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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 875392, 6 pages
http://dx.doi.org/10.1155/2013/875392
Research Article

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

1College of Economics, Lanzhou University of Technology, 287 Langongping Road, Qilihe District, Lanzhou City 730050, China
2Lanzhou University of Finance and Economics, Gansu 730050, China

Received 17 May 2013; Revised 6 July 2013; Accepted 8 July 2013

Academic Editor: Vishal Bhatnagar

Copyright © 2013 Fang Su and HaiYang Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [35]. 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 [810]. Support vector machine based on the statistical learning theory has already outperformed most other prediction algorithms [1114]. 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 [1518] 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 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.

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 , 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 , , are the penalty factors; , denote the weight vector; , 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 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.

It is sufficient to prove the inequality: where

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: where is the dimension of the input vector.

875392.fig.001
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:

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 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.

875392.fig.002
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.
875392.fig.003
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.
875392.fig.004
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.
875392.fig.005
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, 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 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.

tab1
Table 1: The comparison of mean error of economic development prediction values between WPTSVM and SVM.
875392.fig.006
Figure 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.
875392.fig.007
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.

Acknowledgments

This work is supported by (1) Chinese Natural Science Foundation “Research on the Ecological Compensation Mechanism of the Inner River Basins of the Northwestern China—A Case Study of Shiyang River Basin (Grant no. 41171116)”; (2) General Research on Social Sciences of Ministry of Education of China (Grant no. 12YJAZH110).

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