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Stationary Random Functions have been successfully applied in geostatisticalapplications for decades. In some instances, the assumption of a homogeneousspatial dependence structure across the entire domain of interest isunrealistic. A practical approach for modelling and estimating non-stationaryspatial dependence structure is considered. This consists in transforming anon-stationary Random Function into a stationary and isotropic one via abijective continuous deformation of the index space. So far, this approach hasbeen successfully applied in the context of data from several independentrealizations of a Random Function. In this work, we propose an approach fornon-stationary geostatistical modelling using space deformation in the contextof a single realization with possibly irregularly spaced data. The estimationmethod is based on a non-stationary variogram kernel estimator which serves asa dissimilarity measure between two locations in the geographical space. Theproposed procedure combines aspects of kernel smoothing, weighted non-metricmulti-dimensional scaling and thin-plate spline radial basis functions. On asimulated data, the method is able to retrieve the true deformation.Performances are assessed on both synthetic and real datasets. It is shown inparticular that our approach outperforms the stationary approach. Beyond theprediction, the proposed method can also serve as a tool for exploratoryanalysis of the non-stationarity.

Estimation of Space Deformation Model for Non-stationary Random Functions