GaussRF: Gaussian Random Fields

Description

These functions simulate stationary spatial and spatio-temporal Gaussian random fields using turning bands/layers, circulant embedding, direct methods, and the random coin method.

Usage

GaussRF(x, y=NULL, z=NULL, T=NULL, grid=!missing(gridtriple), model, param,
       trend=NULL, method=NULL, n=1, register=0, gridtriple,
       paired=FALSE, ...)
InitGaussRF(x, y=NULL, z=NULL, T=NULL, grid=!missing(gridtriple), model,
            param, trend=NULL, method=NULL, register=0, gridtriple)

Arguments

matrix of coordinates, or vector of x coordinates

vector of y coordinates

vector of z coordinates

vector of time coordinates, may only be given if the random field is defined as an anisotropic random field, i.e. if model=list(list(model=,var=,k=,aniso=),...). T must always be given in the gridtriple f

grid

logical; determines whether the vectors x, y, and z should be interpreted as a grid definition, see Details. grid does not apply for T.

model

string or list; covariance or variogram model, see CovarianceFct, or type PrintModelList() to get the list of a

param

vector or matrix of parameters or missing, see Details and CovarianceFct; The simplest form is that param is vector of the form param=c(NA,variance,nugget,scale,.

trend

trend surface: number (mean) or a vector of length $d+1$ (linear trend $a_0+a_1 x_1 + \ldots + a_d x_d$), or function; you have the choice of using either x, y, z or X1, X2, X3, ... as spatial variables, as time variable T should be chosen

method

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

number of realisations to generate

0:9; place where intermediate calculations are stored; the numbers are aliases for 10 internal registers

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,

paired

logical. If TRUE then the second half of the simulations is obtained by only changing the signs of all the standard Gaussian random variables, on which the first half of the simulations is based. (Antithetic pairs

...

RFparameters that are locally used only.

Value

InitGaussRF returns 0 if no error has occurred and a positive value if failed. The object returned GaussRF and DoSimulateRF depends on the parameters n and grid: 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. 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 space and time. if n > 1 the repetitions make the last dimension

Details

GaussRF can use different methods for the simulation, i.e., circulant embedding, turning bands, direct methods, and random coin method. If method=NULL then GaussRF searches for a valid method. GaussRF may not find the fastest method neither the most precise one. It just finds any method among the available methods. (However it guesses what is a good choice.) See RFMethods for further information. Note that some of the methods do not work for all covariance or variogram models.

An isotropic random field is created byGaussRFwheremodelis the covariance or variogram model and the parameter isparam=c(mean,variance,nugget,scale, ...). Alternatively thetrendcan be given; thenparam=c(variance,nugget,scale, ...).
Nested models can be defined in the same way as a nestedCovarianceFct. If thetrendis not given it is set to 0.
An anisotropic random field (i.e. zonal anisotropy, geometrical anisotropy, separable models, non-separable space-time models) and a random field based on multiplicative or nested models is defined as in the case of an anisotropicCovarianceFct. If thetrendis not given it is set to 0. Themethodmay be specified by the globalmethodor for each model separately, as additional parametermethodfor each entry of the list; note that methods can not be mixed within a multiplicative part.

Ifmodel=list(list(model=,var=,k=,aniso=),...)then a time component might be given. In case ofmodel="nugget",anisomust still be given as a matrix. Namely ifanisois a singular matrix then a zonal nugget effect is obtained. GaussRF calls initially InitGaussRF, which does some basic checks on the validity of the parameters. Then, InitGaussRF performs some first calculations, like the first Fourier transform in the circulant embedding method or the matrix decomposition for the direct methods. Random numbers are not involved. GaussRF then calls DoSimulateRF which uses the intermediate results and random numbers to create a simulation.

When InitGaussRF checks the validity of the parameters, it also checks whether the previous simulation has had the same specification of the random field. If so (and if RFparameters()$STORING==TRUE), the stored intermediate results are used instead of being recalculated. 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))) %\item \code{model="nugget"} is one possibility to create independent %Gaussian random variables. Without loss of efficiency, any %covariance function with parameter vector %\code{c(mean, 0, nugget, scale, ...)} %can also be used. If \code{model="nugget"} is used %the second component of \code{param} must be zero. %% this has to be changed in later versions;

%\item The sum of the components variance and nugget in the argument %\code{param} equals the sill of the variogram.

registeris a parameter which may never be used by most of the users (please let me know if you use it!). In other words, the package will work fine if you ignore this parameter. The parameterregisteris of interest in the following situation. Assume you wish to create sequentially several realisations of two random fields$Z_1$and$Z_2$that have different specifications of the covariance/variogram models, i.e.$Z_1$,$Z_2$,$Z_1$,$Z_2$,... Then, without using different registers, the algorithm will not be able to profit from already calculated intermediate results, as the specifications of the covariance/variogram model change every time. However, using different registers allows for profiting from up to 10 stored intermediate results.

The strings formodelandmethodmay be abbreviated as long as the abbreviations match only one option. See alsoPrintModelList()andPrintMethodList()

Further control parameters for the simulation are set by means ofRFparameters(...).

References

See RFMethods for the references.

Examples

Run this code

#############################################################
 ##                                                         ##
 ## Examples using the symmetric stable model, also called  ##
 ## "powered exponential model"                             ## 
 ##                                                         ##
 #############################################################
 PrintModelList()    ## the complete list of implemented models
 model <- "stable"   
 mean <- 0
 variance <- 4
 nugget <- 1
 scale <- 10
 alpha <- 1   ## see help("CovarianceFct") for additional
              ## parameters of the covariance functions
 step <- 1    ## nicer, but also time consuming if step <- 0.1
 x <- seq(0, 20, step) 
 y <- seq(0, 20, step)     
 f <- GaussRF(x=x, y=y, model=model, grid=TRUE,
              param=c(mean, variance, nugget, scale, alpha))
 image(x, y, f)


 #############################################################
 ## ... using gridtriple
 step <- 1    ## nicer, but also time consuming if step <- 0.1
 x <- c(0, 20, step)  ## note: vectors of three values, not a 
 y <- c(0, 20, step)  ##       sequence
 f <- GaussRF(grid=TRUE, gridtriple=TRUE,
               x=x ,y=y, model=model,  
               param=c(mean, variance, nugget, scale, alpha))
 image(seq(x[1],x[2],x[3]), seq(y[1],y[2],y[3]), f)


 #############################################################
 ## arbitrary points
 x <- runif(100, max=20) 
 y <- runif(100, max=20)
 z <- runif(100, max=20) # 100 points in 3 dimensional space
(f <- GaussRF(grid=FALSE, Print=5,
              x=x, y=y, z=z, model=model, 
              param=c(mean, variance, nugget, scale, alpha)))


 #############################################################
 ## usage of a specific method
 ## -- the complete list can be obtained by PrintMethodList()
 x <- runif(100, max=20) # arbitrary points
 y <- runif(100, max=20)
 (f <- GaussRF(method="dir",  # direct matrix decomposition
              x=x, y=y, model=model, grid=FALSE, 
              param=c(mean, variance, nugget, scale, alpha)))


 #############################################################
 ## simulating several random fields at once
 step <- 1    ## nicer, but also time consuming if step <- 0.1
 x <- seq(0, 20, step)  # grid
 y <- seq(0, 20, step)
 f <- GaussRF(n=3,  # three simulations at once
              x=x, y=y, model=model, grid=TRUE,  
              param=c(mean, variance, nugget, scale, alpha))
 image(x, y, f[,,1])
 image(x, y, f[,,2])
 image(x, y, f[,,3])


        

 #############################################################
 ##                                                         ##
 ##      Examples using the extended definition form        ##
 ##                                                         ##
 ##                                                         ##  
 #############################################################


#% library(RandomFields, lib="~/TMP"); RFparameters(Print=6)
x <- (0:100)/10
m <- matrix(c(1,2,3,4),ncol=2)/5
model <- list("$", aniso=m,
              list("*",              
                   list("power", k=2),
                   list("sph"))
              )
z <- GaussRF(x=x, y=x, grid=TRUE, model=model, me="TBM3")
Print(c(mean(as.double(z)),var(as.double(z))))
image(z,zlim=c(-3,3))

## to know more what GaussRF does, use Print
## TMB can be very slow. To trace the iteration, use every
##
z <- GaussRF(x=x, y=x, grid=TRUE, model=model, me="TBM3",
             Print=3, every=100)
image(z,zlim=c(-3,3))
## here, GaussRF uses `direct decomp' to simulate on the line
##       and the square root of the covariance matrix is
##       calculated by the Cholesky decomposition



## non-separable space-time model applied for two space dimensions
## note that tbm method works in some special cases.
#%    library(RandomFields, lib="~/TMP")
x <- y <- seq(0, 7, if (interactive()) 0.05 else 0.2) 
T <- c(1,32,1) * 10        ## note necessarily gridtriple definition
model <- list("$", aniso=diag(c(3, 3, 0.02)),
             list("nsst", k1=2,
                  list("gauss"),
                  list("genB", k=c(1, 0.5))
              ))
z <- GaussRF(x=x, y=y, T=T, grid=TRUE, model=model,
             method="ci", CE.strategy=1,
             CE.trials=if (interactive()) 4 else 1)
rl <- function() if (interactive()) readline("Press return")
for (i in 1:dim(z)[3]) { image(z[,,i], zlim=range(z)); rl();}
for (i in 1:dim(z)[2]) { image(z[,i,], zlim=range(z)); rl();}
for (i in 1:dim(z)[1]) { image(z[i,,], zlim=range(z)); rl();}




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

x <- seq(0, 10, 1/10)
model <- list("*",
              list("$",  aniso=matrix(nr=2, c(1, 0)),
                   list("fractgauss", k=0.5)),
              list("$",  aniso=matrix(nr=2, c(0, 1)),
                   list("fractgauss", k=0.9))
              )
z <- GaussRF(x, x,  grid=TRUE, gridtriple=FALSE, model=model)
image(x, x, z)




 #############################################################
 ##                                                         ##
 ##                    Brownian motion                      ##
 ##                (using Stein's method)                   ##
 ##                                                         ##  
 #############################################################
# 1d
kappa <- 1   # in [0,2)
z <- GaussRF(x=c(0, 10, 0.001), grid=TRUE, Print=5,
             model=list("fractalB", kappa))
plot(z, type="l")

# 2d
step <- 0.3  ## nicer, but also time consuming if step = 0.1
x <- seq(0, 10, step)
kappa <- 1   # in [0,2)
z <- GaussRF(x=x, y=x, grid=TRUE, model=list("fractalB", kappa))
image(z,zlim=c(-3,3))

# 3d
x <- seq(0, 3, step)
kappa <- 1   # in [0,2)
z <- GaussRF(x=x, y=x, z=x, grid=TRUE,
             model=list("fractalB", kappa)) 
rl <- function() if (interactive()) readline("Press return")
for (i in 1:dim(z)[1]) { image(z[i,,]); rl();}



 #############################################################
 ## This example shows the benefits from stored,            ##
 ## intermediate results: in case of the circulant          ##
 ## embedding method, the speed is doubled in the second    ##
 ## simulation.                                             ##  
 #############################################################

RFparameters(Storing=TRUE)
y <- x <- seq(0, 50, 0.2)
(p <- c(runif(3), runif(1)+1))
ut <- system.time(f <- GaussRF(x=x,y=y,grid=TRUE,model="exponen",
                              method="circ", param=p))
image(x, y, f)cat("system time (first call)", format(ut,dig=3),"")

# second call with the same parameters can be much faster:
ut <- system.time(f <- DoSimulateRF()) 
image(x, y, f)cat("system time (second call)", format(ut,dig=3),"")


 #############################################################
 ##                                                         ##
 ##     Example how the cutoff method can be set            ##
 ##        explicitly using hypermodels                     ##
 ##                                                         ##
 #############################################################

## NOTE: this feature is still in an experimental stage
##       which has not been yet tested intensively;
## further: parameters and algorithms may change in
##       future.

#% library(RandomFields, lib="~/TMP");source("~/R/RF/RandomFields/tests/source.R")
## simuation of the stable model using the cutoff method
#RFparameters(Print=8, Storing=FALSE)
x <- seq(0, 1, 1/24) # nicer pictures with 1/240
scale <- 1.0
model1 <- list("$", scale=scale, list("stable", alpha=1.0))
rs <- get(".Random.seed", envir=.GlobalEnv, inherits = FALSE)
z1 <- GaussRF(x, x,  grid=TRUE, gridtriple=FALSE,
             model=model1, n=1, meth="cutoff", Storing=TRUE)
internal <- GetRegisterInfo()$meth$sub[[1]]$sub[[1]]$S$new
(size <- internal$meth$S$size)
(cut.off.param <- internal$cov$sub[[1]]$sub[[1]]$sub[[1]]$param)

# % Print(GetRegisterInfo(), vec=15)
image(x, x, z1)


## simulation of the same random field using the circulant
## embedding method and defining the hypermodel explicitely
model2 <- list("$", scale = scale,
               list("cutoff", diam=cut.off.param$diam, a=cut.off.param$a,       
                    list("stable", alpha=1.0))
              )
assign(".Random.seed", rs, envir=.GlobalEnv)
z2 <- GaussRF(x, x,  grid=TRUE, gridtriple=FALSE, model=model2,
              meth="circulant", n=1, CE.mmin=size, Storing=TRUE)
#%Print(GetRegisterInfo(), vec=15)
image(x, x, z2)
Print(range(z1-z2)) ## essentially no difference between the fields!




#% library(RandomFields)

 #############################################################
 ## The cutoff method simulates on a torus and a (small)    ##
 ## rectangle is taken as the required simulation.          ##
 ##                                                         ##
 ## The following code shows a whole such torus.            ##
 ## The main part of the code sets local.dependent=TRUE and ##
 ## local.mmin to multiples of the basic rectangle lengths  ##
 #############################################################


# definition of the realisation
RFparameters(CE.useprimes=FALSE)
x <- seq(0, 2, len=4) # better 20
y <- seq(0, 1, len=5) # better 40
grid.size <- c(length(x), length(y))
model <- list("$", var=1.1, aniso=matrix(nc=2, c(2, 1, 0.5, 1)),
              list(model="exp"))

# determination of the (minimal) size of the torus
InitGaussRF(x, y, model=model, grid=TRUE, method="cu")
ce.info.size <- GetRegisterInfo()$meth$sub[[1]]$sub[[1]]$S$new$meth$S$size
blocks <- ceiling(ce.info.size / grid.size / 4) *4
(size <- blocks * grid.size)

# simulation and plot of the torus 
z <- GaussRF(x, y, model=model, grid=TRUE, method="cu", n=prod(blocks) * 2,
             local.dependent=TRUE, local.mmin=size)[,,c(TRUE, FALSE)]
hei <- 8
do.call(getOption("device"),
        list(hei=hei, wid=hei / blocks[2] / diff(range(y)) *
                         blocks[1] * diff(range(x))))

close.screen(close.screen())
sc <- matrix(nc=blocks[1], split.screen(rev(blocks)), byrow=TRUE)
sc <- as.vector(t(sc[nrow(sc):1, ]))

for (i in 1:prod(blocks)) {
  screen(sc[i])
  par(mar=rep(1, 4) * 0.0)
  image(z[,, i], zlim=c(-3, 3), axes=FALSE, col=rainbow(100)) 
}



##############################################################
##	  Simulating with trend (as function)       	    ##
##############################################################

x <- seq(-5,5,0.1)
z <- GaussRF(x=x, y=x, model = "exponential", param=c(1,0,1), grid=TRUE,
             trend=function(x,y) 3*sin(x)*cos(y))
colors=heat.colors(1000)
image(x,x,z,col=colors)




##############################################################
##	  Simulating with linear trend surface    	    ##
##############################################################

x <- seq(-5,5,0.1)
##trend surface: 3x - y
z <- GaussRF(x=x, y=x, model = "cubic", param=c(1,0,2), grid=TRUE,
             trend=c(0,1,-1))
colors=heat.colors(1000)
persp(x,x,z, phi=30, theta=-3)

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