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**Estimating a Non-Stationary Spatial Structure by Simulated Annealing**

PERRIN, Olivier (olivier@math.chalmers.se) INRA - Unite de Biometrie, Domaine Saint-Paul, Site Agroparc, 84914 Avignon Cedex 9, France; and IOVLEFF, Serge, SABRES - IUP de Vannes, Rue Yves Mainguy-Tohannic, 56000 VANNES, France

Key Words: bijection, estimation, prediction

The underlying correlation structure of spatial environmental processes often exhibits non-stationarity. Sampson and Guttorp (1992) model the correlation r(x,y) of the spatial random field Z={Z(x), x belonging to |R^2}, as a function of Euclidean distance between locations in a bijective deformation of the geographic coordinate system. The model for r(x,y) is r(x,y)=K(||f(x)-f(y)||) where ||.|| represents the classical Euclidean norm in |R^2, f represents a bijective bi-continuous deformation and K is a known stationary and isotropic correlation function.

We propose to estimate f with a simulated annealing according to the rules of the Metropolis algorithm, with non-folding constraints, when the random field Z is observed at a finite number of geographical sites. These non-folding constraints ensure the bijection condition of the space deformation f. They consist of building the Delaunay triangulation associated to the geographical sites and of imposing that our algorithm let the topological structure of the triangulation be the same.

Our results are illustrated through spatio-temporal precipitations from 20 sites in the Languedoc-Roussillon region of France. We propose a cross-validation study to demonstrate the improvements in predictions due to the coordinate deformation.