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

Authors: Fang Su; HaiYang Shang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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.


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## Body

## 1. 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 [3–5]. 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 [8–10]. Support vector machine based on the statistical learning theory has already outperformed most other prediction algorithms [11–14]. 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 [15–18] 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 5-dimensional 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 kernel-based 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 Section2; 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.


## 2. 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{(xi,yi)}i=1n⊂Rm×R, where xi and yi denote the input vector and corresponding output, the two optimization problems of primal twin support vector machine can be modified as follows.Minimize(1)T(w1,ξ)=[12(∥w1∥22+b12)+C12×∥f-(θ(A)w1+eb1)∥22+C2e′ξ[12(∥w1∥22+b12)+C12]
subject to
(2)Y-(θ(A)w1+eb1)≥eε1-ξ,ξ≥0.
Minimize
(3)T(w2,η)=[12(∥w2∥22+b22)+C32×∥h-(θ(A)w2+eb2)∥22+C4e′η[12(∥w2∥22+b22)+C32]
subject to
(4)(θ(A)w2+eb2)-Y≥eε2-η,η≥0,
where Ci, i=1,2,3,4, are the penalty factors; w1, w2 denote the weight vector; b1, b2 denote bias term; ξ, η are the positive slack variables; and θ(·) is the mapping function.Then, we can obtain primal twin support vector machine by minimizing the following equation:(5)F(α)=[12(α′Kα+b12)+C12×∥(Kα+eb1)-f∥22+C2e′l·(Kα+eb1-f,μ0)[12(α′Kα+b12)+C12],
where K=K(A,A′) and u0 is a given n-dimensional vector; and α 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:(6)K(x,x′)=∏i=1mcos(1.75×x-x′ai)exp(-∥x-x′∥22ai2),
where ai are the coefficients of wavelet function.It is sufficient to prove the inequality:(7)F[x](w)=(2π)-m/2∫exp(-j(w·x))k(x)dx≥0,
where
(8)k(x)=∏i=1mcos(1.75xai)exp(-∥x∥22ai2)∫exp(-j(w·x))k(x)dx=∏i=1m|ai|2π2(+exp(-(1.75-wiai)22))exp(-(1.75+wiai)22)+exp(-(1.75-wiai)22)).


## 3. 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 kernel-based primal twin support vector machine algorithm can be shown in Figure 1; the training samples are created as follows:
(9)X=[x1x2⋯xn-m]=[a1a2⋯ama2a3⋯am+1⋯⋯⋯an-man-m+1⋯an-1],Y=[y1y2⋯yn-m]=[a1a2⋯an-m],
where m is the dimension of the input vector.Figure 1
The experiment process of the wavelet kernel-based primal twin support vector machine algorithm.If the dimension of the input vector is set to 3, the training samples of this experiment can be described as follows:(10)X=[x1x2x3⋯]=[d1992d1993d1994d1993d1994d1995d1994d1995d1996⋯⋯⋯],Y=[y1y2y3⋯]=[d1995d1996d1997⋯].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.Figure2 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 3-dimensional 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.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
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 4
The comparison of the prediction values between the WPTSVM model and traditional support vector machine model trained by the training samples with 4-dimensional input vector, respectively.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.Finally, Figure6 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 5-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 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 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. 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.Table 1
The comparison of mean error of economic development prediction values between WPTSVM and SVM.
The dimensions of input vector
Mean error of PTSVM
Mean error of SVM
3
0.0253
0.0410
4
0.0194
0.0336
5
0.0283
0.0462Figure 6
The comparison of the prediction values between the WPTSVM model and traditional support vector machine model trained by the training samples with 5-dimensional input vector, respectively.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.


## 4. 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.



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*Source: 875392-2013-08-20.xml*