A kind of "spatial" GEV model can be defined by using response
surfaces for the GEV parameters. For instance, the GEV location
parameters are defined through the following equation:
$$\mu = X_\mu \beta_\mu$$
where \(X_\mu\) is the design matrix and
\(\beta_\mu\) is the vector parameter to be
estimated. The GEV scale and shape parameters are defined accordingly
to the above equation.
The log-likelihood for the GEV spatial model is consequently defined
as follows:
$$\ell(\beta) = \sum_{i=1}^{n.site} \sum_{j=1}^{n.obs} \log
f(y_{i,j}; \theta_i)$$
where \(\theta_i\) is the vector of the GEV parameters for
the \(i\)-th site.
Most often, there will be some dependence between stations. However,
it can be seen from the log-likelihood definition that we supposed
that the stations are mutually independent. Consequently, to get
reliable standard error estimates, these standard errors are estimated
with their sandwich estimates.
There are two different kind of covariates : "spatial" and
"temporal".
A "spatial" covariate may have different values accross station but
does not depend on time. For example the coordinates of the stations
are obviously "spatial". These "spatial" covariates should be used
with the marg.cov and loc.form, scale.form, shape.form.
A "temporal" covariates may have different values accross time but
does not depend on space. For example the years where the annual
maxima were recorded is "temporal". These "temporal" covariates should
be used with the temp.cov and temp.form.loc,
temp.form.scale, temp.form.shape.
As a consequence note that marg.cov must have K rows (K being
the number of sites) while temp.cov must have n rows (n being
the number of observations).