  Stationary Random Functions have been successfully applied in geostatistical
applications for decades. In some instances, the assumption of a homogeneous
spatial dependence structure across the entire domain of interest is
unrealistic. A practical approach for modelling and estimating non-stationary
spatial dependence structure is considered. This consists in transforming a
non-stationary Random Function into a stationary and isotropic one via a
bijective continuous deformation of the index space. So far, this approach has
been successfully applied in the context of data from several independent
realizations of a Random Function. In this work, we propose an approach for
non-stationary geostatistical modelling using space deformation in the context
of a single realization with possibly irregularly spaced data. The estimation
method is based on a non-stationary variogram kernel estimator which serves as
a dissimilarity measure between two locations in the geographical space. The
proposed procedure combines aspects of kernel smoothing, weighted non-metric
multi-dimensional scaling and thin-plate spline radial basis functions. On a
simulated data, the method is able to retrieve the true deformation.
Performances are assessed on both synthetic and real datasets. It is shown in
particular that our approach outperforms the stationary approach. Beyond the
prediction, the proposed method can also serve as a tool for exploratory
analysis of the non-stationarity.
