Discrete sample estimation for gaussian random fields generated by stochastic partial differential equations

Abstract

The article deals with parameter estimation for Gaussian random fields generated by linear stochastic partial differential equations. Using Hilbert space methods and a particular feature of the differential operator on the left side of the equation we derive estimating equations for discrete and continuous observations. Commonly known results obtained by such techniques usually involve eigenvalues, eigenvectors or test functions related to the partial differential operator and its domain. We provide the estimators directly in terms of the observed process and its covariance function. Simulations show that the estimating equations can be effectively applied to fairly small sets of discrete, irregularly sampled data. The equations are easy to evaluate and do not involve the inverse of the covariance matrix nor its determinant which makes them exceptionally useful for computational purposes, especially when the sample size is large.