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Revision c751cbeafa6661ec449a5a53a8b3e659f558be70 authored by Martin Schlather on 27 May 2005, 00:00:00 UTC, committed by Gabor Csardi on 27 May 2005, 00:00:00 UTC
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\title{Conditional Simulation}
  the function returns conditional simulations of a Gaussian random field
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)
  \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 \command{\link{CovarianceFct}}, or
    type \command{\link{PrintModelList}}\code{()} to get all options for
    See \command{\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 \command{\link{CovarianceFct}};
    the value of \code{mean} must be finite
    in the case of simple kriging, and is ignored otherwise.

    See \command{\link{CovarianceFct}} for \code{param} being \code{NULL}
    or list.
  \item{method}{\code{NULL} or string; method used for simulating,
    see \command{\link{RFMethods}}, or
    type \command{\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
  \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
    \command{\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
  \item{err.method}{Only relevant if \code{err.model} is not
    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.}
    The included kriging procedure can be quite time consuming. 
    The character \code{pch} is printed after roughly
    each 80th part of calculation.}
    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.}
  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
    \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),
  The returned object depends on the parameters \code{n} and
    * \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{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.
 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}


## 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"

p <- 1:7
data <- GaussRF(x=p, y=p, grid=TRUE, model=model, param=param)

# 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
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
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
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),
image(x, x, cze, col=colour, xlim=lim, ylim=lim, zlim=zlim)
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