\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 generalised 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}{parameter 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 randomfield.}
\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 parameters; 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{RMgengneiting}} 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. Arxiv.
\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.}
}
}
\author{Martin Schlather, \email{schlather@math.uni-mannheim.de}
}
\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{
RFoptions(seed=0)
%# gamma is mainly a scale effect
model <- RMbigneiting(kappa=2, mu=0.5, gamma=c(0, 3, 6), rhored=1)
x <- seq(0, 10, if (interactive()) 0.02 else 1)
plot(model, ylim=c(0,1))
plot(RFsimulate(model, x=x))
\dontshow{RFoptions(seed=NA)}
}