swh:1:snp:41efaabb19f883462ec3380f3d4c3102b0ed86b4
Tip revision: 41d603eb8a5f4bfe82c56acee957c79e7500bfd4 authored by Martin Schlather on 18 January 2022, 18:12:52 UTC
version 3.3.14
version 3.3.14
Tip revision: 41d603e
RMbigneiting.Rd
\name{RMbigneiting}
\alias{RMbigneiting}
\alias{RMbiwendland}
\title{Gneiting-Wendland Covariance Models}
\description{
\command{\link{RMbigneiting}} is a bivariate stationary isotropic covariance
model family whose elements
are specified by seven parameters.
Let \deqn{\delta_{ij} = \mu + \gamma_{ij} + 1.}
Then,
\deqn{
C_{n}(h) = c_{ij} (C_{n, \delta} (h / s_{ij}))_{i,j=1,2}
}
and \eqn{ C_{n, \delta} }
is the generalized Gneiting model
with parameters \eqn{n} and \eqn{\delta}, see
\code{\link{RMgengneiting}}, i.e.,
\deqn{C_{\kappa=0, \delta}(r) = (1-r)^\beta 1_{[0,1]}(r), \qquad \beta=\delta
+ 2\kappa + 1/2;}{
C_{\kappa=0, \delta}(r) = (1 - r)^\beta 1_{[0,1]}(r), \beta=\delta + 2\kappa + 1/2;}
\deqn{C_{\kappa=1, \delta}(r) = \left(1+\beta r \right)(1-r)^{\beta} 1_{[0,1]}(r),
\qquad \beta = \delta + 2\kappa + 1/2;}{
C_{\kappa=1, \delta}(r) = (1+ \beta r)(1-r)^\beta 1_{[0,1]}(r),
\beta = \delta + 2\kappa + 1/2;}
\deqn{C_{\kappa=2, \delta}(r)=\left( 1 + \beta r + \frac{\beta^{2} -
1}{3}r^{2} \right)(1-r)^{\beta} 1_{[0,1]}(r), \qquad
\beta=\delta + 2\kappa + 1/2;}{
C(_{\kappa=2, \delta}(r) = (1 + \beta r + (\beta^2-1) r^(2)/3)(1-r)^\beta
1_{[0,1]}(r), \beta = \delta + 2\kappa + 1/2;}
\deqn{ C_{\kappa=3, \delta}(r)=\left( 1 + \beta r + \frac{(2\beta^{2}-3)}{5} r^{2}+
\frac{(\beta^2 - 4)\beta}{15} r^{3} \right)(1-r)^\beta 1_{[0,1]}(r),
\qquad \beta=\delta+2\kappa+1/2.}{
C_{\kappa=3, \delta}(r) = (1 + \beta r + (2 \beta^2-3 )r^(2)/5+(\beta^2 - 4) \beta
r^(3)/15)(1-r)^\beta 1_{[0,1]}(r), \beta=\delta + 2\kappa + 1/2.}
}
\usage{
RMbigneiting(kappa, mu, s, sred12, gamma, cdiag, rhored, c, var, scale, Aniso, proj)
}
\arguments{
\item{kappa}{argument that chooses between the four different covariance
models and may take values \eqn{0,\ldots,3}{0,...,3}.
The model is \eqn{k} times
differentiable.}
\item{mu}{\code{mu} has to be greater than or equal to
\eqn{\frac{d}{2}}{d/2} where \eqn{d}{d} is the (arbitrary)
dimension of the random field.}
\item{s}{vector of two elements giving the scale of the models on the
diagonal, i.e. the vector \eqn{(s_{11}, s_{22})}.
}
\item{sred12}{value in \eqn{[-1,1]}. The scale on the offdiagonals is
given by \eqn{s_{12} = s_{21} =}
\code{sred12 *}
\eqn{\min\{s_{11},s_{22}\}}{min{s_{11}, s_{22}}}.
}
\item{gamma}{a vector of length 3 of numerical values; each entry is
positive.
The vector \code{gamma} equals
\eqn{(\gamma_{11},\gamma_{21},\gamma_{22})}.
Note that \eqn{\gamma_{12} =\gamma_{21}}.
}
\item{cdiag}{a vector of length 2 of numerical values; each entry
positive; the vector \eqn{(c_{11},c_{22})}.}
\item{c}{a vector of length 3 of numerical values;
the vector \eqn{(c_{11}, c_{21}, c_{22})}.
Note that \eqn{c_{12}= c_{21}}.
Either
\code{rhored} and \code{cdiag} or \code{c} must be given.
}
\item{rhored}{value in \eqn{[-1,1]}.
See
also the Details for the corresponding value of \eqn{c_{12}=c_{21}}.
}
\item{var,scale,Aniso,proj}{optional arguments; same meaning for any
\command{\link{RMmodel}}. If not passed, the above
covariance function remains unmodified.}
}
\details{
A sufficient condition for the
constant \eqn{c_{ij}} is
\deqn{c_{12} = \rho_{\rm red} \cdot m \cdot \left(c_{11} c_{22}
\prod_{i,j=1,2}
\left(\frac{\Gamma(\gamma_{ij} + \mu + 2\kappa + 5/2)}{b_{ij}^{\nu_{ij} +
2\kappa + 1} \Gamma(1 + \gamma_{ij}) \Gamma(\mu + 2\kappa + 3/2)}
\right)^{(-1)^{i+j}}
\right)^{1/2}
}{
c_{ij} = \rho_r m (c_{11} c_{22})^{1/2}
}
where \eqn{\rho_{\rm red} \in [-1,1]}{\rho_r in [-1,1]}.
The constant \eqn{m} in the formula above is obtained as follows:
\deqn{m = \min\{1, m_{-1}, m_{+1}\}}{m = min\{1, m_{-1}, m_{+1}\}}
Let
\deqn{a = 2 \gamma_{12} - \gamma_{11} -\gamma_{22}}
\deqn{b = -2 \gamma_{12} (s_{11} + s_{22}) + \gamma_{11} (s_{12} +
s_{22}) + \gamma_{22} (s_{12} + s_{11})}
\deqn{e = 2 \gamma_{12} s_{11}s_{22} - \gamma_{11}s_{12}s_{22} -
\gamma_{22}s_{12}s_{11}}
\deqn{d = b^2 - 4ae}
\deqn{t_j =\frac{- b + j \sqrt d}{2 a} }{t_j =(-b + j \sqrt d) / (2 a) }
If \eqn{d \ge0} and \eqn{t_j \not\in (0, s_{12})}{t_j in (0, s_{12})^c} then \eqn{m_j=\infty} else
\deqn{
m_j =
\frac{(1 - t_j/s_{11})^{\gamma_{11}}(1 -
t_j/s_{22})^{\gamma_{22}}}{(1 - t_j/s_{12})^{2 \gamma_{11}}
}{
m_j = (1 - t_j/s_{11})^{\gamma_{11}} (1 -
t_j/s_{22})^{\gamma_{22}} / (1 - t_j/s_{12})^{2 \gamma_{11}}
}
}
In the function \command{\link{RMbigneiting}}, either \code{c} is
passed, then the above condition is checked, or \code{rhored} is passed;
then \eqn{c_{12}} is calculated by the above formula.
}
\value{
\command{\link{RMbigneiting}} returns an object of class \code{\link[=RMmodel-class]{RMmodel}}.
}
\references{
\itemize{
\item Bevilacqua, M., Daley, D.J., Porcu, E., Schlather, M. (2012)
Classes of compactly supported correlation functions for multivariate
random fields. Technical report.
\code{RMbigeneiting} is based on this original work.
D.J. Daley, E. Porcu and M. Bevilacqua have published end of
2014 an article intentionally
without clarifying the genuine authorship of \code{RMbigneiting},
in particular,
neither referring to this original work nor to \pkg{RandomFields},
which has included \code{RMbigneiting} since version 3.0.5 (05 Dec
2013).
\item Gneiting, T. (1999)
Correlation functions for atmospherical data analysis.
\emph{Q. J. Roy. Meteor. Soc} Part A \bold{125}, 2449-2464.
\item Wendland, H. (2005) \emph{Scattered Data Approximation.}
{Cambridge Monogr. Appl. Comput. Math.}
}
}
\me
\seealso{
\command{\link{RMaskey}},
\command{\link{RMbiwm}},
\command{\link{RMgengneiting}},
\command{\link{RMgneiting}},
\command{\link{RMmodel}},
\command{\link{RFsimulate}},
\command{\link{RFfit}}.
}
\keyword{spatial}
\keyword{models}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
%# gamma is mainly a scale effect
model <- RMbigneiting(kappa=2, mu=0.5, gamma=c(0, 3, 6), rhored=1)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))
\dontshow{FinalizeExample()}}