Modelling of Non-Stationary Spatial Structure Using Parametric Radial Basis Deformations

Abstract

Spatial environmental processes often exhibit non-stationarity. Modelling the dispersion d(x, y) = E(Z(x) — Z(y))² , (x, y) ∈ ℝ² x ℝ², of a non-stationary spatial process Z(x) has been proposed in the last decade. It consists of deforming bijectively the geographic coordinate x so that the spatial dispersion structure can be considered stationary and isotropic in terms of a new spatial coordinate system. The model is d(x, y) = _γβ (∥f(y) — f(x)∥) where f represents a bijective transformation and γβ a stationary and isotropic variogram function with parameters β. The non-parametric family of thin-plate splines was used to compute the deformation with two drawbacks: (i) bijection condition is not ensured (ii) the fitting of the model can be a challenging numerical problem. To avoid these disadvantages we propose to use a parametric family of bijective functions we call Radial Basis Deformations. Bijection is ensured by a constraint on parameters and deformation f is a composition of a small number of radial elementary deformations. The paper presents definition and properties of these Radial Basis Deformations. We apply our parametric approach on a precipitation data set.