R: MLE for a spatial GEV model

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,
    temp.cov = NULL, temp.form.loc = NULL, temp.form.scale = NULL,
    temp.form.shape = NULL, ..., start, control = list(maxit = 10000),
    method = "Nelder", warn = TRUE, corr = FALSE)

### 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.
`temp.cov` | Matrix with names columns giving additional *temporal* covariates for the GEV parameters. If `NULL`, no temporal trend are assume for the GEV parameters — see section Details.
`temp.form.loc, temp.form.scale, temp.form.shape` | R formulas defining the temporal trends for the GEV parameters. May be missing. 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.
`warn` | Logical. If `TRUE` (default), users will be warned if the starting values lie in a zero density region.
`corr` | Logical. If `TRUE`, the asymptotic correlation matrix is computed.

### 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) = ∑_(i=1)^(n.site) ∑_(j=1)^(n.obs) log f(y_(i,j);θ_i)

where θ_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).

### 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
    n.site <- 35
    locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
    colnames(locations) <- c("lon", "lat")
    data <- rmaxstab(50, locations, cov.mod = "whitmat", nugget = 0, range =
      3, smooth = 0.5)

    ## 2- Transformation to non unit Frechet margins
    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)
      data[,i] <- frech2gev(data[,i], param.loc[i], param.scale[i],
    param.shape[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(data, locations, form.loc, form.scale, form.shape)

[Package SpatialExtremes version 2.0-5 Index]