A Generalized Convolution Model and Estimation for Non-stationary Random Functions

In this paper, a new model for second order non-stationary random functions as a convolution of an orthogonal random measure with a spatially varying random weighting function is introduced. The proposed model is a generalization of the classical convolution model where a non-random weighting function is considered. For a suitable choice of the random weighting functions family, this model allows to easily retrieve classes of closed-form non-stationary covariance functions with locally varying geometric anisotropy existing in the literature. This offers a clarification of the link between these latter and a convolution representation, thereby allowing a better understanding and interpretation of their parameters. Under a single realization and a local stationarity framework, a parameter estimation procedure of these classes of explicit non-stationary covariance functions is developed. From a local stationary variogram kernel estimator, a weighted local least-squares method in combination with a kernel smoothing method is used to estimate efficiently the parameters. The proposed estimation method is applied on soil and rainfall datasets. It emerges that this non-stationary method outperforms the traditional stationary method, according to several criteria. Beyond the spatial predictions, we also show how conditional simulations can be carried out in this non-stationary framework.