If we dispose of block, e.g., annual, maxima observed at locations
Then conditional on the values of the three Gaussian processes at the weather stations, the block maxima are assumed to follow a GEV distribution
Max-stable processes
Conditional max-stable simulations
Latent Variable
Copula approaches
Miscellaneous
References
Reference Manual
What are Bayesian Hierarchical models?
Hierarchical models have different layers of variations which
must be modelled. When trying to model spatial extremes we can
think of (at least) two layers: a layer that determines the
marginal behaviour of extremes and another layer that controls
the spatial dependence of extremes. Unfortunately because the
likelihood of max-stable processes is not available it prevents
the use of a spatial dependence layer. Consequently this type of
models will never be able to model the dependence between say
two weather stations. However its flexibility for modelling the
marginal behaviour and will be of great interest if one is
interested in modelling only the pointwise distribution of
extreme events.
If we dispose of block, e.g., annual, maxima observed at locations
, univariate extreme value
arguments suggests that these block maxima might be modelled by
a GEV distribution. The key idea of the latent process approach
is to assume that the GEV parameters vary smoothly over space
according to a stochastic
process \})
. The SpatialExtremes package use Gaussian processes
for this and assume that the Gaussian processes related to each
GEV parameter are mutually independent. For instance we take
 = f_\mu(x;
\beta_\mu) + S_\mu(x; \psi_\mu),)
where 
is a
trend surface depending on regression parameters
and 
is a
zero mean stationary Gaussian process with covariance function
depending on
parameters 
. Similar
expressions might be used for the two remaining GEV parameters.
Then conditional on the values of the three Gaussian processes at the weather stations, the block maxima are assumed to follow a GEV distribution
 \mid
\{\mu(x_j); \sigma(x_j), \xi(x_) \} \sim \mbox{GEV}\{\mu(x_j),
\sigma(x_j), \xi(x_j)\},)
independently for each location and
where 
and 
.
If we dispose of block, e.g., annual, maxima observed at locations
Then conditional on the values of the three Gaussian processes at the weather stations, the block maxima are assumed to follow a GEV distribution
Function "latent": Draw a Markov chain from the posterior distribution
The full conditional distributions are easily shown to be
 &\propto& \pi(\mu \mid \beta_\mu,
\psi_\mu) \pi(y \mid \mu, \sigma, \xi)\\
\pi(\psi_\mu \mid \cdots) &\propto& \pi(\psi_\mu)
\pi(\mu \mid \beta_\mu, \psi)\\
\psi(\beta_\mu \mid \cdots) &\propto& \pi(\beta_\mu)
\pi(\mu \mid \beta_\mu, \psi_\mu),
)
where , \pi(\beta_\mu))
are prior distributions. The
full conditional distributions related to the two remaining GEV
parameters give similar expressions.