https://github.com/cran/RandomFields
Tip revision: fd4911aa390fd49ddab92bd139bbbf35422e32e5 authored by Martin Schlather on 06 February 2020, 05:20:37 UTC
version 3.3.8
version 3.3.8
Tip revision: fd4911a
RMparswm.Rd
\name{RMparswm}
\alias{RMparswm}
\alias{RMparswmX}
\title{Parsimonious Multivariate Whittle Matern Model}
\description{
\command{\link{RMparswm}} is a multivariate stationary isotropic
covariance model
whose corresponding covariance function only depends on the distance
\eqn{r \ge 0}{r \ge 0} between
two points and is given for \eqn{i,j \in \{1,2\}}{i,j = 1,2} by
\deqn{C_{ij}(r)= c_{ij} W_{\nu_{ij}}(r).}{C_{ij}(r)=c_{ij} W_{\nu_{ij}}(r).}
Here \eqn{W_\nu} is the covariance of the
\command{\link{RMwhittle}} model.
\command{RMparswmX} ist defined as
\deqn{\rho_{ij} C_{ij}(r)}
where \eqn{\rho_{ij}} is any covariance matrix.
}
\usage{
RMparswm(nudiag, var, scale, Aniso, proj)
RMparswmX(nudiag, rho, var, scale, Aniso, proj)
}
\arguments{
\item{nudiag}{a vector of arbitrary length of positive values; the vector \eqn{(\nu_{11},\nu_{22},...)}.
The offdiagonal elements \eqn{\nu_{ij}} are calculated as
\eqn{0.5 (\nu_{ii} + \nu_{jj})}.}
\item{rho}{any positive definite \eqn{m \times m}{m x m}
matrix;
here, \eqn{m} equals \code{length(nudiag)}.
For the calculation of \eqn{c_{ij}} see Details.
}
\item{var,scale,Aniso,proj}{optional arguments; same meaning for any
\command{\link{RMmodel}}. If not passed, the above
covariance function remains unmodified.
}
}
\details{
In the equation above we have
\deqn{c_{ij} = \rho_{ij} \sqrt{G_{ij}}
}{
c_{ij} = \rho_{ ij} \sqrt G_{ij}}
and
\deqn{G_{ij} = \frac{\Gamma(\nu_{11} + d/2) \Gamma(\nu_{22} + d/2)
\Gamma(\nu_{12})^2}{\Gamma(\nu_{11}) \Gamma(\nu_{22})
\Gamma(\nu_{12}+d/2)^2}
}{
G_{ij} = \Gamma(\nu_{11} + d/2) \Gamma(\nu_{22} + d/2) \Gamma(\nu_{12}) /
(\Gamma(\nu_{11}) \Gamma(\nu_{22}) \Gamma(\nu_{12}+d/2))^2)}
where \eqn{\Gamma} is the Gamma function and \eqn{d} is the dimension
of the space.
Note that the definition of \command{RMparswmX} is
\code{RMschur(M=rho, RMparswm(nudiag, var, scale, Aniso, proj))}.
}
\value{
\command{\link{RMparswm}} returns an object of class \code{\link[=RMmodel-class]{RMmodel}}.
}
\references{
\itemize{
\item Gneiting, T., Kleiber, W., Schlather, M. (2010)
Matern covariance functions for multivariate random fields
\emph{JASA}
}
}
\me
\seealso{
\command{\link{RMbiwm}},
\command{\link{RMwhittle}},
\command{\link{RMmodel}},
\command{\link{RFsimulate}},
\command{\link{RFfit}}.
}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
rho <- matrix(nc=3, c(1, 0.5, 0.2, 0.5, 1, 0.6, 0.2, 0.6, 1))
model <- RMparswmX(nudiag=c(1.3, 0.7, 2), rho=rho)
plot(model)
x.seq <- y.seq <- seq(-10, 10, 0.1)
z <- RFsimulate(model = model, x=x.seq, y=y.seq)
plot(z)
\dontshow{FinalizeExample()}}
\keyword{spatial}
\keyword{models}