covariance: Defines and computes covariance functions

In SpatialExtremes: Modelling Spatial Extremes

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, nugget, sill, range, smooth, smooth2 = NULL, cov.mod =
"whitmat", plot = TRUE, dist, xlab, ylab, col = 1, ...)

Arguments

fitted	An object of class “maxstab”. Most often this will be the output of the fitmaxstab function. May be missing if sill, range, smooth and cov.mod are given.
nugget,sill,range,smooth,smooth2	The nugget, 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", "powexp", "bessel" or "caugen" for the Whittle-Matern, Cauchy, Powered Exponential, Bessel and Generalized Cauchy models. May be missing if fitted is given.
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.
col	The color to be used for the plot.
...	Several option to be passed to the plot function.

Details

Currently, four covariance functions are defined: the Whittle-Matern, powered exponential (also known as "stable"), Cauchy and Bessel models. These covariance functions are defined as follows for h > 0

Whittle-Matern

γ(h) = σ 2^(1-κ) / Γ(κ) (h/λ)^κ K_κ(h / λ)

Powered Exponential

γ(h) = σ exp[-(h/λ)^κ]

Cauchy

γ(h) = σ [1 + (h/λ)^2]^(-κ)

Bessel

γ(h) = σ (2 λ / h)^(κ) Gamma(κ + 1) J_κ(h / λ)

Generalized Cauchy

γ(h) = σ [1 + (h / λ)^κ_2]^(-κ / κ_2)

where σ, λ and κ are the sill, the range and shape parameters, Γ is the gamma function, K_κ and J_κ are both modified Bessel functions of order κ. In addition a nugget effect can be set that is there is a jump at the origin, i.e., γ(o) = ν + σ, where ν is the nugget effect.

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.

Author(s)

Mathieu Ribatet

Examples

## 1- Calling covariance using fixed covariance parameters
covariance(nugget = 0, sill = 1, range = 1, smooth = 0.5, cov.mod = "whitmat")
covariance(nugget = 0, 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
##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
data <- rmaxstab(30, locations, cov.mod = "whitmat", nugget = 0, range =
3, smooth = 1)

##Fit a max-stable model
fitted <- fitmaxstab(data, locations, "whitmat", nugget = 0)
covariance(fitted, ylim = c(0, 1))
covariance(nugget = 0, sill = 1, range = 3, smooth = 1, 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)

SpatialExtremes documentation built on April 19, 2022, 5:06 p.m.