Observation-State Representation of Non Stationary Spatial Processes

Let {Z(x):x\in D} be a stochastic process in a domain D\subset R^d, d>=1. To apply statistical procedures, it is often necessary to decompose the process into several parts: mean trend structures with large-scale variations, second-order stationary processes with smooth small-scale and micro-scale variations and eventually white noise (measurement error). Although this additive partitioning is of considerable utility it also has several drawbacks.
In this talk, I present a new decomposition motivated by the state-space representation of time series. Let Z(x)=W(x)+\epsilon(x), (observation equation) and W(x)=\int_D k(x,s)W(s)ds+Y(x), x\in D, (state equation), where the kernel k(x,s) is a sufficiently regular function, Y(x) is a second-order stationary spatial process and \epsilon(x) is a zero-mean white-noise. The state at the point x is then a weighted mean of its neighborhood states - expressed as a Fredholm integral equation of second kind - plus a second-order stationary spatial process. For several classes of kernels the state equation has an explicit solution. The parameters of the kernel and the second order structure of Y(x) can be expressed in terms of the second order moments of Z(x).
Other existing decompositions can be reconstructed by the new representation. The new model takes account of diverse shapes of trends and one does not have to decide whether the process is stationary or not. I develop and discuss estimates of the parameters of the covariogram of Y(x) based on a high dimensional nonlinear optimization. I discuss the efficiency of the proposed method and compare the results with those of other common models. The new method performs particularly well when a small non stationary component is added to the second-order process. Whereas classical methods break down because they are unable to detect the non stationarity.
A Matlab interface allows the user to execute simulations of spatial processes in one or two dimensions and to treat real data.
I illustrate the method by applying it to a data set containing measurements of precipitation in Switzerland and show that the new approach furnishes a competitive model despite its complexity.

Keywords: Spatial decomposition, non parametric trend estimation, covariance estimation, integral equations, nonlinear optimization.