CondSimu: Conditional Simulation

Description

the function returns conditional simulations of a Gaussian 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

krige.method

Assumptions on the random field which corresponds to the respective kriging method; currently 'S' (simple kriging) and 'O' (ordinary kriging) are implemented.

matrix or vector of x coordinates; points to be kriged.

vector of y coordinates.

vector of z coordinates.

vector in grid triple form for the time coordinates.

grid

logical; determines whether the vectors x, y, and z should be interpreted as a grid definition, see Details.

gridtriple

logical. Only relevant if grid=TRUE. If gridtriple=TRUE then x, y, and z are of the form c(start,end,step); if gridtriple=FALSE then x

model

string; covariance model of the random field. See CovarianceFct, or type PrintModelList() to get all options f

param

parameter vector: 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

method

NULL or string; method used for simulating, see RFMethods, or type PrintMethodList() to get all options.

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.

data

the values measured.

trend

Not programmed yet. (used by universal kriging)

number of realisations to generate. If paired=TRUE then n must be even.

0:9; place where intermediate calculations are stored; the numbers are aliases for 10 internal registers; see GaussRF for further details.

err.model

covariance function for the error model. String or list. See model for details.

err.param

parameters for the error model. See also param.

err.method

Only relevant if err.model is not NULL. Then it must be given if and only if method is given; see method for details.

err.register

see register for details.

tol

considered only if grid=TRUE; tolerated distances of a given point to the nearest grid point to be regarded as being zero; see Details.

pch

character. The included kriging procedure can be quite time consuming. The character pch is printed after roughly each 80th part of calculation.

paired

logical. logical. If TRUE then every second simulation is obtained by only changing the signs of the standard Gaussian random variables, the simulation is based on (antithetic pairs).

na.rm

logical. If TRUE then NAs are removed from the given data.

Value

The returned object depends on the parameters n and grid: n=1: * grid=FALSE. A vector of simulated values is returned (independent of the dimension of the random field) * grid=TRUE. An array of the dimension of the random field is returned. n>1: * grid=FALSE. A matrix is returned. The columns contain the realisations. * grid=TRUE. An array of dimension $d+1$, where $d$ is the dimension of the random field as given by x, y, and z, is returned. The last dimension contains the realisations.

Details

The same way as GaussRF the function 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 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 x, y, z, and the locations of given. Comments on specific parameters

grid=FALSE: the vectorsx,y, andzare interpreted as vectors of coordinates
(grid=TRUE) && (gridtriple=FALSE): the vectorsx,y, andzare increasing sequences with identical lags for each sequence. A corresponding grid is created (as given byexpand.grid).
(grid=TRUE) && (gridtriple=TRUE): the vectorsx,y, andzare triples of the form (start,end,step) defining a grid (as given byexpand.grid(seq(x$start,x$end,x$step), seq(y$start,y$end,y$step), seq(z$start,z$end,z$step)))

References

Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.

Cressie, N.A.C. (1993) Statistics for Spatial Data. New York: Wiley. Goovaerts, P. (1997) Geostatistics for Natural Resources Evaluation. New York: Oxford University Press. Wackernagel, H. (1998) Multivariate Geostatistics. Berlin: Springer, 2nd edition.

Examples

Run this code

## 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)
for (i in 1:3) do.call(getOption("device"), list(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
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)

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