估计灰色Verhulst模型参数的LS-SVM方法及应用

Estimation of Grey Verhulst Model Parameter Based on LS-SVM Method and Its Application

估计灰色Verhulst模型中的参数通常采用最小二乘法,这种基于大样本理论的经验风险最小化方法无法保证小样本预测下模型的推广性能.为提高灰色Verhulst模型的预测精度,本文提出了基于LS-SVM算法估计模型参数的方法.首先根据Verhulst灰色差分方程的特点,通过构造以背景值序列和原始序列为训练样本的LS-SVM模型,将一维样本空间里的Verhulst模型转化为一个二维特征空间里的LS-SVM模型,进而将Verhulst模型的灰参数的估计问题转化为一个LS-SVM模型的回归系数估计问题.然后通过核函数构造法,结合模型特点合理构造了LS-SVM模型的核函数,基于LS-SVM算法求解回归系数,进而得到Verhulst模型的参数估计.实验结果表明该方法是可行的有效的,可保证Verhulst模型具有良好推广性,相比于传统参数估计方法本文预测精度更高.

The grey Verhulst model is modeled for small sample data.To use the grey Verhulst model for prediction, the grey parameters in the model must be determined first. The prediction accuracy of the model is directly affected by the quality of grey parameter estimation. It is estimated that the parameters in the grey Verhulst model usually use the least squares method. However, this empirical risk minimization method based on the large sample theory can not guarantee the generalization performance of the model under small sample prediction.In order to enhance the prediction accuracy of grey Verhulst model, an estimation of grey Verhulst model parameter based on LS-SVM method is presented. First, according to the characteristics of Verhulst grey difference equation, the Verhulst model in one-dimensional sample space is transformed into a LS-SVM model in a two-dimensional feature space by background sequence and original sequence as training samples. There is a corresponding relationship between the coefficients of the LS-SVM model and the parameters of the Verhulst model. In this way, the estimation of the grey parameter of the Verhulst model is transformed into the regression coefficient estimation problem of an LS-SVM model. Then, based on the characteristics of the above LS-SVM model, a kernel function K(x,y)=〈x,y〉+〈x,y〉;is constructed. The kernel function adapts to the characteristics of the saturated data sequence. At last the above function is used as the kernel function of the LS-SVM algorithm to solve the regression coefficients, and then the parameter estimation of the Verhulst model is obtained.In our numerical analysis, the settlement monitoring is chosen for South-to-North Water Diversion as an example. And the after test rule is used to verify the effect prediction. The experiment results show that the results by the proposed method reach the first order accuracy, which verified the method is feasible and effective;On the other hand, based on the proposed method, by adjusting the factor λ in the LS-SVM model, the complexity and generalization of the model can be compromised, which is helpful to improve the prediction effect of the model. This method in this paper extends the application scope of the grey Verhulst model.