¼1 aiyi¼ 0. ð9Þ
Note the constraints must be revised for using in a non-linear soft margin classifier SVM. The only difference these constraints and those of the separable non-linear classifier are in the upper bound C on the Lagrange multipliers ai.
Consequently, the constraints of the optimization problem become
Subject to CP aiP0; i¼ 1; . . . ; l; ð10Þ
Xl i¼1
aiyi¼ 0. ð11Þ
In this way, the influence of the training data point will be limited and remained on the wrong side of a separating non-linear hypersurface. The decision hypersurface d(x) and the indicator function, which were determined by the non-linear SVM classifier, are as follows:
dðxÞ ¼X l i¼1 yiaiKðxi; xjÞ þ b; ð12Þ iFðxÞ ¼ signðdðxÞÞ ¼ sign Xl i¼1 yiaiKðxi; xjÞ þ b ! . ð13Þ
Depending upon the chosen kernel, the bias term b may implicitly be a part of the kernel function. For example, the bias term b is not required when Gaussian RBFs are used as kernels. When the bias term b is included within other ker-nel functions, the non-linear SVM classifier is as follows: iFðxÞ ¼ signðdðxÞÞ ¼ sign Xl i¼1 yiaiKðxi; xjÞ þ b ! ¼ sign X number of SVs s¼1 ysasKðx; xsÞ ! . ð14Þ
2.2. Real-valued genetic algorithm (RGA)
Recently, genetic algorithms (GAs) have been widely and successfully applied to various optimization problems (Fogel, 1994; Goldberg, 1989; Grefenstette, 1986). GAs are well suited to the concurrent manipulating of models with varying resolutions and structures since they can search non-linear solution spaces without requiring gradi-ent information or a priori knowledge about model charac-teristics (McCall & Petrovski, 1999). The problem existing in the binary coding lies in the fact that a long string always occupies the computer memory even though only a few bits are actually involved in the crossover and mutation opera-tions. This is particularly the case when a lot of parameters are needed to be adjusted in the same problem and a higher precision is required for the final result. To overcome the inefficient occupation of the computer memory, the under-lying real-valued crossover and mutation algorithms are employed (Huang & Huang, 1997). In contrast to the binary genetic algorithm (BGA), the real-valued genetic algorithm (RGA) uses a real value as a parameter of the chromosome in populations without performing coding and encoding process before calculates the fitness values of individuals
(Haupt & Haupt, 1998). Namely, RGA is more straightfor-ward, faster and more efficient than BGA. Since this study is concerned with finding optimal values of SVM parameters whose precise values are unknown, the aforementioned properties of RGA are highly advantageous.
2.3. Parameter optimization
To design an effective SVM model, values of parameters in SVM have to be chosen carefully in advance (Duan, Keerthi, & Poo, 2003; Lin, 2001; Min & Lee, 2005). These parameters include the following: (1) regularization param-eter C, which dparam-etermines the tradeoff cost between mini-mizing the training error and minimini-mizing the complexity of the model; (2) parameter sigma (r or d) of the kernel function which defines the non-linear mapping from the input space to some high-dimensional feature space. This investigation only considers only the Gaussian kernel, the variance of whose function is sigma squared r2; (3) a kernel function used in SVM, which constructs a non-linear deci-sion hypersurface in an input space.
Cristianini, Shawe-Taylor, and Campell (1998)
pro-posed the Kernel-Adatron Algorithm which can automati-cally select models without testing on a validation data. Unfortunately, this algorithm is ineffective if the data have a flat ellipsoid distribution (Campbell, 2002). Therefore, one possible way to solve the problem is to consider the dis-tribution of the data. Interestingly, various specific func-tions in SVM, after the learning stage, can create the decision hypersurfaces of the same type (Kecman, 2001).
To solve the problem,Lin (2001)provided a systematic method for selecting SVM parameters. His systematic design for selecting parameters of support vector regression was adopted the concept of the sampling theory into Gaussian Filter. Min and Lee (2005) proposed a grid-search technique using 5-fold crossvalidation to find out the optimal parameters values of kernel function of SVM. In contrast to abovementioned methods of parameter optimization on SVM, this study develops a new method, named GA-SVM, for optimizing the two SVM parameters (C and r2) simultaneously. The first parameter, C, deter-mines the trade-off between the fitting error minimization and model complexity. The second parameter, r2, is the bandwidth of the radial basis function (RBF) kernel. 2.4. Overview of statistical approaches for predicting bankruptcy
The corporate distress literature includes several diverse methodologies for discriminating between failed and non-failed firms, following Beaver’s univariate comparison of financial ratios in 1966. Extensive studies in this area have applied statistical and AI approaches over the last three decades. The well-known multivariate models used in this area include multiple discriminate analysis (MDA) (Altman, 1968; Altman, Haldeman, & Narayanan, 1977), logit analysis (Ohlson, 1980; Platt & Platt, 1990; Tseng &
Lin, 2005; Zavgren, 1985), and probit analysis (Zmijewski, 1984). Most recently, AI approaches, such as neural net-work approaches (Lee et al., 1996; Lee, Booth, & Alam, 2005; Odom & Sharda, 1990; Tam, 1991; Yang et al., 1999) or SVM (Min & Lee, 2005; Salcedo-Sanz et al., 2005) have shown promise as classification tools.
3. Designing a genetic-based SVM model for predicting bankruptcy
In this section, we describe the design of our proposed model, a genetic-based SVM model, for predicting bank-ruptcy. The approach of combining real-valued genetic algorithm with SVM is introduced in the first section. Research data and description of samples are described in the next section. Modeling and the parameter settings of BPN and SVM are presented in Section3. Chromosome representations, the design of fitness function and genetic operators in this study are discussed in the final sections. 3.1. Our proposed approach
In the proposed GA-SVM model, the SVM parameters are dynamically optimized by implementing the RGA evo-lutionary process and the SVM model then performs the prediction task using these optimal values. Namely, the RGA tries to search the optimal values to enable SVM to fit various datasets. The process of GA-SVM was illus-trated inFig. 1. The optimal values of SVM’s parameters are searching by GAs with a randomly generated initial populations consisting of chromosomes. The values of the two parameters, C and r2, are directly coded in the chromosomes with real-valued data. The proposed model can implement either the roulette-wheel method or the tournament method for selecting chromosomes. Ade-wuya’s crossover method and boundary mutation method were used to modify the chromosome. The single best chro-mosome in each generation is survives to the succeeding generation. The proposed model was developed and imple-mented in the MATLAB v6.5 environment. The major tool for training and validating the SVM were those developed byPelckmans et al. (2002). The proposed model is able to handle huge data sets and easily be combined with the real-valued genetic algorithm in the MATLAB environment.
Predicting bankruptcy (or financial distress) has been a major research issue in accounting and finance over in the last three decades (Lee et al., 1996). Therefore, the genetic-based SVM (GA-SVM) model was applied to the problem of financial distress in Taiwan to verify its accu-racy and generalization ability, must be shown to be more accurate than the traditional multivariate statistical models and neural network technique.
3.2. Research data
Financial-statement data of the failed and non-failed firms were obtained from the database of the Taiwan
Eco-nomic Journal (TEJ), covering in cases of 3 years prior to failure and 1 year after failing. ‘‘Failure’’ is defined as the inability of a firm to pay its financial obligations as they mature. A firm is specifically said to have failed when any of the following events have occurred: bankruptcy, default on bonds, the overdrawing of a bank account, or non-payment of a preferred stock dividend (Beaver, 1966). This study defined the firms in financial distress as those whose listed securities have been classified as the cat-egory of alter-trading-method.1 When any of the events exists in aforementioned events occurs in the operating rules, this Corporation may place the listed securities under the category of altered-trading-method.
According to the definition of Beaver (1966), the ‘‘first year before failure’’ is defined as that year included in the most recent financial statement prior to the year in which the firm is reported to have failed. The data sample consists of firms in Taiwan that failed in the period from 1998 to 2002. The failed firms were selected from the lists of bank-rupt companies by the Taiwan stock exchange (TSE) and the database of TEJ. A failed firm was paired with a non-failed firm by (1) industry, (2) products, (3) capitalization, and (4) values of assets. Table 1 presents the description of samples. Failed companies were paired with non-failed firms in a similar industry, dealing in similar products, with similar capitalization, and with similar values of assets.
The size of matched sample was 88 firms, including 22 failed firms and 66 non-failed firms. In the simulated sam-ple, the total sample size was 44 companies, including 22 failed firms and 22 non-failed firms. The holdout sample comprises of all corporations listed on the TSE and OTC market from 2001 to 2002. The sample size for 2001 was 538 firms, including 373 firms on the TSE and OTC market in 2001. The sample size for 2002 was 534 firms, including 356 firms on the TSE and OTC market.
Table 1also presents further details about the matched sample in this study. The matched sample was paired according to industry, primary product, capitalization and values of assets. A lower matching ratio (e.g., 1:1 or 1:2) corresponds to higher bias in the selection in choice-based, which leads to oversampling (Platt & Platt, 2002;
Zmijewski, 1984). Using the matching rule that has been
proposed bySu (2000), this study adopted one financially distressed firm was paired to three non-failed firms (1:3 ratio) to avoid the problem of oversampling and bias in the choice-based sample (Platt & Platt, 2002).
3.3. Modeling
The 19 financial variables are those which have been found or actually used in previous research to be significant in predicting bankruptcy. These ratios can be grouped into
1 This method is according to Articles 49, 50, and 50-1 of the Operating
four categories, including liquidity, profitability, asset man-agement, and financial structure. Initially, financial vari-ables are selected for used in the bankruptcy prediction model. A list of tested financial ratios is summarized in
Table 2.
3.3.1. Neural network
The feed-forward back-propagation neural network (BPN) applied to the experimental sample includes 19 input neurons in the input layer, seven neurons in the hidden layer, and one in the output layer. This study constructed a three-layer network and employed the
‘‘TRAINLM algorithm’’, ‘‘LEARNGDM’’, and ‘‘MSE-REG’’ as the training function, the adaptive learning func-tion, and the performance funcfunc-tion, respectively. The transfer function was set to the ‘‘TANSIG function’’ and the ‘‘PURELIN function’’ for hidden layer and output layer, respectively. The number of epochs was set to 300 and the learning rate was set to 0.05 in each epoch. Table 3 presents the parameter settings.
3.3.2. The SVM model
When data sets are noisy and exhibit a large overlap between data classes, error variables ei> 0 are introduced
Fig. 1. The process of this approach.
Table 1
Description of samples
Samples Paired rate Number of observations Time period
(F vs. NF) F NF Total
Matched sample 1:3 22 66 88 (training and forecasting) 1998–2000
Simulated sample (random sampling by bootstrap) 1:1 15 15 30 (training) 1998–2000
7 7 14 (forecasting)
Holdout sample None 7 531 538 (forecasting for TSE in 2001) 2001–2002
14 520 534 (forecasting for TSE in 2002) 5 368 373 (forecasting for OTC in 2001) 5 351 356 (forecasting for OTC in 2002) Note: F = failed firms; NF = non-failed firms.
to allow the output of the outlier to be locally corrected, constraining the range of the Lagrange multiplier aifrom
0 to C. C is a constant penalty function designed to prevent outliers from affecting the optimal hyperplane. Hence, the non-linear objective function is
Maximize: WðaÞ ¼X l i¼1 ai 1 2 Xl i;j¼1 aiajyiyjðKðxi; xjÞÞ ð15Þ Subject to 0 6 ai6C; i¼ 1; . . . ; l; ð16Þ Xl i¼1 aiyi¼ 0. ð17Þ
The optimal weight w*and bias are determined by
solv-ing the quadratic programmsolv-ing problem. w¼X
l
i¼1
aiyixi; ð18Þ
b¼ yi wTxi ð19Þ
The optimal decision function is as follows: fðxÞ ¼ sign X l i¼1 yia iKðx; xiÞ þ b ! . ð20Þ
In machine learning theories, popular kernel functions, such as the Gaussian kernel function, have been found to provide good generalization capabilities (Campbell, 2002; Kecman, 2001). Accordingly, the Gaussian kernel function is employed as the kernel function in this work. The Gauss-ian kernel function is given by
GaussianðRBFÞ kernel: Kðxi; xjÞ ¼ exp
kxi xjk2
2r2
! . ð21Þ A kernel function K(x, z), satisfying Mercer’s condition, performs a high dimensional mapping u : RN7!F and be used as a substitute for (u(x)Æu(z)) which replaces (xÆz)
(Vapnik, 1995). Consequently, the optimal hyperplane
classification function is obtained by the SVM model to fit an optimal hyperplane between two classes in a training data set.
3.4. Chromosome representations
Unlike the traditional BGA, the RGA used to solve optimization problems, directly codes all of the corre-sponding parameters or variables in a chromosome. Hence, the representation of the chromosome is straightforward in the RGA. The two parameters, C and r, of SVM were directly coded to form the chromosome in the proposed method. The chromosome X is represented as X = {p1, p2}, where p1 and p2 denote the regularization
parameter C and sigma r (the parameter of the kernel func-tion), respectively.
3.5. The fitness function
A fitness function, assessing the performance of each chromosome, must be designed before starts to search opti-mal values of SVM parameters. Several measurement indi-cators have been developed and applied to evaluate the predictive accuracy of models; they include the hit ratio, MAPE, RMSE, and the maximum error. The hit ratio is used herein as the indicator of model performance to com-pare the results achieved by the proposed model with those obtained using other models (traditional SVM, discrimi-nant analysis, logit analysis, probit regression, and NN). The hit ratio denotes the value of the fitness function in GA-SVM.
3.6. Genetic operators
The real-valued genetic algorithm uses selection, cross-over, and mutation operators to generate the offspring of the existing population.
Table 2
List of tested financial ratios
Section Financial ratios
Liquidity Current ratio
Quick ratio Cash flow ratio
Profitability Net income to sales
Gross profit to sales Net income to total assets Net income to stockholder’s equity Operating income to sales Earning per share (EPS) Growth ratio of sales
Asset management Total asset turnover
Fixed assets turnover Inventory turnover Receivables turnover
Financial structure Debt ratio
Long-term liabilities to fixed assets Degree of financial leverage (DFL) Liabilities to stockholder’s equity Interest coverage ratio
Table 3
Parameter settings used in BPN
Parameters of NN Values
Network type FEED-FORWARD
BACK-PROPAGATION
Training function TRAINLM
Adaptive learning function LEARNGDM
Performance function MSEREG
Number of layers 3
Neurons in hidden layer 7
Transfer function of hidden layer TANSIG Transfer function of output layer PURELIN
Epochs 300
3.6.1. Selection
The proposed GA-SVM model incorporates two well-known selection methods – the roulette wheel method and the tournament method. The tournament selection method is adopted here to decide whether a chromosome can survive to the next generation. The chromosomes that survive to the next generation are placed in a matting pool for crossover and mutation operations.
3.6.2. Crossover
Once a pair of chromosomes has been selected for cross-over, one or more randomly selected positions are assigned to the to-be-crossed chromosomes. The newly crossed chromosomes are then combined with the rest of the chro-mosomes to generate a new population. However, over-loading problem frequently occurs when the RGA is used to optimize values. This study uses the method proposed
byAdewuya (1996)to prevent overload of post-crossover
when genetic algorithm with real-valued chromosomes are applied. Xold1 ¼ fx11; x12; . . . ; x1ng; Xold2 ¼ fx21; x22; . . . ; x2ng ð22Þ move closer: Xnew1 ¼ Xold 1 þ rðX old 1 X old 2 Þ; ð23Þ Xnew2 ¼ Xold 2 rðX old 1 X old 2 Þ ð24Þ move away: Xnew1 ¼ Xold 1 þ rðX old 2 X old 1 Þ; ð25Þ Xnew 2 ¼ X old 2 rðX old 2 X old 1 Þ. ð26Þ
Xold1 and Xold2 represent the pair of populations before crossover operation; Xnew1 and Xnew2 represent the pair of new populations after crossover operation. In addition, r is a random micro number that controls the variance of each crossover operations.
3.6.3. Mutation
The mutation operation follows the crossover operation and determines whether a chromosome should be mutated in the next generation. In this study, uniform mutation method is applied and designed in the presented model. Consequently, researchers can select the method of muta-tion in GA-SVM best suited to their problems of interest.
Uniform mutation
Xold¼ fx1; x2; . . . ; xng; ð27Þ
xnewk ¼ LBkþ rðUBk LBkÞ; ð28Þ
Xnew¼ fx
1; x2; . . . ; xnewk ; . . . ; xng; ð29Þ
where n denotes the number of parameters; r represents a random number in the range (0, 1), and k is the position of the mutation. LB and UB are the low and upper bounds on the parameters, respectively. LBk and UBk denote the
low and upper bound at location k. Xold represents the population before mutation operation; Xnew represents the new population following mutation operation.
4. Empirical analysis for predicting bankruptcy
Empirical analysis is divided into five sections: (1) data analysis, (2) normality testing, (3) predictive accuracy of matched samples, (4) predictive accuracy of holdout sam-ple, and (5) predictive of simulated (bootstrap) sample. 4.1. Data analysis
Experiments were performed to examine three kinds of validation: (1) internal validation (matched sample), (2) external validation (holdout sample prediction) and (3) external validation (simulated sample prediction). Besides the accuracy of the predictions of bankruptcy, Type I and Type II errors were analyzed among these experiments. Type I error was defined as the probability that a firm pre-dicted not to fail will in fact fail, while the Type II error was defined as the probability that a firm predicted to fail will not in fact fail (Blum, 1974). The SVM model is applied with fix values of parameters (Fig. 2).
The bankruptcy models in this investigation employed 19 financial variables (see Table 2), selected in previous research on financial distress, as input variables. These variables were organized into four groups, according to whether they related to liquidity, profitability, asset man-agement or financial structure. The input variables of all the models are the same. The hit ratio of classification is the indicator used to evaluate the predictive accuracy of model. The bootstrap technique has been widely used in financial research to evaluate the external validity of model in prediction. Additional evidence of the stationary of the models was obtained by other samples. Thus, this study not only evaluates the within-sample predictive capacity (internal validity) but also employs the bootstrap technique to evaluate the predictive ability in simulated sample (external validity). Table 4 describes the matched sample in this study. A total of listed 88 firms were obtained from the literature on financial distress in Taiwan.
4.2. Normality test
Most multivariate models assume that the data are nor-mally distributed. Thus, the normality of the input vari-ables must be tested before these models can be applied. This study employed the Kolmogorov–Smirnov Z test (KS Z) to test the data for 1 year prior to failure, 2 years prior to failure, and 3 years prior to fail, to determine the distribution of the experimental data. The result of normal-ity test was illustrated inTable 5. AsTable 5shows, most of financial ratios were not normally distributed as has been stated in previous research.
4.3. Predictive accuracy of matched samples
The average predictive accuracy of the failing company model is 92.61% in the first year before failure, 91.19% in the second year, and 83.81% in the third. With reference
to the first year before failure, the failed of firms predicted to non-failure (Type I error) is rarer than the non-failed of firms predicted to failure (Type II error). Artificial intelli-gence models (NN, SVM, and GA-SVM) are able to per-fectly predict bankruptcy (100% accuracy), in the first, second and third years before failure. DA exhibited the lowest predictive accuracy of all the models. DA and probit model yielded the highest Type II and Type I errors, respectively, in all years before failure (Table 6).
In practice, the cost of misclassifying a failed firm is likely to be much greater than that of misclassifying a non-failed firm. Type I is the probability of misclassifying a failed while Type II error is the probability of misclassifying a non-failed firms. In the first year before failure, predictions of failed firms not to fail (Type I error) were greater than predictions of non-failed companies to fail (Type II error). In the first year before failure, the average Type I and II errors are 4.9% and 2.6%, respectively. In the second year before failure, the average Type I and II errors are 6.5% and 2.3%, respectively. In the third year before failure, the average Type I and II errors are 11.4% and 4.8%, respec-tively. Probit model and logit model yield the highest Type I error, while DA has the highest Type II error.
4.4. Predictive accuracy of holdout sample
The holdout method, sometimes called test sample esti-mation, partitions the data into two mutually exclusive subsets called a training set and a test set, or a holdout set. Two thirds of the data are commonly used as the Fig. 2. Analysis steps.
Table 4
Description of the matched sample
Industry 1998 Number of firms 1999 Number of firms 2000 Number of firms Total Foods 8 4 4 16 Plastics 0 12 0 12
Fiber and textile 0 8 0 8
Electric, engineering, and machine
4 0 0 4
Electric wire and cable 0 4 0 4
Glass and ceramic 0 4 0 4
Steel 4 8 4 16 Motor 4 0 0 4 Electronics 0 8 0 8 Construction 4 4 0 8 Metal 4 0 0 4 Total 28 52 8 88 Table 5 Normality test Ratio Year 1 KS Z Year 2 KS Z Year 3 KS Z Liquidity Current ratio 2.120*** 1.816*** 1.546* Quick ratio 2.326*** 1.941*** 1.803**
Cash flow ratio 1.155 1.003 1.899**
Profitability
Net income to sales 2.735*** 2.689*** 2.935***
Gross profit to sales 0.579 1.046 1.046
Net income to total assets 1.976*** 2.494*** 2.747*** Net income to stockholder’s equity 3.522*** 1.074 2.466*** Operating income to sales 1.260 1.339 1.080 Earning per share (EPS) 2.477*** 1.344 3.822***
Growth ratio of sales 1.824*** 1.835*** 1.147
Asset management
Total asset turnover 1.190 1.387 1.457*
Fixed assets turnover 2.157*** 2.212*** 1.781**
Inventory turnover 1.508 1.783** 1.370* Receivables turnover 2.252*** 2.096*** 2.190*** Financial structure Debt ratio 0.816 0.500 0.606 Long-term liabilities to fixed assets 2.285*** 2.840*** 2.353*** Degree of financial leverage (DFL) 4.142*** 4.273*** 4.058** Liabilities to stockholder’s equity 2.831*** 1.476* 4.466***
Interest coverage ratio 4.211*** 4.187*** 4.504***
KS Z denotes Kolmogorov–Smirnov Z test.
* Denotes a < 0.1. **
Denotes a < 0.05.
training set and the remaining one third are then used as the test set. The training set is given to the inducer, and the induced classifier is tested on the test set. In this study, the holdout sample is the group of data unused when the financial distress models are computed. This sample is used to validate applicability of the financial bankruptcy models to a separate sample of respondents, also called the valida-tion sample (Hair, Anderson, Tatham, & Black, 1998). The holdout sample herein consists of all firms listed on the TSE and OTC in 2001 and 2002. The holdout sample there-fore includes four data files. The file (2001TSE.txt) con-tained 605 records but 67 records were deleted because data were missing. For the same reason, 69 records were removed from the file (2002TSE.txt). The file (2001OTC) included 373 records, after 33 of the original 406 records were removed and file (2002OTC) included 356 records, after 50 of the original 356 records were removed. The models used to predict financial distress were trained using the preceding year’s data. Results revealed that the pro-posed model, GA-SVM, outperformed the other bank-ruptcy models.
AsTable 7 shows, the average predictive accuracy was
87.33% in predicting companies in the TSE market that failed in 2001, and 74.9% in 2002. The proposed model,
GA-SVM, outperformed other bankruptcy models in 2001 and 2002 years. The GA-SVM had the highest predic-tive accuracy, the highest Type I error and the lowest Type II error. DA had the worst predictive accuracy of all mod-els but it had the lowest Type I error. Table 8 presents results concerning the prediction of bankruptcy in the OTC market in 2001–2002. The average predictive accu-racy is 79.50% in predicting companies in the OTC market that failed in 2001, and 78.47% in 2002. The financial dis-tress model performed better for the OTC market than the TSE market in 2001–2002. The overall predictive accu-racies exceeded that for the TSE market. The GA-SVM still outperformed among the other financial distress mod-els in 2001 and 2002 years. The GA-SVM exhibited the highest predictive accuracy, the lowest Type I and Type II errors for the 2001 OTC market. DA and probit model had the lowest predictive accuracy for the OTC market in both 2001 and 2002.
4.5. Predictive accuracy of simulated sample
The bootstrap technique was introduced by Efron and is fully elucidated in Efron and Tibshirani (1993). Given a dataset of size n, a bootstrap sample is constructed by
Table 7
Predictive accuracies of models in holdout sample (the TSE market)
Models 2001 TSE (538 firms) 2002 TSE (534 firms)
Year + 1 Year + 2
Accuracy Type I error Type II error Accuracy Type I error Type II error
DA 0.7937 0.000 0.206 0.7846 0.009 0.206 Logit 0.8773 0.007 0.115 0.6985 0.015 0.287 Probit 0.8922 0.004 0.104 0.5524 0.019 0.429 NN 0.8271 0.006 0.167 0.5131 0.017 0.470 SVM 0.8662 0.006 0.128 0.9738 0.026 0.000 GA-SVM 0.9833 0.013 0.004 0.9738 0.026 0.000 Average 0.8733 0.006 0.121 0.7494 0.019 0.232
Note: DA denotes discriminant analysis; NN denotes neural network. SVM refers to a support vector machine with fixed values of parameters.
GA-SVM denotes a support vector machine with values of parameters optimized by RGA. Type I error represents the probability of misclassifying a failed firm.
Type II error represents the probability of misclassifying a non-failed firm. Table 6
Accuracies of models on financial distress prediction
Models Year 1 Year 2 Year 3
Accuracy Type I error Type II error Accuracy Type I error Type II error Accuracy Type I error Type II error
DA 0.8750 0.023 0.102 0.8522 0.057 0.091 0.7500 0.080 0.170 Logit 0.9205 0.080 0.000 0.8978 0.102 0.000 0.8068 0.182 0.011 Probit 0.9090 0.091 0.000 0.8977 0.102 0.000 0.7955 0.193 0.011 NN 1.0000 0.000 0.000 1.0000 0.000 0.000 1.0000 0.000 0.000 SVM 1.0000 0.000 0.000 1.0000 0.000 0.000 1.0000 0.000 0.000 GA-SVM 1.0000 0.000 0.000 1.0000 0.000 0.000 1.0000 0.000 0.000 Average 0.9261 0.049 0.026 0.9119 0.065 0.023 0.8381 0.114 0.048
Note: DA denotes discriminant analysis; NN denotes neural network. SVM refers to a support vector machine with fixed values of parameters.
GA-SVM denotes a support vector machine with values of parameters optimized by RGA. Type I error represents the probability of misclassifying a failed firm.
randomly sampling n instances uniformly from the original data. In this section, the simulated sample is constructed by bootstrap techniques. The original sample of 88 enterprises
is divided into a training sample of 60 enterprises, and the remaining 28 enterprises are used as the validation sample. The ratio of the size of the training sample to that of the validation sample is designed to be 2:1. The financial dis-tress models are predicting for varying samples, by boot-strapping from 50 to 1000 times, to evaluate the reliability of validation.
Table 9 and Fig. 3 present and plot, respectively, the predictive accuracies of the bankruptcy models obtained after bootstrapping various numbers of times. As results shown, GA-SVM performed well when applied to the sim-ulated sample when bootstrapping was performed various numbers of times. Traditional SVM with fixed values of parameters had the lowest predictive accuracy, regardless of the number of bootstrapping times. The results implied that the predictive accuracies of the SVM were dramati-cally increased by using the optimal parameters. Addition-ally, GA-SVM and SVM run with fixed values (Fix-SVM) have the lowest Type I error, but that the Fix-SVM exhib-ited the highest Type II error for simulated sample (Table 10).
5. Discussions
This study develops a novel model to search the optimal values of SVM parameters, to increase the accuracy of pre-diction and ability of generalization. DA, logit, probit, NN, SVM, and the proposed model (GA-SVM) were applied to a dataset on bankruptcies in Taiwan. First, this study found that the RGA yields different optimal values of the parameters of SVM given various datasets (paired sam-Table 8
Predictive accuracies of models in holdout sample (the OTC market)
Models 2001 OTC (373 firms) 2002 OTC (356 firms)
Year + 1 Year + 2
Accuracy Type I error Type II error Accuracy Type I error Type II error
DA 0.5899 0.000 0.410 0.7416 0.006 0.253 Logit 0.7837 0.000 0.216 0.6067 0.011 0.382 Probit 0.6938 0.000 0.306 0.5056 0.011 0.483 NN 0.7135 0.000 0.287 0.8820 0.014 0.104 SVM 0.9944 0.000 0.006 0.9860 0.014 0.000 GA-SVM 0.9944 0.000 0.006 0.9860 0.014 0.000 Average 0.7950 0.000 0.2051 0.7847 0.012 0.204 Table 9
Predictive accuracies of bankruptcy models for simulated samples Bootstrap times Predictive accuracy of bankruptcy models
DA Logit Probit NN Fix-SVM* GA-SVM
50 0.68 0.69 0.69 0.69 0.64 0.72 100 0.71 0.69 0.70 0.68 0.66 0.75 200 0.71 0.69 0.69 0.70 0.65 0.75 300 0.72 0.69 0.69 0.70 0.64 0.75 400 0.72 0.70 0.71 0.70 0.65 0.76 500 0.71 0.69 0.69 0.70 0.65 0.76 1000 0.71 0.69 0.69 0.70 0.65 0.75
Note: Predictive accuracies of bankruptcy models are represented as percentages.
*
Fix-SVM denotes that SVM is run with fix values of SVM parameters.
Predictive Accuracies of Bankruptcy Models
0.6 0.65 0.7 0.75 0.8 50 100 200 300 400 500 1000 Times Predictive Accuracy DA Logit Probit NN Fix-SVM GA-SVM
Fig. 3. Predictive accuracies of bankruptcy models for simulated sample.
Table 10
Type I and II errors of bankruptcy models for simulated samples
Bootstrap times Type I error % Type II error %
DA Logit Probit NN Fix SVM GA SVM DA Logit Probit NN Fix SVM GA SVM
50 0.19 0.18 0.19 0.17 0.08 0.13 0.12 0.13 0.12 0.14 0.27 0.15 100 0.16 0.18 0.19 0.17 0.11 0.13 0.13 0.13 0.12 0.15 0.23 0.12 200 0.16 0.18 0.19 0.17 0.11 0.12 0.12 0.13 0.12 0.12 0.23 0.12 300 0.16 0.18 0.19 0.16 0.10 0.12 0.12 0.14 0.13 0.14 0.26 0.13 400 0.16 0.17 0.18 0.16 0.11 0.12 0.12 0.12 0.11 0.14 0.24 0.13 500 0.16 0.17 0.19 0.17 0.11 0.12 0.13 0.14 0.13 0.14 0.24 0.12 1000 0.17 0.18 0.19 0.17 0.11 0.12 0.13 0.14 0.13 0.14 0.24 0.13
ple, holdout sample, and simulated sample). The optimal values of SVM parameters are not constant but fall in the range and the range of each parameter differs with the dataset. This work found that the optimal values of the two parameters of SVM differed with the dataset, and changed at each time. The results reveal that the optimal values have a range, and are not constants which provide a direction for future investigation. Secondly, properly determining the values of SVM parameters were drastically increased the accuracy of the prediction of bankruptcies. The optimal kernel parameters of SVM can be automati-cally determined via the proposed approach.
Thirdly, artificially intelligent models (GA-SVM, SVM, and NN) are more accurate in predicting financial distress than other multivariate statistical models. Experimental results show that the GA-SVM model performs the best, implying that the hybrid system has a high potential to dra-matically increase the predictive accuracy when integrating GA with traditional SVM model. Fourthly, most financial ratios did not satisfy the normality assumption for multi-variate statistical models such as the MDA and the probit model. Thus, MDA exhibited the worst predictive accuracy and the largest errors of all models tested herein. Finally, the structural risk minimization principle (SRM) has been shown to be superior to traditional empirical risk minimi-zation principle (ERM) which employed by conventional neural networks, was embodied in SVM. SRM is able to minimize an upper bound of generalization error as opposed to ERM that minimizes the error on training data. Thus, the solution of SVM may be global optimum while other neural network models tend to fall into a local opti-mal solution, and overfitting is unlikely to occur with SVM (Cristianini et al., 1998; Hearst, Dumais, Osuna, Platt, & Scholkopf, 1998; Kim, 2003). Based on these reasons, we can conclude that the proposed model more accurately pre-dicts bankruptcy than the other tested models of bank-ruptcy. Additionally, the results of this work demonstrate that the predictive accuracy of the SVM in forecasting the financial distress of corporations is significantly increased by optimizing its parameters.
6. Conclusions
This study pioneered on applying the GA-SVM to finan-cial distress prediction. Therefore, the primary goal of this study is to apply this new model to increase the predictive accuracy of financial failure. Empirical results reveal that the proposed GA-SVM model is a very promising hybrid SVM model for predicting bankruptcy in terms of both predictive accuracy and generalization ability. The pro-posed GA-SVM model can be automated to determine the optimal values of SVM parameters and exhibits increased predictive accuracy in given various datasets.
Grice and Ingram (2001) reported that Altman’s Z-score
model declined when applied to various industries. In other words, the Z-score model was sensitive to industry classifi-cations. In addition, both theories and experiments have
shown that benefits do not always accrue as the computa-tional cost is increased, especially if the relative accuracies are more important that the exact values. The results imply that the models with high predictive accuracies in sample do not guarantee to the same high accuracies in holdout sample prediction. The contribution of this study demon-strated that the proposed model performed well when applied in the holdout sample, revealing the generalizabil-ity of this model to forecast financial distressed firms in various industries. Thus, the forecasting technique (GA-SVM) can be developed and coded as a commercial pack-age for investors. The result of the GA-SVM (financial distress pre-warning system) can provide a guide of invest-ment for investors and governinvest-ment.
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