Maximum Likelihood Estimation For A Gaussian Process With Special Covariance Functions

The Gaussian process is a kind of common and important stochastic processes,it can be viewed as a collection of random variables,in the collection,any combinations of the set of random variables obey the joint Gaussian distribution.The Gaussian process has many strong points,it is adopted in many fields,such as in machine learning,it is simple and practical,simultaneously,it has high applicability and accuracy for prediction.Common Gaussian processes include linear Gaussian process,Brownian motion,exponential Gaussian process,and symmetrical Gaussian process,and so on.In general cases,the Gaussian process is determined by the mean function and the covariance function.In most applications,the mean function is given,so the Gaussian process is determined by the covariance function uniquely,the Gaussian process corresponds to the covariance function,therefore the study for the Gaussian process is the research for the covariance function.In most cases,some parameters of the covariance function are unknown,we need to use datas for estimation,common methods for estimation include the maximum likelihood estimation,Bayesian estimation,and so on.In this paper,we study the Gaussian process in a stable and real-value stochastic space,the mean function is given and we need to estimate parameters of the covariance function.For estimating unknown parameters of the covariance function,we take the maximum likelihood estimation,in the process of solving parameters,we employ Newton iterative method to find the approximate solution.More specifically,we choose the native form and the tensor product form to study,at the same time,we use Newton iterative method to estimate approximate solutions for unknown parameters of them,finally,we conduct the numerical simulation and the analysis for parameters of the native covariance function.