Revision 513ad2c4d40f9e25102134188eb154b18140f6a2 authored by Martin Schlather on 16 April 2017, 09:57:35 UTC, committed by cran-robot on 16 April 2017, 09:57:35 UTC
1 parent 1b4f8cd
RMgencauchy.Rd
\name{RMgencauchy}
\alias{RMgencauchy}
\title{Generalized Cauchy Family Covariance Model}
\description{
\command{\link{RMgencauchy}} is a stationary isotropic covariance model
belonging to the generalized Cauchy family.
The corresponding covariance function only depends on the distance \eqn{r \ge 0}{r \ge 0} between
two points and is given by
\deqn{C(r) = (1 + r^\alpha)^(-\beta/\alpha)}{C(r) = (1 + r^\alpha)^(-\beta/\alpha)}
where \eqn{\alpha \in (0,2]}{0 < \alpha \le 2} and \eqn{\beta > 0}{\beta > 0}.
See also \command{\link{RMcauchy}}.
}
\usage{
RMgencauchy(alpha, beta, var, scale, Aniso, proj)
}
\arguments{
\item{alpha}{a numerical value; should be in the interval (0,2]
to provide a valid covariance function for a random field of any dimension.}
\item{beta}{a numerical value; should be positive to provide a valid
covariance function for a random field of any dimension.}
\item{var,scale,Aniso,proj}{optional arguments; same meaning for any
\command{\link{RMmodel}}. If not passed, the above
covariance function remains unmodified.}
}
\details{
This model has a smoothness parameter \eqn{\alpha}{\alpha} and a
paramater \eqn{\beta}{\beta} which determines the asymptotic power law.
More precisely, this model admits simulating random fields where fractal dimension
\emph{D} of the Gaussian sample and Hurst coefficient \emph{H}
can be chosen independently (compare also with \command{\link{RMlgd}}.): Here, we have
\deqn{ D = d + 1 - \alpha/2, \alpha \in (0,2]}{ D = d + 1 - \alpha/2, 0
< \alpha \le 2}
and
\deqn{ H = 1 - \beta/2, \beta > 0.}{ H = 1 - \beta/2, \beta > 0.}
I. e. the smaller \eqn{\beta}{\beta}, the longer the long-range
dependence.
The covariance function is very regular near the origin, because its
Taylor expansion only contains even terms and reaches its sill slowly.
Each covariance function of the Cauchy family is a normal scale mixture.
Note that the Cauchy Family (see \command{\link{RMcauchy}}) is included
in this family for the choice \eqn{\alpha = 2}{\alpha = 2} and
\eqn{\beta = 2 \gamma}{\beta = 2 \gamma}.
}
\value{
\command{\link{RMgencauchy}} returns an object of class \code{\link[=RMmodel-class]{RMmodel}}
}
\references{
Covariance function
\itemize{
% \item Gneiting, T. (2002) Compactly supported correlation
% functions. \emph{J. Multivariate Anal.} \bold{83} 493--508.
\item Gneiting, T. and Schlather, M. (2004)
Stochastic models which separate fractal dimension and Hurst effect.
\emph{SIAM review} \bold{46}, 269--282.
}
Tail correlation function (for \eqn{\alpha \in (0,1]}{0 < \alpha \le 1})
\itemize{
\item Strokorb, K., Ballani, F., and Schlather, M. (2014)
Tail correlation functions of max-stable processes: Construction
principles, recovery and diversity of some mixing max-stable
processes with identical TCF.
\emph{Extremes}, \bold{} Submitted.
}
}
\author{Martin Schlather, \email{schlather@math.uni-mannheim.de}
}
\seealso{
\command{\link{RMcauchy}},
\command{\link{RMcauchytbm}},
\command{\link{RMmodel}},
\command{\link{RFsimulate}},
\command{\link{RFfit}}.
}
\keyword{spatial}
\keyword{models}
\examples{
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
\dontshow{StartExample()}
model <- RMgencauchy(alpha=1.5, beta=1.5, scale=0.3)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))
\dontshow{FinalizeExample()}
}
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