  We develop a novel multi-factor copula model for multivariate spatial
extremes, which is designed to capture the different combinations of marginal
and cross-extremal dependence structures within and across different spatial
random fields. Our proposed model, which can be seen as a multi-factor copula
model, can capture all possible distinct combinations of extremal dependence
structures within each individual spatial process while allowing flexible
cross-process extremal dependence structures for both upper and lower tails. We
show how to perform Bayesian inference for the proposed model using a Markov
chain Monte Carlo algorithm based on carefully designed block proposals with an
adaptive step size. In our real data application, we apply our model to study
the upper and lower extremal dependence structures of the daily maximum air
temperature (TMAX) and daily minimum air temperature (TMIN) from the state of
Alabama in the southeastern United States. The fitted multivariate spatial
model is found to provide a good fit in the lower and upper joint tails, both
in terms of the spatial dependence structure within each individual process, as
well as in terms of the cross-process dependence structure. Our results suggest
that the TMAX and TMIN processes are quite strongly spatially dependent over
the state of Alabama, and moderately cross-dependent. From a practical
perspective, this implies that it may be worthwhile to model them jointly when
interest lies in a computing spatial risk measures that involve both
quantities.
