Karhunen-Loeve Expansions And Positive Definite Of Covariance Function Of Gaussian Process

A random process is an important means of researching an accidental phenomenon. Gaussian process is the most important random process. It is the important tool of signal processing, closely link to kernel function, compact operators and small deviation probability. And there are very good characteres: the linear combination and covariance function is positive definite, independent is equivalent to unrelated, etc.In this dissertation, the gaussian process are the core, we discussed several types of the covariance function positive definite, and Karhunen-Loeve expansions, finally constructed corresponding reproducing kernel Hilbert spaces with its covariance function. This paper mainly researches the following problems:(1) researches the positive definite function's definition, character, discriminant method on complex field. By using the inner product of the decomposition and half a positive definite matrices, we researched positive definite function's discriminant, and according to the positive definite matrix constructed the positive definite function.(2) researches the positive definite of gaussian process covariance function, and obtained some kinds of gaussian process's Karhunen-Loeve expansion.(3) according to the gaussian process Karhunen-Loeve expansion contact with reproducing kernel Hilbert spaces, constructed the reproducing kernel Hilbert spaces correspond with the gaussian process.This paper's structure is as follows: In the first chapter, we introduced this topic's background, present situation and other relevant circumstances; in the second chapter, we systematically researched the positive definite of binary function; in the third chapter, we given several types of Karhunen-Loeve expansion of gaussian process; in the fourth chapter, we structured corresponding reproducing kernel Hilbert space with gaussian process which is yielded in third chapter.