\name{Kriging}
\alias{Kriging}
\title{Kriging methods}
\description{
  The functions allow for different methods of kriging.
}
\usage{
Kriging(method, x, y=NULL, z=NULL, grid, gridtriple=FALSE,
        model, param, given, data)
}
%- maybe also `usage' for other objects documented here.
\arguments{
  \item{method}{kriging method; currently only \code{"S"} (simple
    kriging) and \code{"O"} (ordinary kriging) implemented}
  \item{x}{\eqn{(n \times d)}{(n x d)} matrix or vector of \code{x} coordinates; coordinates of
    \eqn{n} points to be kriged}
  \item{y}{vector of \code{y} coordinates}
  \item{z}{vector of \code{z} 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, see \code{\link{CovarianceFct}}, or
    type \code{\link{PrintModelList}()} to get all options}
  \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 \link{CovarianceFct}.
    The value of \code{mean} must be finite
    in the case of simple kriging, and is ignored otherwise.}
  \item{given}{matrix or vector of points where data are available}
  \item{data}{the data values given at \code{given}; it might be a
    vector or a matrix. If a matrix is given multivariate data are
    assumed which are kriged \emph{separately}.}
}
\details{
  \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==FALSE)} : 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{
  \code{Kriging} returns a vector or matrix
  of kriged values corresponding to the
  specification of \code{x}, \code{y}, \code{z}, and
  \code{grid}, and \code{data}.

     \code{data} a vector or matrix with one column:\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 (according which specification
    given by \code{x}, \code{y}, and \code{z}).\cr
    
    \code{data} a matrix with at least two columns:\cr
    * \code{grid==FALSE}.  A matrix with the \code{ncol(data)} columns
    is returned.\cr
    * \code{grid==TRUE}.  An array of dimension
    \eqn{d+1}{d+1}, where \eqn{d}{d} is the dimension of
    the random field, is returned (according which specification
    given by \code{x}, \code{y}, and \code{z}).  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{Martin.Schlather@uni-bayreuth.de}
  \url{http://www.geo.uni-bayreuth.de/~martin}}
%\note{}

\seealso{
  \code{\link{CondSimu}},
  \code{\link{CovarianceFct}},
  \code{\link{EmpiricalVariogram}},
  \code{\link{RandomFields}},
}

\examples{

## creating random variables first
## here, a grid is chosen, but does not matter
step <- 0.25 
x <-  seq(0,7,step)
param <- c(0,1,0,1)
model <- "exponential"
RFparameters(PracticalRange=FALSE)$null
p <- 1:7
points <- as.matrix(expand.grid(p,p))
data <- GaussRF(points, grid=FALSE, model=model, param=param)

## visualise generated spatial data
zlim <- c(-2.6,2.6)
colour <- rainbow(100)
image(p, p, xlim=range(x), ylim=range(x),
      matrix(data,ncol=length(p)),
      col=colour,zlim=zlim)

## now: kriging
method <- "O" ## ordinary kriging
z <-  Kriging(method=method,
              x=x, y=x, grid=TRUE,
              model=model, param=param,
              given=points, data=data)
image(x,x,z,col=colour,zlim=zlim)
}
\keyword{spatial}%-- one or more ...