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\name{RFcov}
\alias{RFcov}
\title{(Cross-)Covariance function}
\description{
Calculates both the empirical and the theoretical (cross-)covariance function.
}
\usage{
RFcov(model, x, y = NULL, z = NULL, T=NULL, grid, params, distances, dim, ...,
data, bin=NULL, phi=NULL, theta = NULL, deltaT = NULL, vdim=NULL)
}
\arguments{
\item{model,params}{\argModel }
\item{x}{\argX}
\item{y,z}{\argYz}
\item{T}{\argT}
\item{grid}{\argGrid}
\item{data}{\argData}
\item{bin}{\argBin}
\item{phi}{\argPhi}
\item{theta}{\argTheta}
\item{deltaT}{\argDeltaT}
\item{distances,dim}{\argDistances}
\item{vdim}{\argVdim}
\item{...}{\argDots}
}
\details{ \command{\link{RFcov}} computes the empirical
cross-covariance function for given (multivariate) spatial data.
The empirical
(cross-)covariance function of two random fields \eqn{X}{X} and \eqn{Y}{Y} is given by
\deqn{\gamma(r):=\frac{1}{N(r)} \sum_{(t_{i},t_{j})|t_{i,j}=r}
(X(t_{i})Y(t_{j})) - m_{X} m_{Y}}{\gamma(r):=1/N(r)
\sum_{(t_{i},t_{j})|t_{i,j}=r} (X(t_{i})Y(t_{j})) - m_{X} m_{Y}}
where \eqn{t_{i,j}:=t_{i}-t_{j}}{t_{i,j}:=t_{i}-t_{j}}, \eqn{N(r)}{N(r)}
denotes the number of pairs of data points with distancevector
\eqn{t_{i,j}=r}{t_{i,j}=r} and where
\eqn{m_{X} := \frac{1}{N(r)} \sum_{(t_{i},t_{j})|t_{i,j}=r}
X_{t_{i}}}{m_{X} := \frac{1}{N(r)} \sum_{(t_{i},t_{j})|t_{i,j}=r}
X_{t_{i}}} and \eqn{m_{Y} := \frac{1}{N(r)} \sum_{(t_{i},t_{j})|t_{i,j}=r}
Y_{t_{i}}}{m_{Y} := 1/N(r) \sum_{(t_{i},t_{j})|t_{i,j}=r}
Y_{t_{i}}} denotes the mean of data points with distancevector
\eqn{t_{i,j}=r}{t_{i,j}=r}.
The spatial coordinates \code{x}, \code{y}, \code{z}
should be vectors. For random fields of
spatial dimension \eqn{d > 3} write all vectors as columns of matrix x. In
this case do neither use y, nor z and write the columns in
\code{gridtriple} notation.
If the data is spatially located on a grid a fast algorithm based on
the fast Fourier transformed (fft) will be used.
As advanced option the calculation method can also be changed for grid
data (see \command{\link{RFoptions}}.)
It is also possible to use \command{\link{RFcov}} to calculate
the pseudocovariance function (see \command{\link{RFoptions}}).
}
\value{
\command{\link{RFcov}} returns objects of class
\command{\link[=RFempVariog-class]{RFempVariog}}.
}
\references{
Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp,
P. (eds.) (2010) \emph{Handbook of Spatial Statistics.}
Boca Raton: Chapman & Hall/CRL.
Stein, M. L. (1999) \emph{Interpolation of Spatial Data.}
New York: Springer-Verlag
}
\author{Jonas Auel; Sebastian Engelke; Johannes Martini; \martin}
\seealso{
\command{\link{RFvariogram}},
\command{\link{RFmadogram}},
\command{\link{RMstable}},
\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
n <- 1 ## use n <- 2 for better results
## isotropic model
model <- RMexp()
x <- seq(0, 10, 0.02)
z <- RFsimulate(model, x=x, n=n)
emp.vario <- RFcov(data=z)
plot(emp.vario, model=model)
## anisotropic model
model <- RMexp(Aniso=cbind(c(2,1), c(1,1)))
x <- seq(0, 10, 0.05)
z <- RFsimulate(model, x=x, y=x, n=n)
emp.vario <- RFcov(data=z, phi=4)
plot(emp.vario, model=model)
## space-time model
model <- RMnsst(phi=RMexp(), psi=RMfbm(alpha=1), delta=2)
x <- seq(0, 10, 0.05)
T <- c(0, 0.1, 100)
z <- RFsimulate(x=x, T=T, model=model, n=n)
emp.vario <- RFcov(data=z, deltaT=c(10, 1))
plot(emp.vario, model=model, nmax.T=3)
## multivariate model
model <- RMbiwm(nudiag=c(1, 2), nured=1, rhored=1, cdiag=c(1, 5),
s=c(1, 1, 2))
x <- seq(0, 20, 0.1)
z <- RFsimulate(model, x=x, y=x, n=n)
emp.vario <- RFcov(data=z)
plot(emp.vario, model=model)
## multivariate and anisotropic model
model <- RMbiwm(A=matrix(c(1,1,1,2), nc=2),
nudiag=c(0.5,2), s=c(3, 1, 2), c=c(1, 0, 1))
x <- seq(0, 20, 0.1)
dta <- RFsimulate(model, x, x, n=n)
ev <- RFcov(data=dta, phi=4)
plot(ev, model=model, boundaries=FALSE)
\dontshow{\dontrun{
######################################################
# NOTE: distinguish the different uses of x and y
x <- c(1,2,1)
y <- c(4,5,6)
# coordinate space 1-dimensional, evaluated at 3 points:
RFcov(model=model, x=as.matrix(x), y=as.matrix(y))
# coordinate space is 3-dimensional, evaluated at a pair of points
RFcov(model=model, x=t(x), y=t(y))
}}
\dontshow{FinalizeExample()}}
\keyword{spatial}
\keyword{models}
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