A Wavelet Kernel-Based Primal Twin Support Vector Machine for Economic Development Prediction

Economic development forecasting allows planners to choose the right strategies for the future. This study is to propose economic development prediction method based on the wavelet kernel-based primal twin support vector machine algorithm. As gross domestic product (GDP) is an important indicator to measure economic development, economic development prediction means GDP prediction in this study. The wavelet kernel-based primal twin support vector machine algorithm can solve two smaller sized quadratic programming problems instead of solving a large one as in the traditional support vector machine algorithm. Economic development data of Anhui province from 1992 to 2009 are used to study the prediction performance of the wavelet kernel-based primal twin support vector machine algorithm. The comparison of mean error of economic development prediction between wavelet kernel-based primal twin support vector machine and traditional support vector machine models trained by the training samples with the 3–5 dimensional input vectors, respectively, is given in this paper. The testing results show that the economic development prediction accuracy of the wavelet kernel-based primal twin support vector machine model is better than that of traditional support vector machine.


Introduction
As policy makers examine the future economic plans for their regions, economic development forecasting allows planners to choose the right strategies for the future [1,2].Gross domestic product (GDP) is an important indicator to measure economic development.Thus, economic development prediction means GDP prediction in this study.Artificial neural networks are the popular prediction algorithms, which have high parallel processing and error tolerance ability [3][4][5].Bildirici et al. applied artificial neural networks to economic development prediction, and according to tests of equal forecast accuracy, the results suggest obvious advantages of artificial neural networks compared to regression analysis algorithm [6].Kim et al. presented early warning system of economic crisis based on artificial neural networks; the experimental results indicated that artificial neural networks can predict economic growth effectively [7].However, artificial neural networks have the shortcomings of local extremum and overfitting [8][9][10].Support vector machine based on the statistical learning theory has already outperformed most other prediction algorithms [11][12][13][14].This study is to propose economic development prediction method based on the wavelet kernel-based primal twin support vector machine algorithm (WPTSVM).The wavelet kernel-based primal twin support vector machine algorithm can solve two smaller sized quadratic programming problems instead of solving a large one as in the traditional support vector machine algorithm.In the wavelet kernel-based primal twin support vector machine, Morlet wavelet function [15][16][17][18] can be used as its kernel function.
Economic development data of Anhui province from 1992 to 2009 are used to study the prediction performance of the wavelet kernel-based primal twin support vector machine algorithm.In this experiment, we employ the training samples with different dimensional input vector to train the wavelet kernel-based primal twin support vector machine algorithm.The comparison of the prediction values between the wavelet kernel-based primal twin support vector machine model and traditional support vectors machine model trained by the training samples with 3-, 4-, and 5dimensional input vectors, respectively, is given; and the comparison of the prediction error between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the training samples with 3-, 4-, and 5-dimensional input vectors, respectively, is given.And the comparison of mean error of economic development prediction between wavelet kernelbased primal twin support vector machine and traditional support vector machine model trained by the training samples with the 3-5 dimensional input vectors, respectively, is given.It can be seen that the economic development prediction accuracy of the wavelet kernel-based primal twin support vector machine model is better than that of traditional support vector machine.
The organization of this paper has been described as follows: wavelet kernel-based primal twin support vector machine has been introduced in Section 2; experimental analysis of economic development prediction method based on the wavelet kernel-based primal twin support vector machine algorithm is described in Section 3; and Section 4 gives the conclusions.

The Proposed Wavelet Kernel-Based Primal Twin Support Vector Machine
The wavelet kernel-based primal twin support vector machine algorithm can solve two smaller sized quadratic programming problems instead of solving a large one as in the traditional support vector machine algorithm.Given a set of training sets {(  ,   )}  =1 ⊂   × , where   and   denote the input vector and corresponding output, the two optimization problems of primal twin support vector machine can be modified as follows. Minimize subject to Minimize subject to where   ,  = 1, 2, 3, 4, are the penalty factors;  1 ,  2 denote the weight vector;  1 ,  2 denote bias term; ,  are the positive slack variables; and (⋅) is the mapping function.
Then, we can obtain primal twin support vector machine by minimizing the following equation: where  = (,   ) and  0 is a given -dimensional vector; and  is the Lagrangian multiplier.
It is well known that primal twin support vector machine with appropriate structure is able to gain excellent nonlinear regression function.In the study, wavelet kernel function can be used to deal with input variables of primal twin support vector machine.
Here, Morlet wavelet function can be selected as the kernel function of the proposed primal twin support vector machine, which can be described as follows: where   are the coefficients of wavelet function.

Experimental Analysis
Economic development data of Anhui province from 1992 to 2009 [19] are used to study the prediction performance of the wavelet kernel-based primal twin support vector machine  algorithm.The experiment process of the wavelet kernelbased primal twin support vector machine algorithm can be shown in Figure 1; the training samples are created as follows: where  is the dimension of the input vector.
If the dimension of the input vector is set to 3, the training samples of this experiment can be described as follows: In this experiment, we employ the training samples with different dimensional input vectors to train the wavelet kernel-based primal twin support vector machine algorithm.
Figure 2 gives the comparison of the prediction values between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the training samples with 3-dimensional input vector, respectively; and the comparison of the prediction error between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the training samples with 3dimensional input vector, respectively is given in Figure 3.Then, the comparison of the prediction values between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the training samples with 4-dimensional input vector, respectively, is given in Figure 4; and Figure 5 gives the comparison of the prediction error between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the regression training samples with 4-dimensional input vector, respectively.
Finally, Figure 6 gives the comparison of the prediction values between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the training samples with 5dimensional input vector, respectively; and the comparison of the prediction error between the wavelet kernel-based primal twin support vector machine model and traditional support vector machine model trained by the regression training samples with 5-dimensional input vector, respectively, is given in Figure 7. Table 1 gives the comparison of mean error of economic development between wavelet kernelbased primal twin support vector machine and traditional support vector machine models trained by the training samples with the 3-5-dimensional input vectors, respectively.It can be seen that the economic development prediction accuracy of the wavelet kernel-based primal twin support vector machine model is better than that of traditional support vector machine.

Conclusions
The wavelet kernel-based primal twin support vector machine algorithm is proposed to predict economic development in the paper.The WPTSVM algorithm can solve two smaller sized quadratic programming problems instead of solving a large one as in the traditional support vector machine algorithm.Morlet wavelet function is employed to construct the primal twin support vector machine.Then, we use multistep prediction mode to indicate the practicability and stability of the wavelet kernel-based primal twin support vector machine algorithm.In this experiment, we employ the training samples with different dimensional input vectors to train the wavelet kernel-based primal twin support vector machine algorithm.The comparison of mean error of economic development prediction between wavelet kernel-based primal twin support vector machine and traditional support vector machine modelstrained by the training samples with the 3-5-dimensional input vectors, respectively is given in this paper.The testing results show that the economic development prediction accuracy of the WPTSVM model is better than that of traditional SVM.

Figure 1 :
Figure 1: The experiment process of the wavelet kernel-based primal twin support vector machine algorithm.

Figure 2 :
Figure 2: The comparison of the prediction values between the WPTSVM model and traditional support vector machine model trained by the training samples with 3-dimensional input vector, respectively.

Figure 3 :Figure 4 :
Figure 3: The comparison of the prediction error between the WPTSVM model and traditional support vector machine model trained by the training samples with 3-dimensional input vector, respectively.

Figure 5 :Figure 6 :
Figure 5: The comparison of the prediction error between the WPTSVM model and traditional support vector machine model trained by the regression training samples with 4-dimensional input vector, respectively.

Figure 7 :
Figure 7: The comparison of the prediction error between the WPTSVM model and traditional support vector machine model trained by the regression training samples with 5-dimensional input vector, respectively.

Table 1 :
The comparison of mean error of economic development prediction values between WPTSVM and SVM.