\name{CondSimu}
\alias{CondSimu}
\title{Conditional Simulation}
\description{
the function returns conditional simulations of a random field
}
\usage{
CondSimu(krige.method, x, y=NULL, z=NULL, T=NULL, grid,
gridtriple=FALSE, model, param, method=NULL, given, data,
trend, n=1, register=0,
err.model=NULL, err.param=NULL, err.method=NULL,
err.register=1, tol=1E-5, pch=".", paired=FALSE, na.rm=FALSE)
}
\arguments{
\item{krige.method}{Assumptions on the random field which corresponds to
the respective kriging method;
currently 'S' (simple
kriging) and 'O' (ordinary kriging) are implemented.}
\item{x}{matrix or vector of \code{x} coordinates; points to be kriged.}
\item{y}{vector of \code{y} coordinates.}
\item{z}{vector of \code{z} coordinates.}
\item{T}{vector in grid triple form for the time coordinates.}
\item{grid}{logical; determines whether the vectors \code{x},
\code{y}, and \code{z} should be
interpreted as a grid definition, see Details.}
\item{gridtriple}{logical. Only relevant if \code{grid==TRUE}.
If \code{gridtriple==TRUE}
then \code{x}, \code{y}, and \code{z} are of the
form \code{c(start,end,step)}; if
\code{gridtriple==FALSE} then \code{x}, \code{y}, and \code{z}
must be vectors of ascending values.}
\item{model}{string; covariance model of the random field.
See \code{\link{CovarianceFct}}, or
type \code{\link{PrintModelList}()} to get all options for
\code{model}.
See \code{\link{CovarianceFct}} for \code{model} being a list.
}
\item{param}{parameter vector:
\code{param=c(mean, variance, nugget, scale,...)};
the parameters must be given
in this order; further parameters are to be added in case of a
parametrised class of covariance functions,
see \code{\link{CovarianceFct}};
the value of \code{mean} must be finite
in the case of simple kriging, and is ignored otherwise.
See \code{\link{CovarianceFct}} for \code{param} being \code{NULL}
or list.
}
\item{method}{\code{NULL} or string; method used for simulating,
see \code{\link{RFMethods}}, or
type \code{\link{PrintMethodList}()} to get all options.}
\item{given}{matrix or vector of locations where data are available;
note that it is not possible to give the points in form of a grid
definition.}
\item{data}{the values measured.}
\item{trend}{Not programmed yet. (used by universal kriging)}
\item{n}{number of realisations to generate. If \code{paired=TRUE}
then \code{n} must be even.}
\item{register}{0:9; place where intermediate calculations are stored;
the numbers are aliases for 10 internal registers; see
\code{\link{GaussRF}} for further details.}
\item{err.model}{covariance function for the error model. String or list.
See \code{model} for details.
}
\item{err.param}{parameters for the error model. See also
\code{param}.
}
\item{err.method}{Only relevant if \code{err.model} is not
\code{NULL}.
Then it must be given if and only if \code{method} is given;
see \code{method} for details.}
\item{err.register}{see \code{register} for details.}
\item{tol}{considered only if \code{grid=TRUE};
tolerated distances of a given point to the nearest grid point to
be regarded as being zero; see Details.}
\item{pch}{character.
The included kriging procedure can be quite time consuming.
The character \code{pch} is printed after roughly
each 80th part of calculation.}
\item{paired}{logical.
logical. If \code{TRUE} then every second simulation is obtained by
only changing the signs of the standard Gaussian random variables, the
simulation is based on (\dQuote{antithetic pairs}).
}
\item{na.rm}{logical. If \code{TRUE} then \code{NA}s are removed from
the given data.}
}
\details{
The same way as \code{GaussRF} the function
\code{CondSimu} allows for simulating on grids or arbitrary
locations. However simulation on a grid is sometimes performed
as if the points were at arbitrary locations, what may
imply a great reduction in speed. This happens when the \code{given}
locations do not lay on the specified grid, since in an intermediate
step simulation has to be performed simultaneously on both the grid
defined by \code{x}, \code{y}, \code{z}, and the locations
of \code{given}.\cr
Comments on specific parameters
\itemize{
\item \code{grid==FALSE} : the vectors \code{x}, \code{y},
and \code{z} are interpreted as vectors of coordinates
\item \code{(grid==TRUE) && (gridtriple==FALSE)} : the vectors
\code{x}, \code{y}, and \code{z}
are increasing sequences with identical lags for each sequence.
A corresponding
grid is created (as given by \code{expand.grid}).
\item \code{(grid==TRUE) && (gridtriple==TRUE)} : the vectors
\code{x}, \code{y}, and \code{z}
are triples of the form (start,end,step) defining a grid
(as given by \code{expand.grid(seq(x$start,x$end,x$step),
seq(y$start,y$end,y$step),
seq(z$start,z$end,z$step))})
}
}
\value{
The returned object depends on the parameters \code{n} and
\code{grid}:\cr
\code{n==1}:\cr
* \code{grid==FALSE}. A vector of simulated values is
returned (independent of the dimension of the random field)\cr
* \code{grid==TRUE}. An array of the dimension of the
random field is returned.\cr
\code{n>1}:\cr
* \code{grid==FALSE}. A matrix is returned. The columns
contain the repetitions.\cr
* \code{grid==TRUE}. An array of dimension
\eqn{d+1}{d+1}, where \eqn{d}{d} is the dimension of
the random field as given by \code{x}, \code{y}, and \code{z},
is returned. The last dimension contains the repetitions.
}
\references{
Chiles, J.-P. and Delfiner, P. (1999)
\emph{Geostatistics. Modeling Spatial Uncertainty.}
New York: Wiley.
Cressie, N.A.C. (1993)
\emph{Statistics for Spatial Data.}
New York: Wiley.
Goovaerts, P. (1997) \emph{Geostatistics for Natural Resources
Evaluation.} New York: Oxford University Press.
Wackernagel, H. (1998) \emph{Multivariate Geostatistics.} Berlin:
Springer, 2nd edition.
}
\author{Martin Schlather, \email{schlath@hsu-hh.de}
\url{http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html}}
%\note{}
\seealso{
\code{\link{CovarianceFct}},
\code{\link{GaussRF}},
\code{\link{Kriging}}
\code{\link{RandomFields}},
}
\examples{
## creating random variables first
## here, a grid is chosen, but any arbitrary points for which
## data are given are fine. Indeed if the data are given on a
## grid, the grid has to be expanded before calling `CondSimu',
## see below.
## However, locations where values are to be simulated,
## should be given in form of a grid definition whenever
## possible
param <- c(0, 1, 0, 1)
model <- "exponential"
RFparameters(PracticalRange=FALSE)
p <- 1:7
data <- GaussRF(x=p, y=p, grid=TRUE, model=model, param=param)
get(getOption("device"))(height=4,width=4);
get(getOption("device"))(height=4,width=4);
get(getOption("device"))(height=4,width=4);
# another grid, where values are to be simulated
step <- 0.25 # or 0.3
x <- seq(0, 7, step)
# standardisation of the output
lim <- range( c(x, p) )
zlim <- c(-2.6, 2.6)
colour <- rainbow(100)
## visualise generated spatial data
dev.set(2)
image(p, p, data, xlim=lim, ylim=lim, zlim=zlim, col=colour)
#conditional simulation
krige.method <- "O" ## random field assumption corresponding to
## those of ordinary kriging
%source("/home/schlather/article/R/NEW.RF/RandomFields/R/modelling.R")
%source("/home/schlather/article/R/NEW.RF/RandomFields/R/rf.R")
cz <- CondSimu(krige.method, x, x, grid=TRUE,
model=model, param=param,
given=expand.grid(p,p),# if data are given on a grid
# then expand the grid first
data=data)
range(cz)
dev.set(3)
image(x, x, cz, col=colour, xlim=lim, ylim=lim, zlim=zlim)
#conditional simulation with error term
cze <- CondSimu(krige.method, x, x, grid=TRUE,
model=model, param=c(0, 1/2, 0, 1),
err.model="gauss", err.param=c(0, 1/2, 0, 1),
given=expand.grid(p,p),
data=data)
range(cze)
dev.set(4)
image(x, x, cze, col=colour, xlim=lim, ylim=lim, zlim=zlim)
}
\keyword{spatial}
