RFsimulateAdvanced: Simulation of Random Fields -- Advanced

Description

This function simulates unconditional random fields:

univariate and multivariat, spatial and spatio-temporal Gaussian random fields
stationary Poisson fields
Chi2 fields
t fields
Binary fields
stationary max-stable random fields.

It also simulates conditional random fields for

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

For basic simulation of Gaussian random fields, see RFsimulate. See RFsimulate.more.examples and RFsimulate.sophisticated.examples for further examples.

Arguments

model

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

matrix of coordinates, or vector of x coordinates, or object of class GridTopology or raster; if matrix, ncol(x)

y

optional vector of y coordinates, ignored if x
 is a matrix

z

optional vector of z coordinates, ignored if x
 is a matrix

T

optional vector of time coordinates, 
 T must always be an equidistant vector or given in a
 gridtriple format (see argument grid); for each component
 of T, the random field is simulated at all location points

grid

logical; determines whether the vectors x,
 y, and z or the columns of x should be
 interpreted as a grid definition (see Details). If grid=TRUE,
 either x, y, and

data

matrix, data.frame or object of class RFsp;
 coordinates and response values of
 measurements in case that conditional simulation is to
 be performed;
 if a matrix or a data.frame, the first c

err.model

same as model; gives the model of the
 measurement errors for the measured data (which must be given
 in this case!), see Details,
 err.model=NULL (default) corresponds to error-free
 measurements, the most common al

distances

object of class dist representing
 the upper trianguar part of the matrix of Euclidean distances
 between the points at which the field is to be simulated; only
 applicable for stationary and isotropic models;

dim

integer; space or space-time dimension of the field

n

number of realizations to generate

...

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
 RFspatialGridDataFrame
 if $[space-time-dimension > 1]$ and the coordinates are on a grid,
 result is of class 
 RFgridDataFrame
 if $[space-time-dimension = 1]$ and the coordinates are on a grid,
 result is of class
 RFspatialPointsDataFrame
 if $[space-time-dimension > 1]$ and the coordinates are not on a grid,
 result is of class 
 RFpointsDataFrame
 if $[space-time-dimension = 1]$ and the coordinates are not on a
 grid.
 The output format can be switched to the "old" array format using
 RFoptions, either by globally setting
 RFoptions(spConform=FALSE) or by passing spConform=FALSE
 in the call of RFsimulate.
 Then the object returned by RFsimulate
 depends on the arguments n and grid in the following way:
if vdim > 1 the vdim-variate vector makes the first dimension
 if grid=TRUE an array of the dimension of the
 random field makes the next dimensions. Here, the dimensions
 are ordered in the sequence x, y, z, T
 (if given).
 
 Else if no time component is given, then the values are passed as a
 single vector. Else if the time component is given the next 2
 dimensions give the space and the time, respectively.
 
 if n > 1 the repetitions make the last dimension
 Note: Conversion between the sp
 format and the conventional format can be
 done using the method RFspDataFrame2conventional and the
 function conventional2RFspDataFrame. 
 
 InitRFsimulate returns 0 if no error has occurred and a positive value
 if failed.

`Details`

RFsimulate simulates different classes of random fields,
 controlled by the wrapping model.
 If the wrapping function of the model argument is a covariance
 or variogram model (i.e., one of list obtained by
 RFgetModelNames(type="negative definite",
 group.by="type"), by default, a Gaussian field 
 with the corresponding covariance structure is simulated.
 By default, the simulation method is chosen automatically through internal algorithms.
 The simulation method can be set explicitly by enclosing the
 covariance function with a method specification.
 If other than Gaussian fields are to be simulated, the model
 argument must be enclosed by a function specifying the type of the
 random field.

 There are different possibilities of passing the locations at which
 the field is to be simulated. If grid=FALSE, all coordinate
 vectors (except for the time component $T$) must have the same
 length and the field is only simulated at the locations given by the
 rows of $x$ or of cbind(x, y, z). If $T$ is not
 missing, the field is simulated for all combinations
 $(x[i, ], T[k])$ or $(x[i], y[i], z[i], T[k])$,
 $i=1, ...,$nrow(x), $k=1, ...,$length(T),
 even if model is not explicitly a space-time model. 
 
If grid=TRUE, the vectors x,
 y, z and T or the columns of x and
 T are
 interpreted as a grid definition, i.e. the field is simulated at all
 locations $(x_i, y_j, z_k, T_l)$, as given by
 expand.grid(x, y, z, T). 
 Here, grid means equidistant in each direction, i.e. all
 vectors must be equidistant and in ascending order.
 In case of more than 3 space dimensions, the coordinates must be
 given in matrix notations. To enable different grid lengths for each
 direction in combination with the matrix notation, the
 gridtriple notation c(from, stepsize, len) is used:
 If x, y,
 z, T or the columns of x are of length 3, they
 are internally replaced by seq(from=from,
 to=from+(len-1)*stepsize, by=stepsize) , i.e. the field
 is simulated at all locations
expand.grid(seq(x$from, length.out=x$len, by=x$stepsize),
 seq(y$from, length.out=y$len, by=y$stepsize),
 seq(z$from, length.out=z$len, by=z$stepsize),
 seq(T$from, length.out=T$len, by=T$stepsize))
 
 If data is passed, conditional simulation 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`

General
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.
 Original work:
 
Circulant embedding:
 
 Chan, G. and Wood, A.T.A. (1997)
 An algorithm for simulating stationary Gaussian random fields.J. R. Stat. Soc., Ser. C46, 171-181.
 
 Dietrich, C.R. and Newsam, G.N. (1993)
 A fast and exact method for multidimensional Gaussian
 stochastic simulations.Water Resour. Res.29, 2861-2869. 
 
 Dietrich, C.R. and Newsam, G.N. (1996)
 A fast and exact method for multidimensional {G}aussian stochastic
 simulations: Extensions to realizations conditioned on direct and
 indirect measurementWater Resour. Res.32, 1643-1652. 
 Wood, A.T.A. and Chan, G. (1994)
 Simulation of stationary Gaussian processes in$[0,1]^d$J. Comput. Graph. Stat.3, 409-432.
 The code used inRandomFieldsis based on
 Dietrich and Newsam (1996).
Intrinsic embedding and Cutoff embedding:
 
 Stein, M.L. (2002)
 Fast and exact simulation of fractional Brownian surfaces.J. Comput. Graph. Statist.11, 587--599.
 
 
 Gneiting, T., Sevcikova, H., Percival, D.B., Schlather, M. and
 Jiang, Y. (2005)
 Fast and Exact Simulation of Large Gaussian Lattice Systems in$R^2$: Exploring the LimitsJ. Comput. Graph. Statist.Submitted.
Markov Gaussian Random Field:
 
 Rue, H. (2001) Fast sampling of Gaussian Markov random fields.J. R. Statist. Soc., Ser. B,63(2), 325-338.
 
 Rue, H., Held, L. (2005)Gaussian Markov Random Fields:
 Theory and Applications.Monographs on Statistics and Applied Probability, no104,
 Chapman \& Hall.
Turning bands method (TBM), turning layers:
 
 Dietrich, C.R. (1995) A simple and efficient space domain implementation
 of the turning bands method.Water Resour. Res.31,
 147-156.
 
 Mantoglou, A. and Wilson, J.L. (1982) The turning bands method for
 simulation of random fields using line generation by a spectral
 method.Water. Resour. Res.18, 1379-1394.
 Matheron, G. (1973)
 The intrinsic random functions and their applications.Adv. Appl. Probab.5, 439-468.
 Schlather, M. (2004)
 Turning layers: A space-time extension of turning bands.Submitted
Random coins:
 
 Matheron, G. (1967)Elements pour une Theorie des Milieux
 Poreux. Paris: Masson.

`See Also`

RFoptions,
 RMmodel,
 RFgui,
 methods for simulating Gaussian random fields,
 RFfit,
 RFempiricalvariogram,
 RFsimulate.more.examples,
 RFsimulate.sophisticated.examples,
 RPgauss,

`Examples`

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

\dontrun{

#############################################################
##                                                         ##
## Example 1: Specification of simulation method           ##
##                                                         ##
#############################################################

 ## usage of a specific method
 ## -- the complete list is obtained by RFgetMethodNames()
 model <- RMstable(alpha=1.5)
 x <- runif(100, max=20) 
 y <- runif(100, max=20) # 100 points in 2 dimensional space
 simulated <- RFsimulate(model = RPdirect(model), x=x, y=y, 
 cPrint=3) # direct matrix decomposition
 plot(simulated)



#############################################################
##                                                         ##
## Example 2: Turnung band with different number of lines  ##
##                                                         ##
#############################################################
model <- RMstable(alpha=1.5)
x <- seq(0, 10, if (interactive()) 0.01, 1)
z <- RFsimulate(model = RPtbm(model), x=x, y=x)
plot(z)



#############################################################
##                                                         ##
## Example 3: Shot noise fields (random coins)             ##
##                                                         ##
#############################################################

 x <- GridTopology(0, .1, 500)

 z <- RFsimulate(model=RPpoisson(RMgauss()), x=x, mpp.intensity = 100)

 plot(z)
 par(mfcol=c(2,1))
 plot(z@data[,1:min(length(z@data), 1000)], type="l")
 hist(z@data[,1])
 
 
 z <- RFsimulate(x=x, model=RPpoisson(RMball()), mpp.intensity = 0.1)
 
 plot(z)
 par(mfcol=c(2,1))
 plot(z@data[,1:min(length(z@data), 1000)], type="l")
 hist(z@data[,1])
 



 #############################################################
 ##                                                         ##
 ## Example 4: a 2d random field based on                   ##
 ## covariance functions valid in 1d only                   ##
 ##                                                         ##
 #############################################################

x <- seq(0, 2, 0.1)
model <- RMfbm(alpha=0.5, Aniso=matrix(nrow=1, c(1, 0))) + 
         RMfbm(alpha=0.9, Aniso=matrix(nrow=1, c(0, 1)))
z <- RFsimulate(x, x, model=model)
plot(z)



#############################################################
##                                                         ##
## Example 5 : Brownian sheet                              ##
## (using Stein's method)                                  ##
##                                                         ##
#############################################################
 
# 2d
step <- 0.3 ## nicer, but also time consuming if step = 0.1
x <- seq(0, 5, step)
alpha <- 1 # in [0,2)
z <- RFsimulate(x=x, y=x, model=RMfbm(alpha=alpha))
plot(z)


# 3d
z <- RFsimulate(x=x, y=x, z=x,
 model=RMfbm(alpha=alpha))
}
FinalizeExample()
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