SpatialExtremes (version 1.2-0)

fitspatgev: MLE for a spatial GEV model

Description

This function derives the MLE of a spatial GEV model.

Usage

fitspatgev(data, covariables, loc.form, scale.form, shape.form, ...,
start, control = list(maxit = 10000), method = "Nelder", std.err.type =
"score", warn = TRUE)

Arguments

data

A matrix representing the data. Each column corresponds to one location.

covariables

Matrix with named columns giving required covariates for the GEV parameter models.

loc.form, scale.form, shape.form

R formulas defining the spatial models for the GEV parameters. See section details.

start

A named list giving the initial values for the parameters over which the pairwise likelihood is to be minimized. If start is omitted the routine attempts to find good starting values - but might fail.

...

Several arguments to be passed to the optim functions. See section details.

control

The control argument to be passed to the optim function.

method

The method used for the numerical optimisation procedure. Must be one of BFGS, Nelder-Mead, CG, L-BFGS-B or SANN. See optim for det

std.err.type

Character string. Must be one of "score", "grad" or "none". If none, no standard errors are computed.

warn

Logical. If TRUE (default), users will be warned if the starting values lie in a zero density region.

Value

• An object of class spatgev. Namely, this is a list with the following arguments:
• fitted.values
  The parameter estimates.
• param
  All the parameters e.g. parameter estimates and fixed parameters.
• std.err
  The standard errors.
• var.cov
  The asymptotic MLE variance covariance matrix.
• counts,message,convergence
  Some information about the optimization procedure.
• logLik,deviance
  The log-likelihood and deviance values.
• loc.form, scale.form, shape.form
  The formulas defining the spatial models for the GEV parameters.
• covariables
  The covariables used for the spatial models.
• ihessian
  The inverse of the Hessian matrix of the negative log-likelihood.
• jacobian
  The variance covariance matrix of the score.

Details

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.

Examples

## 1- Simulate a max-stable random field
require(RandomFields)
n.site <- 35
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")
ms0 <- MaxStableRF(locations[,1], locations[,2], grid=FALSE, model="wh",
                   param=c(0,1,0,3, .5), maxstable="extr",
                   n = 50)
## 2- Transformation to non unit Frechet margins
ms1 <- t(ms0)
param.loc <- -10 + 2 * locations[,2]
param.scale <- 5 + 2 * locations[,1]
param.shape <- rep(0.2, n.site)
for (i in 1:n.site)
  ms1[,i] <- param.scale[i] * (ms1[,i]^param.shape[i] - 1) /
  param.shape[i] + param.loc[i]

## 3- Fit a ''spatial GEV'' mdoel to data with the following models for
##    the GEV parameters
form.loc <- loc ~ lat
form.scale <- scale ~ lon
form.shape <- shape ~ 1

fitspatgev(ms1, locations, form.loc, form.scale, form.shape)