A support vector machine is a new learning machine; it is based on the statistics learning theory
and attracts the attention of all researchers. Recently, the support vector machines SVMs  have
been applied to the problem of financial early-warning prediction Rose, 1999. The SVMs-based
method has been compared with other statistical methods and has shown good results. But the
parameters of the kernel function which influence the result and performance of support vector
machines have not been decided. Based on genetic algorithms, this paper proposes a new scientific
method to automatically select the parameters of SVMs for financial early-warning model. The
results demonstrate that the method is a powerful and flexible way to solve financial early-warningproblem.
1. Intro duction
The development of the financial early-warning prediction model has long been regarded as
an important and widely studied issue in the academic and business community. Statistical
methods and data mining techniques have been used for developing more accurate financial
early-warning models. The statistical methods include regression, logistic models, and factor
analysis. The data mining techniques include decision trees, neural networks NNs, fuzzy
logic, genetic algorithm GA, and support vector machines SVMs  etc 1 . However, the
application of statistics was limited in the real world because of the strict assumptions.
Recently, SVM, which was developed byVapnik Vapnik1995  , is one of the methods
that is receiving increasing attention with remarkable results. In financial applications, time
series prediction such as stock price indexing and classification such as credit rating andfinancial warning are main areas with SVMs 2. However, as SVMs are applied for pattern
classification problems, it is important to select the parameters of SVMs.
This paper applies the proposed evolutionary support vector machine based on
genetic algorithms model to the financial early-warning problem using a real data set from
the companies which come into market in China.
2. Th eoretical Backgro und
2.1. Genetic Algorithm (GA)
A GA is a flexible optimization technique inspired by evolutionary notions and natural
selection. A GA is based on an iterative and parallel procedure that consists of a population
of individuals each one representing an attempted solution to the problem  which is
improved in each iteration by means of crossover and mutation, generating new individuals
or attempted solutions which are then tested by3 .
There are three main questions that have become in relevant topics in GA design
research: 1 encoding; 2 operators; 3 control parameters. The GA starts to work by
selecting a sample randomly or by means of any other procedure of potential solutions
to the problem to be solved—previously the problem has to be formulated in chromosomes
notation. In a second step the fitness value of every chromosome potential solution —in
accordance with an objective function that classifies the solutions from the best to the worst-is computed 4 . The third step applies the reproduction operator to the initial set of potential
solutions. The individuals with higher fitness values are more largely reproduced. One of the
most common methods which is used in this paper is the “roulette wheel.” This method is
equivalent to a fitness-proportionate selection method for population large enough. There are
two essential actions in the GA procedure: 1 the creation of attempted solutions or ideas to
solve the problem through recombination and mutation;2 the elimination of errors or bad
solutionsafter testing them  by selecting the better adapted ones or the closer to the truth.
2.2. Support Vector Machines (SVMs)
Since SVMs were introduced from statistical learning theory by Vapnik, a number of studies
have been announced concerning its theory and applications 5. A simple description of the
SVMs algorithm is provided as follows.
Given a training setT  {x1,y1 , x2,y2,..., xl,yl}∈ X, Y lwith input vectorsxix1
i,x2i,...,xni ∈ Rnand target labels yi ∈{−1, 1}, the support vector machinesSVMs  classifier, according to Vapnik’s theory, finds an optimal separating hyper planewhich satisfies the following conditions:H  {x ∈ Rn, ω · x   b  0},ω∈ Rn,b∈ R. 2.1with the decision functionf x  sign  ω · x b .
To find the optimal hyper plane: ω · x b  0, the norm of the vector needs to be
minimized, on the other hand, the margin 1/ω  should be maximized between two classes.
The solution of the primal problem is obtained after constructing the Lagrange.
From the conditions of optimality, one obtains a quadratic programming problem withDiscrete Dynamics in Nature and Society 3Lagrange multipliersαi’s. A multiplier αiexists for each training data instance. Data instancescorresponding to nonzero αi’s are called support vectors 6 .On the other hand, the above primal problem can be converted into the following dualproblem with objective function and constraints:
Min :12k i,j  1αiαjyiyjxixj−k i 1αis . t.αi ≥ 0,i 1, 2,...,k,k i 1αiyi  02.2
with the decision functionf x   signk i 1αi· yix · xi  b
Most of classification problems are, however, linearly nonseparable in the real world.
In the nonlinear case, we first mapped the data to a high-dimensional space, using a mapping,
φ : Rd→ H . Then instead of the form of dot products, “kernel function” K is issued suchhat K xi,xj φ xi · φ xj. We will find the optimal hyper plane: ω · φ x   b  0withthedecision function f x sign αi· y i φ x φ xi b .
In this paper, RBF kernel functions are used as follows: K x,y  e
− x − y 1/σ2.Usinghe dual problem, the quadratic programming problems can be rewritten as
Min :12k i,j  1αiαjyiyj Kxixj−k i 1αi
s. t. 0 ≤ αi ≤ C, i 1, 2,...,k,k i 1αiyi  0.2.4
3. Th e F inancial-Warning Model o f Chinese Companies
3.1. Evolutionary Support Vector Machines Based on Genetic Algorithms
As SVMs are applied for pattern classification problems, it is important to select the
parameters of SVMs7. This paper applies genetic algorithms to define the parameters of
SVMs. The steps of evolutionary support vector machines based on genetic algorithms are
given as follows.
Step 1. Define the string or chromosome and encode parameters of SVMs into chromo-
somes. In this paper, the radial basis function RBF  is used as the kernel function for financial
warning prediction. There are two parameters while using RBF kernels: C and δ2.Inthisstudy, C and δ2are encoded as binary strings and optimized by GA. In addition, the lengthof the GA chromosome used in this paper is 18. The first 9 bits represent C and the remaining9 bits represent δ2.
Step 2. Define population size and generate binary-coded initial population of chromosomes
randomly. The initial random population size is 40.
Step 3. Define probability of crossoverPc and probability of mutation Pm anddothe
operation of GAselection, crossover and mutation  .
Generate o ff spring population by performing crossover and mutation on parent pairs.
There are di ff erent selection methods to perform reproduction in the GA to choose the
individuals that will create o ff spring for the next generation 8 . One of the most common
method and the one used in this paper is the “roulette wheel.”
Step 4. Decode the chromosomes to obtain the corresponding parameters of SVMs.
Step 5. Apply the corresponding parameters to the SVMs model to compute the output ok.
Each new chromosome is evaluated by sending it to the SVMs model.
Step 6. Evaluate fitness of the chromosomes usingokfitness function: predictive accuracy
and judge whether stopping condition is true, if true end; if false, turn to Step 3 . The fitness
of an individual of the population is based on the performance of SVMs.
Considering the real problem, we define the predictive accuracy of the testing set as
the fitness function. It is represented mathematically as follows:
fitness − function n i 1Yin, 3.1
where Yi
is one, if the actual output equals the predicted value of the SVMs model, otherwise
Yiis zero.
3.2. The Selection of Input Variables
There are many financial ratios which can represent the profitability of company, and
the di ff erences between industries are obviously, such as, household appliances and
pharmaceutical industry. So the horizontal comparability of many financial ratios is not
reasonable. This paper focuses on the profitability of company, and then selects six ratios
as the input variables: 1 Sell profit rate; 2 Assets earning ratio;3 Net asset earning ratio;
4 Profit growth rate of main business; 5 Net profit growth rate; 6 Total profit growth rate
9 .
3.3. The Selection of Output Variable
We assume that the economy environment is similar, and select ROE Rate of Return on
Common Stockholders’ Equity as the standard of selection of output variable because ROE
is one of the important ratios which are used to judge the profitability 10 . The method is
represented as follows. We select those companies whose ROE is greater than 0 in the year
n − 1 and the year n, and we distinguish those companies into two kinds by judging the
numerical of ROE in yearn  1: the first kind, ROE is greater than 0, and the output is y  1;
the second kind, ROE is equal or less than 0, and the output is y  −1.
Ta b l e 1 : The training set.
Va r i a b l e sCompaniesSell profitrateAssetsearningratioNet assetearningratioProfitgrowth rateof mainbusinessNet profitgrowthrateTo t a lprofitgrowthrateN  1year’sROEOutputChenmingPaper
0.09132 0 .03615 0 .0415 −0.3056983 −0.429 −0.3424 0.1077 1Foshanelectrical andlighting0.24647 0 .09228 0 .0909 0 .29356763 0 .0757 0 .109821 0 .1044 1Huali Group 0.30593 0 .13124 0 .1806 −0.1730657 −0.148 −0.11433 0.1561 1GreeElectricappliances0.04546 0 .04738 0 .1568 −0.1580513 0.0949 0 .100019 0 .1617 1ZhuhaiZhongfu0.19179 0 .05514 0 .0661 −0.2462878 −0.086 −0.11713 0.1011 1Zijiangenterprise0.26843 0 .10353 0 .1346 0 .46888081 0 .3058 0 .320772 0 .1621 1Qingdao Haier 0.23414 0 .14571 0 .1253 0 .12413238 0 .5833 0 .546907 0 .078 1Fujian Nanzhi 0 .12411 0 .05107 0 .0734 0 .13528976 0 .1961 0 .050202 0 .0619 1ST Swan 0 .03803 0 .02276 0 .0012 −0.7855768 −0.987 −0.69645 −0.326 −1ST Macro 0 .08378 0 .01404 0 .2073 −0.0473601 −0.85 −0.89666 −3.943 −1
ST Tianyi 0 .05505 0 .01348 0 .0096 −0.2675278 −0.865 −0.83182 −0.281 −1
ST Jizhi 0 .08603 0 .01643 0 .005 −0.3048749 −0.575 −0.73226 −0.24 −1
ST Hushan 0 .16272 0 .04433 0 .0503 0 .09753527 −0.799 −0.81446 −0.256 −1
ST Jiangzhi 0 .30537 0 .09123 0 .0813 0 .41089893 0 .1747 0 .220284 −0.86 −1
ST Ziyi 0 .10243 0 .03277 0 .0014 −0.1932007 −0.986 −0.96686 −0.32 −1
Xiaxinelectronic0.04935 0 .05256 0 .0552 −0.428435 −0.765 −0.78969 −0.335 −1
Chunlan Gufen 0.21223 0 .09518 0 .0787 −0.3200746 −0.121 −0.11254 0.0405 1
ShangfengIndustrial0.44223 0 .101126 0 .0598 . −0.5987261 −0.281 −0.30444 0.0327 1
Aucma 0.10466 0 .03811 0 .0363 0 .3780161 0 .6449 0 .720158 0 .0286 1
XinjiangTianhong0.10348 0 .04947 0 .0451 −0.0591081 −0.155 −0.25771 0.0406 1
Jincheng Paper 0 .1038 0 .03736 0 .051 −0.4548067 −0.406 −0.48903 0.0145 1
WuzhongYibiao0.33955 0 .06541 0 .0631 0 .26631751 0 .046 0 .007402 0 .0125 1
Qingshan Paper0.24708 0 .06499 0 .071 −0.0290705 −0.137 −0.13544 0.0088 1
Hakongtiao 0.29178 0 .12776 0 .1819 0 .20166437 0 .9176 0 .578143 0 .0104 1
3.4. Research Data and Experiments
The research data we employ is from the companies which come into market in China, and
consists of 50 medium-size firms from 1999 to 2001. The data set is arbitrarily split into two
subsets; about 50% of the data is used for a training set and 50% for a testing set. The training
data for SVMs is totally used to construct the model. The testing data is used to test the results
with the data that is not utilized to develop the model. The training set is shown atTable 1.
Ta b l e 2 : Classification accuracies of various parameters in the first model.
δ
2
1 10305080
CtrainTesttraintesttraintesttraintesttraintest
1 86.87 87.50 80.00 70.83 66.67 70.83 66.67 70.83 66.67 70.83
10 93.33 75.50 86.67 70.83 66.67 70.83 66.67 70.83 66.67 70.83
30 93.33 75.50 86.67 70.83 66.67 70.83 66.67 70.83 66.67 75.00
50 100 79.17 86.67 79.17 66.67 70.83 66.67 70.83 66.67 75.00
90 100 79.17 93.33 87.50 66.67 70.83 66.67 70.83 66.67 75.00
100 100 79.17 93.33 79.17 66.67 70.83 66.67 79.17 66.67 70.83
150 100 79.17 86.67 79.17 66.67 70.83 66.67 79.17 66.67 70.83
200 100 79.17 86.67 75.00 66.67 70.83 66.67 70.83 66.67 70.83
250 100 79.17 86.67 79.17 66.67 70.83 66.67 70.83 66.67 70.83
Ta b l e 3 : Prediction accuracy of the second model.
Training Testing
93.33 87.50
Additionally, to evaluate the eff ectiveness of the proposed model, we compare two
di ff erent models.
1 The first model, with arbitrarily selected values of parameters, varies the
parameters of SVMs to select optimal values for the best prediction performance.
2 We design the second model as a new scientific method to automatically select the
parameters optimized by GA.
4. Experimental Results
4.1. Classification Accuracies of Various Parameters in
the First Model (Table 2)
Based on the results proposed by Tay and Cao 2001 , we set an appropriate range of
parameters δ2as follows: a range for kernel parameter is between 1 and 100 and a range for C
is between 1 and 250. Test results for this study are summarized in Table 2. Each cell of Table 2
contains the accuracy of the classification techniques. The experimental result also shows that
the prediction performance of SVMs is sensitive to the various kernel parameters δ2and the
upper bound C. InTable 2, the results of SVMs show the best prediction performances when
δ2is 10 and C is 90 on the most data set of various data set sizes. Figure 1 gives the results of
SVMs with various C where δ2is fixed at 10.
4.2. Results of Evolutionary Support Vector Machines Based on
Genetic Algorithms
Table 3describes the prediction accuracy of the second model. In Pure SVMs, we use the best
parameters from the testing set out of the results. In Table 3, the proposed model shows better
performance than that of the first model.
Discrete Dynamics in Nature and Society
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Accuracy of training set
Accuracy of testing set
Figure 1: The results of SVMs with various C where δ2is fixed at 10.
The results in Table 3show that the overall prediction performance of the second
model on the testing set is consistently good. Moreover, the accuracy and the generalization
using evolutionary support vector machines are better than that of the first model.
5. Conclusions
In this paper, we applied evolutionary support vector machines based on genetic algorithms
to financial early-warning problem and showed its attractive prediction power compared
to the pure SVMs method. In this paper we utilize genetic algorithms in order to choose
optimal values of the upper bound C and the kernel parameter δ2that are most important in
SVMs model selection. To validate the prediction performance of this evolutionary support
vector machines based on genetic algorithms model, we statistically compared its prediction
accuracy with the pure SVMs model, respectively. The results of empirical analysis showed
that proposed model outperformed the other methods.
In a classification problem, the selection of features is important for many reasons:
good generalization performance, running time requirements, and constraints imposed by
the problem itself 11  . While this study used six ratios as a feature subset of SVMs model,
it should be noted that the appropriate features can be problem-specific; hence it remains
an interesting topic for further study to select proper features according to the types of
classification problems.
Obviously, after the application of genetic algorithms, there is a significant improve-
ment in the accuracy. That is just what we need in the selection of parameters of SVMs for
financial early-warning model.
References
1 J. Wang, “The statistical properties of the interfaces for the lattice Widom-Rowlinson model,” Applied
Mathematics L etters, vol. 19, no. 3, pp. 223–228, 2006.
2 V. A. Kholodnyi, “Valuation and hedging of European contingent claims on power with spikes: a
non-Markovian approach,”Journal of E ngineering Mathematics , vol. 49, no. 3, pp. 233–252, 2004.
3 T. M. Liggett,Interacting Particle S ystems , vol. 276 of Grundlehren der Mathematischen Wissenschaften,
Springer, New York, NY, USA, 1985.
4 V. A. Kholodnyi, “Universal contingent claims in a general market environment and multiplicative
measures: examples and applications,”Nonlinear Analysis: T heory, Methods & Applications, vol. 62, no.
8, pp. 1437–1452, 2005.
5 D. Lamberton and B. Lapeyre,Introduction to Stochastic Calculus Applied t o F inance , Chapman &
Hall/CRC, Boca Raton, Fla, USA, 2000.
6 V. A. Kholodnyi, “Modeling power forward prices for power with spikes: a non-Markovian
approach,” Nonlinear Analysis: T heory, Methods & Applications, vol. 63, no. 5–7, pp. 958–965, 2005.
7 J. Wang, “Supercritical Ising model on the lattice fractal—the Sierpinski carpet,” Modern Physics L etters
B, vol. 20, no. 8, pp. 409–414, 2006.
8 P. Billingsley, Convergence of Probability Measure s , John Wiley & Sons, New York, NY, USA, 1968.
9 R. S. Ellis,Entropy, Large D eviations, and Statistical Mechanics, vol. 271 of Grundlehren der Mathematischen
Wissenschaften, Springer, New York, NY, USA, 1985.
10  F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of P olitical Economy ,
vol. 81, pp. 637–654, 1973.
11  Y. Higuchi, J. Murai, and J. Wang, “The Dobrushin-Hryniv theory for the two-dimensional lattice
Widom-Rowlinson model,” inStochastic Analysis on Large S cale Interacting S ystems,vol.39of Advanced
Studies in P ure Mathematics , pp. 233–281, Mathematical Society of Japan, Tokyo, Japan, 2004.