fitspatgev {SpatialExtremes}R Documentation

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 details.
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.

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:

μ = X_μ β_μ

where X_μ is the design matrix and β_μ 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:

llik(β) = 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.

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.

Author(s)

Mathieu Ribatet

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)

[Package SpatialExtremes version 1.1-1 Index]