Spatial prediction

Abstract

In this chapter, we consider the problem of using the available data to predict aspects of the realised, but unobserved, signal S(·). More formally, our target for prediction is the realised value of a random variable T = T (S), where S denotes the complete set of realised values of S(x) as x varies over the spatial region of interest, A. The simplest example of this general problem is to predict the value of the signal, T = S(x), at an arbitrary location x, using observed data Y = (Y₁, ..., Y_n), where each Y_i represents a possibly noisy version of the corresponding S(x_i). Other common targets T include the integral of S(x) over a prescribed sub-region of A or, more challengingly, a non-linear functional such as the maximum of S(x), or the set of locations for which S(x) exceeds some prescribed value. In this chapter, we ignore the problem of parameter estimation, in effect treating all model parameters as known quantities.