  A stationary spatial model is an idealization and we expect that the true
dependence structures of physical phenomena are spatially varying, but how
should we handle this non-stationarity in practice? We study the challenges
involved in applying a flexible non-stationary model to a dataset of annual
precipitation in the conterminous US, where exploratory data analysis shows
strong evidence of a non-stationary covariance structure.
  The aim of this paper is to investigate the modelling pipeline once
non-stationarity has been detected in spatial data. We show that there is a
real danger of over-fitting the model and that careful modelling is necessary
in order to properly account for varying second-order structure. In fact, the
example shows that sometimes non-stationary Gaussian random fields are not
necessary to model non-stationary spatial data.
