Signal Regression Based On Least Square Reproducing Kernel Support Vector Machine

The support vector machine is a kind of learning mechanism which is based on the statistical learning theory and is a new technology based on the kernel which is used to solve the problems of learning from the samples. Different support vector machine which is based on different kernel can solve different practical problems, and the reproducing kernel and its corresponding reproducing kernel Hilbert space play an important role on the approximation of function and the regularization theory, thereby a kernel function which can reflect the features of a approximating function in the reproducing kernel Hilbert space has a very important practical significance. In this paper, on one hand a new kernel function is given by the reproducing kernel which is on the reproducing kernel Hilbert space to realize a signal regression of least square support vector machine, on the other hand uses Littlewood-Paley wavelet to consider the approximation of solution of the Laplace equation's initial value problem. The main results of the paper are the following:First, based on the conditions of kernel function of support vector machine and the reproducing kernel on the Sobolev Hilbert space H 1( R; a , b ), the paper gives a support vector machine kernel function which is produced by the reproducing kernel on the Sobolev Hilbert space H 1( R; a , b ), and on the theory proves that this kernel function satisfies the conditions of kernel function of support vector machine. At the same time the thesis combines the above kernel function with the least square support vector machine, provides a new regression model which is called the least square reproducing kernel support vector machine, and applies the regression model to the simulation experiment of the signal regression. The experiment shows that the reproducing kernel which the least square support vector machine adopts is feasible, and better than the usual Gaussian kernel function.Second, with the help of the Littlewood-Paley wavelet, the paper discusses the approximation of solution of the Laplace equation's initial value problem. The paper uses multi-resolution analysis method of the wavelet analysis and with the help of high frequency decay property in the Littlewood-Paley wavelet, projects the solution which is produced under the boundary condition of the Laplace function to the compactly supported function space to be a regularization, gives regular solution of the Laplace equation's initial value problem, and proves that the regular solution uniformly converges into the exact solution, which supplies a new method for further discussing the exact solution of the ill-posed equations.