  Recent years have seen a huge development in spatial modelling and prediction
methodology, driven by the increased availability of remote-sensing data and
the reduced cost of distributed-processing technology. It is well known that
modelling and prediction using infinite-dimensional process models is not
possible with large data sets, and that both approximate models and, often,
approximate-inference methods, are needed. The problem of fitting simple global
spatial models to large data sets has been solved through the likes of
multi-resolution approximations and nearest-neighbour techniques. Here we
tackle the next challenge, that of fitting complex, nonstationary, multi-scale
models to large data sets. We propose doing this through the use of
superpositions of spatial processes with increasing spatial scale and
increasing degrees of nonstationarity. Computation is facilitated through the
use of Gaussian Markov random fields and parallel Markov chain Monte Carlo
based on graph colouring. The resulting model allows for both distributed
computing and distributed data. Importantly, it provides opportunities for
genuine model and data scaleability and yet is still able to borrow strength
across large spatial scales. We illustrate a two-scale version on a data set of
sea-surface temperature containing on the order of one million observations,
and compare our approach to state-of-the-art spatial modelling and prediction
methods.
