\name{Kriging} \alias{Kriging} \title{Kriging methods} \description{ The function allows for different methods of kriging. } \usage{ Kriging(krige.method, x, y=NULL, z=NULL, T=NULL, grid, gridtriple=FALSE, model, param, given, data, trend, pch=".", return.variance=FALSE, internal=FALSE) } %- maybe also `usage' for other objects documented here. \arguments{ \item{krige.method}{kriging method; currently only 'S' (simple kriging) and '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{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, 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}.} \item{trend}{not programmed yet (will be used in case of universal kriging)} \item{pch}{Kriging procedures are quite time consuming in general. The character \code{pch} is printed after roughly each 80th part of calculation.} \item{return.variance}{logical. If \code{FALSE} the kriged field is returned. If \code{TRUE} a list of two elements, \code{estim} and \code{var}, i.e. the kriged field and the kriging variances, is returned.} \item{internal}{ \code{FALSE}. \code{internal} should not be set to \code{TRUE} by the user. (In case \code{internal=TRUE}, various consistency checks for the input variables are not performed. Further, \code{grid} must be \code{FALSE}, and \code{model} must be given in the output format of \code{\link{PrepareModel}}.) } } \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==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{ If \code{variance.return=FALSE} \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}.\cr \code{data}: a vector or matrix with \emph{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 to the specification of \code{x}, \code{y}, and \code{z}).\cr \code{data}: a matrix with \emph{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 to the specification of \code{x}, \code{y}, and \code{z}). The last dimension contains the repetitions. If \code{variance.return=TRUE} a list of two elements, \code{estim} and \code{var}, i.e. the kriged field and the kriging variances, is returned. The format of \code{estim} is the same as described above. The format of \code{var} is accordingly. } \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{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) 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 krige.method <- "O" ## ordinary kriging z <- Kriging(krige.method=krige.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 ...