  Environmental processes resolved at a sufficiently small scale in space and
time will inevitably display non-stationary behavior. Such processes are both
challenging to model and computationally expensive when the data size is large.
Instead of modeling the global non-stationarity explicitly, local models can be
applied to disjoint regions of the domain. The choice of the size of these
regions is dictated by a bias-variance trade-off; large regions will have
smaller variance and larger bias, whereas small regions will have higher
variance and smaller bias. From both the modeling and computational point of
view, small regions are preferable to better accommodate the non-stationarity.
However, in practice, large regions are necessary to control the variance. We
propose a novel Bayesian three-step approach that allows for smaller regions
without compromising the increase of the variance that would follow. We are
able to propagate the uncertainty from one step to the next without issues
caused by reusing the data. The improvement in inference also results in
improved prediction, as our simulated example shows. We illustrate this new
approach on a data set of simulated high-resolution wind speed data over Saudi
Arabia.
