\name{RMhyperbolic}
\alias{RMhyperbolic}
\title{Generalized Hyperbolic Covariance Model}
\description{
\command{\link{RMhyperbolic}} is a stationary isotropic covariance model
called \dQuote{generalized hyperbolic}.
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) = \frac{(\delta^2+r^2)^{\nu/2}
K_\nu(\xi(\delta^2+r^2)^{1/2})}{\delta^\nu K_\nu(\xi
\delta)}}{C(r) = \delta^(-\nu) (K_\nu(\nu \delta))^{-1}
(\delta^2+r^2)^{\nu/2} K_\nu(\xi(\delta^2+r^2)^{1/2})}
where \eqn{K_{\nu}}{K_\nu} denotes the modifies Bessel function of
second kind.
}
\usage{
RMhyperbolic(nu, lambda, delta, var, scale, Aniso, proj)
}
\arguments{
\item{nu, lambda, delta}{numerical values; should either satisfy\cr
\eqn{\delta \ge 0}{\delta \ge 0}, \eqn{\lambda > 0}{\lambda > 0}
and \eqn{\nu > 0}{\nu > 0}, or\cr
\eqn{\delta > 0}{\delta > 0}, \eqn{\lambda > 0}{\lambda > 0} and
\eqn{\nu = 0}{\nu = 0}, or\cr
\eqn{\delta > 0}{\delta > 0}, \eqn{\lambda \ge 0}{\lambda \ge 0}
and \eqn{\nu < 0}{\nu < 0}.}
\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 class is over-parametrized, i.e. it can be reparametrized by
replacing the three parameters \eqn{\lambda}{\lambda},
\eqn{\delta}{\delta} and scale by two other parameters. This means
that the representation is not unique.
Each generalized hyperbolic covariance function is a normal scale
mixture.
The model contains some other classes as special cases;
for \eqn{\lambda = 0}{\lambda = 0} we get Cauchy covariance function
(see \command{\link{RMcauchy}}) with \eqn{\gamma =
-\frac{\nu}2}{\gamma = -\nu/2} and scale=\eqn{\delta}{\delta};
the choice \eqn{\delta = 0}{\delta = 0} yields a covariance model of type
\command{\link{RMwhittle}} with smoothness parameter \eqn{\nu}{\nu}
and scale parameter \eqn{\lambda^{-1}}{1/\lambda}.
}
\value{
\command{\link{RMhyperbolic}} returns an object of class \code{\link[=RMmodel-class]{RMmodel}}
}
\references{
\itemize{
\item Shkarofsky, I.P. (1968) Generalized turbulence space-correlation and
wave-number spectrum-function pairs. \emph{Can. J. Phys.} \bold{46},
2133-2153.
\item Barndorff-Nielsen, O. (1978) Hyperbolic distributions and
distributions on hyperbolae. \emph{Scand. J. Statist.} \bold{5}, 151-157.
\item Gneiting, T. (1997). Normal scale mixtures and dual
probability densities. \emph{J. Stat. Comput. Simul.} \bold{59}, 375-384.
}
}
\author{Martin Schlather, \email{schlather@math.uni-mannheim.de}
}
\seealso{
\command{\link{RMcauchy}},
\command{\link{RMwhittle}},
\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
model <- RMhyperbolic(nu=1, lambda=2, delta=0.2)
x <- seq(0, 10, if (interactive()) 0.02 else 1)
plot(model, ylim=c(0,1))
plot(RFsimulate(model, x=x))
\dontshow{FinalizeExample()}
}