\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}