Simulation of Random Fields

Description

This function simulates unconditional random fields:

• univariate and multivariat, spatial and spatio-temporal Gaussian random fields

• fields based on Gaussian fields such as Chi2 fields or Binary fields, see RP.

• stationary Poisson fields

• stationary max-stable random fields.

It also simulates conditional random fields for

• univariate and multivariat, spatial and spatio-temporal Gaussian random fields

Here, only the simulation of Gaussian random fields is described. For other kind of random fields (binary, max-stable, etc.) or more sophisticated approches see RFsimulateAdvanced.

Usage

RFsimulate(model, x, y=NULL, z=NULL, T=NULL, grid=NULL,
  distances, dim, data, given=NULL, err.model, n=1, ...)

Arguments

Details

By default, all Gaussian random fields have zero mean. Simulating with trend can be done by including RMtrend in the model, see the examples below.

If data is passed, conditional simulation based on simple kriging is performed:

• if of class RFsp, ncol(data@coords) must equal the dimension of the index space. If data@data contains only a single variable, variable names are optional. If data@data contains more than one variable, variables must be named and model must be given in the tilde notation resp ~ ... (see RFformula) and "resp" must be contained in names(data@data).

• If data is a matrix or a data.frame, either ncol(data) equals (dimension of index space + 1) and the order of the columns is (x, y, z, T, response) or, if data contains more than one response variable (i.e. ncol(data) > (dimension of index space + 1)), colnames(data) must contain colnames(x) or those of "x", "y", "z", "T" that are not missing. The response variable name is matched with model, which must be given in the tilde notation. If "x", "y", "z", "T" are missing and data contains NAs, colnames(data) must contain an element which starts with ‘data’; the corresponding column and those behind it are interpreted as the given data and those before the corresponding column are interpreted as the coordinates.

• if x is missing, RFsimulate searches for NAs in the data and performs a conditional simulation for them.

Specification of err.model: In geostatistics we have two different interpretations of a nugget effect: small scale variability and measurement error. The result of conditional simulation usually does not include the measurement error. Hence the measurement error err.model must be given separately. For sake of generality, any model (and not only the nugget effect) is allowed. Consequently, err.model is ignored when unconditional simulation is performed.

Value

By default, an object of the virtual class RFsp; result is of class RMmodel.

• RFspatialGridDataFrame if the space-time dimension is greater than 1 and the coordinates are on a grid,

• RFgridDataFrame if the space-time dimension equals 1 and the coordinates are on a grid,

• RFspatialPointsDataFrame if the space-time dimension is greater than 1 and the coordinates are not on a grid,

• RFpointsDataFrame if the space-time dimension equals 1 and the coordinates are not on a grid.

In case of a multivariate

If n > 1 the repetitions make the last dimension.

See RFsimulateAdvanced for additional options.

Note

Several advanced options can be found in sections ‘General options’ and ‘coords’ of RFoptions. In particular, option spConform=FALSE leads to a simpler (and faster!) output, see RFoptions for details.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de http://ms.math.uni-mannheim.de/de/publications/software

References

Gneiting, T. and Schlather, M. (2013) Statistical modeling with covariance functions. In preparation.

Lantuejoul, Ch. (2002) Geostatistical simulation. New York: Springer.

Schlather, M. (1999) An introduction to positive definite functions and to unconditional simulation of random fields. Technical report ST 99-10, Dept. of Maths and Statistics, Lancaster University.

See RFsimulateAdvanced for more specific literature.

See Also

RFempiricalvariogram, RFfit, RFgetModelInfo, RFgui, RMmodel, RFoptions, RFsimulateAdvanced, RFsimulate.more.examples

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again


#############################################################
## ##
## ONLY TWO VERY BASIC EXAMPLES ARE GIVEN HERE ##
## see ##
## ?RMsimulate.more.examples ##
## and ##
## ?RFsimulateAdvanced ##
## for more examples ##
## ##
#############################################################

#############################################################
## ##
## Unconditional simulation ## 
## ##
#############################################################

## first let us look at the list of implemented models
RFgetModelNames(type="positive definite", domain="single variable",
                iso="isotropic") 

## our choice is the exponential model;
## the model includes nugget effect and the mean:
model <- RMexp(var=5, scale=10) + # with variance 4 and scale 10
 RMnugget(var=1) + # nugget
 RMtrend(mean=0.5) # and mean
 
## define the locations:
from <- 0
to <- 20
x.seq <- seq(from, to, length=200) 
y.seq <- seq(from, to, length=200)

simu <- RFsimulate(model, x=x.seq, y=y.seq)
plot(simu)



#############################################################
## ##
## Conditional simulation ## 
## ##
#############################################################

# first we simulate some random values at a
# 100 random locations:
n <- 100
x <- runif(n=n, min=-1, max=1)
y <- runif(n=n, min=-1, max=1)
data <- RFsimulate(model = RMexp(), x=x, y=y, grid=FALSE)
plot(data)

# let simulate a field conditional on the above data
x.seq.cond <- y.seq.cond <- seq(-1.5, 1.5, length=n)
model <- RMexp()
cond <- RFsimulate(model, x=x.seq.cond, y=y.seq.cond, data=data)
plot(cond, data)