Nonlinear Analysis Discuss Based On Kernel Function

Nonlinear data processing methods based on kernel trick had made great progress and are used gradually. In the paper, Firstly, on the base of kernel-function's theory, principal characteristics of kernel-based methods are described. The main idea of kernel methods is original input space data are mapped into high dimension feature spaces through nonlinear mapping, Data is applied to class or regression in the feature spaces. Its key is inducted into kernel function, that scalar product operation in high dimension feature is transformed into kernel function compute in input space, so non-linearization is achieved in input spaces. Secondly, some traditional linear methods(principal component analysis,canonical correlation analysis,fisher discriminate analysis) which is reconstructed into kernel principal component analysis,kernel canonical correlation analysis,kernel fish linear discriminate analysis through kernel methods is introduced, and analyze its work principle and mathematics model. Meanwhile, we have used principal component analysis and kernel principal component analysis of China's 30 provinces and municipalities in 1993 peasant family consumption is analyzed . Come to use traditional principal component analysis, The first principal component's contribution rate of 40. 37%, The anterior four eigenvalues's cumulative contribution is rate of 88.45%, The first principal component to use kernel principal component analysis on the contribution rate of 95%,kenel principal component analysis shows a better results than principal components analysis in dimensions reduction. Thirdly, kernel methods is introduced in regression analysis, they are support vector regression,kernel partial least squares regression. kernel partial least squares regression is the expansion through nonlinear mapping in feature space of linear partial least squares regression, partial least squares validity of data processing is based on the linear relationship between data, When the nonlinear relationship between variables, we can map data into high-dimensional feature space, So their relationship is the linear, This can be used in the feature space deal with data through partial least squares, the so-called kernel partial least squares methods .Lastly, classical Tikhonov regularized problem have a kernel simple form.