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\name{Smith}
\alias{RPsmith}
\alias{mixed moving maxima}
\alias{moving maxima}
\alias{M2}
\alias{M3}
\title{(Mixed) Moving Maxima}
\description{
\command{RPsmith} defines a moving maximum process or a mixed moving
maximum process with finite number of shape functions.
}
\usage{
RPsmith(shape, tcf, xi, mu, s)
}
\arguments{
\item{shape}{an \command{\link{RMmodel}} giving the spectral function}
\item{tcf}{an \command{\link{RMmodel}} specifying the
extremal correlation function; either \code{shape} or \code{tcf} must
be given. If \code{tcf} is given a shape function is tried to be
constructed via the \command{\link{RMm2r}} construction of
deterministic, monotone functions.
}
\item{xi,mu,s}{the extreme value index, the location parameter and the
scale parameter, respectively, of the generalized extreme value
distribution. See Details.
}
}
\note{
IMPORTANT: For consistency reasons with the geostatistical definitions
in this package the scale argument differs froms the original
definition of the Smith model! See the example below.
\command{RPsmith} depends on \command{\link{RRrectangular}}
and its arguments.
Advanced options
are \code{maxpoints} and \code{max_gauss}, see
\command{\link{RFoptions}}.
}
\details{
\GEV
It simulates max-stable processes \eqn{Z} that are referred to as
\dQuote{Smith model}.
\deqn{Z(x) = \max_{i=1}^\infty X_i Y_i(x-W_i),
}{Z(x) = max_{i=1, 2, ...} X_i * Y_i(x - W_i),}
where \eqn{(W_i, X_i)} are the points of a Poisson point process on
\eqn{\R^d \times (0, \infty)}{R^d x (0, \infty)} with intensity
\eqn{dw * c/x^2 dx} and \eqn{Y_i \sim Y}{Y_i ~ Y} are iid measurable
random functions with
\eqn{E[\int \max(0, Y(x)) dx] < \infty}{E[int max(0, Y(x)) dx ] < \infty}.
The constant \eqn{c} is chosen such that \eqn{Z} has standard Frechet
margins.
}
\me
\references{
\itemize{
\item Haan, L. (1984)
A spectral representation for max-stable processes.
\emph{Ann. Probab.}, \bold{12},
1194-1204.
\item Smith, R.L. (1990) Max-stable processes and spatial extremes
Unpublished Manuscript.
}
}
\seealso{
\command{\link{Advanced RMmodels}},
\command{\link{Auxiliary RMmodels}},
\command{\link{RMmodel}},
\command{\link{RPbernoulli}},
\command{\link{RPgauss}},
\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
model <- RMball()
x <- seq(0, 1000, 0.2)
z <- RFsimulate(RPsmith(model, xi=0), x)
plot(z)
hist(z@data$variable1, 50, freq=FALSE)
curve(exp(-x) * exp(-exp(-x)), from=-3, to=8, add=TRUE)
## 2-dim
x <- seq(0, 10, 0.1)
z <- RFsimulate(RPsmith(model, xi=0), x, x)
plot(z)
## original Smith model
x <- seq(0, 10, 0.05)
model <- RMgauss(scale = sqrt(2)) # !! cf. definition of RMgauss
z <- RFsimulate(RPsmith(model, xi=0), x, x)
plot(z)
\dontshow{\dontrun{
## original Smith model at arbitrary locations
n <- 200
model <- RMgauss(scale = sqrt(2)) # !! cf. definition of RMgauss
z <- RFsimulate(RPsmith(model, xi=0), runif(n, 0, 10), runif(0, 0, 10))
plot(z)
}}
## for some more sophisticated models see 'maxstableAdvanced'
\dontshow{FinalizeExample()}}
|
