RandomFields (version 3.1.12)

RFsimulate: Simulation of Random Fields

Description

This function simulates unconditional random fields:

• univariate and multivariat, spatial and spatio-temporalGaussian random fields
• fields based on Gaussian fields such asChi2 fieldsorBinary fields, seeRP.
• 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

model

object of class RMmodel, RFformula or formula; specifies the model to

x

vector of x coordinates, or object of class GridTopology or raster; For more options see

y

optional vector of y coordinates

z

optional vector of z coordinates

T

optional vector of time coordinates, T must always be an equidistant vector. Instead of T=seq(from=From, by=By, len=Len) one may also write T=c(From, By, Len).

grid

logical; RandomFields can find itself the correct value in nearly all cases, so that usually grid need not be given. See also RFsimulateAdvanced.

distances

another alternative to pass the (relative) coordinates, see RFsimulateAdvanced.

dim

Only used if distances are given.

data

For conditional simulation and random imputing only. If data is missing, unconditional simulation is performed. Matrix, data.frame or object of class RFsp; coordinates and res

given

optional, matrix or list. If given matrix then the coordinates can be given separately, namely by given where, in each row, a single location is given. If given is a list, it may consist of x

err.model

For conditional simulation and random imputing only. Usually err.model=RMnugget(var=var), or not given at all (error-free measurements).

n

number of realizations to generate. For a very advanced feature, see the notes in RFsimulateAdvanced.

...

for advanced use: further options and control arguments for the simulation that are passed to and processed by RFoptions

Value

• By default, an object of the virtual class RFsp; result is of class RMmodel.
  • RFspatialGridDataFrameif the space-time dimension is greater than 1 and the coordinates are on a grid,
  • RFgridDataFrameif the space-time dimension equals 1 and the coordinates are on a grid,
  • RFspatialPointsDataFrameif the space-time dimension is greater than 1 and the coordinates are not on a grid,
  • RFpointsDataFrameif 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.

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 classRFsp,ncol(data@coords)must equal the dimension of the index space. Ifdata@datacontains only a single variable, variable names are optional. Ifdata@datacontains more than one variable, variables must be named andmodelmust be given in the tilde notationresp ~ ...(seeRFformula) and"resp"must be contained innames(data@data).
• % Beschreibung hier stimmt nicht so ganz mit Examples unten ueberein Ifdatais a matrix or a data.frame, eitherncol(data)equals$(dimension of index space + 1)$and the order of the columns is (x, y, z, T, response) or, ifdatacontains more than one response variable (i.e.ncol(data) > (dimension of index space + 1)),colnames(data)must containcolnames(x)or those of"x", "y", "z", "T"that are not missing. The response variable name is matched withmodel, which must be given in the tilde notation. If"x", "y", "z", "T"are missing anddatacontainsNAs,colnames(data)must contain an element which starts withdata; the corresponding column and those behind it are interpreted as the given data and those before the corresponding column are interpreted as the coordinates.
• ifxis missing,RFsimulatesearches forNAs 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.

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

Run this code

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

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