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\name{BrownResnick}
\alias{RPbrownresnick}
\alias{Brown-Resnick}
\alias{Brown-Resnick process}
\title{Brown-Resnick process}
\description{
\command{RPbrownresnick} defines a Brown-Resnick process.
}
\usage{
RPbrownresnick(phi, tcf, xi, mu, s)
}
\arguments{
\item{phi}{specifies the covariance model or variogram, see
\link{RMmodel} and \link{RMmodelsAdvanced}.
}
\item{tcf}{the extremal correlation function; either \code{phi} or
\code{tcf} must be given.}
\item{xi, mu, s}{the extreme value index, the location parameter and the
scale parameter, respectively, of the generalized extreme value
distribution. See Details.}
}
\details{
\GEV
The functions \code{RPbrorig}, \code{RPbrshifted} and \code{RPbrmixed}
perform the simulation of a Brown-Resnick process, which is defined by
\deqn{Z(x) = \max_{i=1}^\infty X_i \exp(W_i(x) - \gamma^2),
}{Z(x) = max_{i=1, 2, ...} X_i * exp(W_i(x) - gamma^2),}
where the \eqn{X_i} are the points of a Poisson point process on the
positive real half-axis with intensity \eqn{x^{-2} dx}{1/x^2 dx},
\eqn{W_i \sim W}{W_i ~ Y} are iid centered Gaussian processes with
stationary increments and variogram \eqn{\gamma}{gamma} given by
\code{phi}.
For simulation, internally, one of the methods
\command{\link{RPbrorig}}, \command{\link{RPbrshifted}} and
\command{\link{RPbrmixed}} is chosen automatically.
}
\author{\marco; \martin}
\references{
\itemize{
\item
Brown, B.M. and Resnick, S.I. (1977).
Extreme values of independent stochastic
processes. \emph{J. Appl. Probab.} \bold{14}, 732-739.
\item
Buishand, T., de Haan , L. and Zhou, C. (2008).
On spatial extremes: With application to
a rainfall problem. \emph{Ann. Appl. Stat.} \bold{2}, 624-642.
\item Kabluchko, Z., Schlather, M. and de Haan, L (2009)
Stationary max-stable random fields associated to negative
definite functions \emph{Ann. Probab.} \bold{37}, 2042-2065.
\item Oesting, M., Kabluchko, Z. and Schlather M. (2012)
Simulation of {B}rown-{R}esnick Processes, \emph{Extremes},
\bold{15}, 89-107.
}
}
% TO DO: ueberall diese notes einfuegen
\note{Advanced options
are \code{maxpoints} and \code{max_gauss}, see
\command{\link{RFoptions}}.
Further advanced options related to the
simulation methods \command{\link{RPbrorig}},
\command{\link{RPbrshifted}} and \command{\link{RPbrmixed}} can be
found in the paragraph \sQuote{Specific method options for Brown-Resnick
Fields} in \command{\link{RFoptions}}.}
\seealso{
\command{\link{RPbrorig}},
\command{\link{RPbrshifted}},
\command{\link{RPbrmixed}},
\command{\link{RMmodel}},
\command{\link{RPgauss}},
\command{\link{maxstable}},
\command{\link{maxstableAdvanced}}.
}
\keyword{spatial}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
\dontshow{\dontrun{
model <- ~ RPbrownresnick(RMfbm(alpha=A), xi=0)
x <- seq(0, 10, 0.2)
z <- RFsimulate(model=model, x, x, n=4, A=0.9) # about 1 min on a fast machine
plot(z)
z <- RFsimulate(model, x=x, n=4, A=1.9)
plot(z)
## basic model in Buishand, de Haan, Zhou (2008)
model <- RMfbm(proj=1, alpha=1, var=0.5) + RMfbm(proj=2, alpha=1, var=0.5)
x <- seq(0, 5, 0.05)
z <- RFsimulate(RPbrownresnick(model, xi=0), x, x, every=1000)
plot(z)
}}
## for some more sophisticated models see 'maxstableAdvanced'
\dontshow{FinalizeExample()}}
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