SpatialExtremes (version 1.2-0)

covariance: Defines and computes covariance functions

Description

This function defines and computes several covariance function either from a fitted ``max-stable'' model; either by specifying directly the covariance parameters.

Usage

covariance(fitted, sill, range, smooth, cov.mod = "whitmat", plot = TRUE,
dist, xlab, ylab, ...)

Arguments

fitted

An object of class ``maxstab''. Most often this will be the output of the fitmaxstab function. May be missing if scale and smooth are given.

sill,range,smooth

The sill, scale and smooth parameters for the covariance function. May be missing if fitted is given.

cov.mod

Character string. The name of the covariance model. Must be one of "whitmat", "cauchy" or "powexp" for the Whittle-Matern, Cauchy and Powered Exponential models.

plot

Logical. If TRUE (default) the covariance function is plotted.

dist

A numeric vector corresponding to the distance at which the covariance function should be evaluated. May be missing.

xlab,ylab

The x-axis and y-axis labels. May be missing.

...

Several option to be passed to the plot function.

Value

• This function returns the covariance function. Eventually, if dist is given, the covariance function is computed for each distance given by dist. If plot = TRUE, the covariance function is plotted.

Details

Currently, three covariance functions are defined: the Whittle-Matern, powered exponential (also known as "stable") and the Cauchy models. These covariance functions are defined as follows:

[object Object],[object Object],[object Object] where $\Gamma$ is the gamma function and $K_{smooth}$ is the modified Bessel function of order $smooth$.

Examples

## 1- Calling covariance using fixed covariance parameters
covariance(sill = 1, range = 1, smooth = 0.5, cov.mod = "whitmat")
covariance(sill = 0.5, range = 1, smooth = 0.5, cov.mod = "whitmat",
  dist = seq(0,5, 0.2), plot = FALSE)

## 2- Calling covariance from a fitted model
require(RandomFields)

##Define the coordinate of each location
n.site <- 30
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")

##Simulate a max-stable process - with unit Frechet margins
ms0 <- MaxStableRF(locations[,1], locations[,2], grid=FALSE, model="wh",
                   param=c(0,1,0,3, .5), maxstable="extr",
                   n = 30)
ms1 <- t(ms0)

##Now define the spatial model for the GEV parameters
param.loc <- -10 + 2 * locations[,2]
param.scale <- 5 + 2 * locations[,1] + locations[,2]^2
param.shape <- rep(0.2, n.site)
##Transform the unit Frechet margins to GEV 
for (i in 1:n.site)
  ms1[,i] <- param.scale[i] * (ms1[,i]^param.shape[i] - 1) /
  param.shape[i] + param.loc[i]

##Define a model for the GEV margins to be fitted
##shape ~ 1 stands for a constant GEV shape parameter
##over the region
loc.form <- loc ~ lat
scale.form <- scale ~ lon + I(lat^2)
shape.form <- shape ~ 1

fitted <- fitmaxstab(ms1, locations, "whitmat", loc.form, scale.form,
               shape.form)
covariance(fitted, ylim = c(0, 1))
covariance(sill = 1, range = 3, smooth = .5, cov.mod = "whitmat", add = TRUE, col
               = 3)
title("Whittle-Matern covariance function")
legend("topright", c("Theo.", "Fitted"), lty = 1, col = c(3,1), inset =
           .05)