Optimal linear prediction (also known as kriging) of a random field Z indexed by a compact metric space can be obtained if its second-order structure, that is the mean and the covariance function of Z, are known. In this talk, I will consider the problem of predicting the value of Z(x*) at some location x* based on n observations at locations x_1, …, x_n, which accumulate at x* as n increases – or, more generally, predicting f(Z) based on f_1(Z), …, f_n(Z) for linear functionals f, f_1, …, f_n. The main result of my talk characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure, without any restrictive assumptions such as stationarity. Specifically, I will present necessary and sufficient conditions on the mean and covariance function for asymptotic optimality of the corresponding linear predictor.
In the second part of my talk, I will discuss several applications of these general results including the class of generalized Whittle–Matérn Gaussian random fields. For this class of random fields, the covariance operators are negative fractional powers of elliptic second-order differential operators formulated on a bounded Euclidean domain and augmented with homogeneous Dirichlet boundary conditions. These outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.
This talk is based on joint work with David Bolin (King Abdullah University of Science and Technology).