# grf

##### Simulation of Gaussian Random Fields

`grf()`

generates (unconditional)
simulations of Gaussian random fields for
given covariance parameters.
`geoR2RF`

converts model specification used by geoR
to the correponding one in RandomFields.

##### Usage

```
grf(n, grid = "irreg", nx, ny, xlims = c(0, 1), ylims = c(0, 1),
borders, nsim = 1, cov.model = "matern",
cov.pars = stop("missing covariance parameters sigmasq and phi"),
kappa = 0.5, nugget = 0, lambda = 1, aniso.pars,
mean = 0, method, RF=TRUE, messages)
```geoR2RF(cov.model, cov.pars, nugget = 0, kappa, aniso.pars)

##### Arguments

- n
number of points (spatial locations) in each simulations.

- grid
optional. An \(n \times 2\) matrix with coordinates of the simulated data.

- nx
optional. Number of points in the X direction.

- ny
optional. Number of points in the Y direction.

- xlims
optional. Limits of the area in the X direction. Defaults to \([0,1]\).

- ylims
optional. Limits of the area in the Y direction. Defaults to \([0,1]\).

- borders
optional. Typically a two coluns matrix especifying a polygon. See DETAILS below.

- nsim
Number of simulations. Defaults to 1.

- cov.model
correlation function. See

`cov.spatial`

for further details. Defaults to the*exponential*model.- cov.pars
a vector with 2 elements or an \(n \times 2\) matrix with values of the covariance parameters \(\sigma^2\) (partial sill) and \(\phi\) (range parameter). If a vector, the elements are the values of \(\sigma^2\) and \(\phi\), respectively. If a matrix, corresponding to a model with several structures, the values of \(\sigma^2\) are in the first column and the values of \(\phi\) are in the second.

- kappa
additional smoothness parameter required only for the following correlation functions:

`"matern"`

,`"powered.exponential"`

,`"cauchy"`

and`"gneiting.matern"`

. More details on the documentation for the function`cov.spatial`

.- nugget
the value of the nugget effect parameter \(\tau^2\).

- lambda
value of the Box-Cox transformation parameter. The value \(\lambda = 1\) corresponds to no transformation, the default. For any other value of \(\lambda\) Gaussian data is simulated and then transformed.

- aniso.pars
geometric anisotropy parameters. By default an isotropic field is assumed and this argument is ignored. If a vector with 2 values is provided, with values for the anisotropy angle \(\psi_A\) (in radians) and anisotropy ratio \(\psi_A\), the coordinates are transformed, the simulation is performed on the isotropic (transformed) space and then the coordinates are back-transformed such that the resulting field is anisotropic. Coordinates transformation is performed by the function

`coords.aniso`

.- mean
a numerical vector, scalar or the same length of the data to be simulated. Defaults to zero.

- method
simulation method with options for

`"cholesky"`

,`"svd"`

,`"eigen"`

,`"RF"`

. Defaults to the*Cholesky*decomposition. See section`DETAILS`

below.- RF
logical, with defaults to TRUE, indicating whether the algorithm should try to use the function

`GaussRF`

from the package RandomFields in case of`method`

is missing and the number of points is greater than 500.- messages
logical, indicating whether or not status messages are printed on the screen (or output device) while the function is running. Defaults to

`TRUE`

.

##### Details

For the methods `"cholesky"`

, `"svd"`

and `"eigen"`

the
simulation consists of multiplying a vector of standardized
normal deviates by a square root of the covariance matrix.
The square root of a matrix is not uniquely defined. These
three methods differs in the way they compute the
square root of the (positive definite) covariance matrix.

The previously available
method `method = "circular.embedding"`

is no longer available
in geoR. For simulations in a fine grid and/or with a large number
of points use the package RandomFields.

The option `"RF"`

calls internally the function
`GaussRF`

from the package RandomFields.

The argument `borders`

, if provides takes a
polygon data set following argument `poly`

for the splancs' function `csr`

, in case of
`grid="reg"`

or `gridpts`

, in case of
`grid="irreg"`

. For the latter the simulation will have
*approximately* “n” points.

##### Value

`grf`

returns a list with the components:

an \(n \times 2\) matrix with the coordinates of the simulated data.

a vector (if `nsim = 1`

) or a matrix with the
simulated values. For the latter each column corresponds to one
simulation.

a string with the name of the correlation function.

the value of the nugget parameter.

a vector with the values of \(\sigma^2\) and \(\phi\), respectively.

value of the parameter \(\kappa\).

value of the Box-Cox transformation parameter \(\lambda\).

a vector with values of the anisotropy parameters, if provided in the function call.

a string with the name of the simulation method used.

a string "1d" or "2d" indicating the spatial dimension of the simulation.

the random seed by the time the function was called.

messages produced by the function describing the simulation.

the function call.

geoR2grf returns a list with the components:

RandomFields name of the correlation model

RandomFields parameter vector

##### References

Wood, A.T.A. and Chan, G. (1994) Simulation of stationary Gaussian
process in \([0,1]^d\).
*Journal of Computatinal and Graphical Statistics*, **3**, 409--432.

Schlather, M. (1999) *Introduction to positive definite functions
and to unconditional simulation of random fields*. Tech. Report
ST--99--10, Dept Maths and Stats, Lancaster University.

Schlather, M. (2001) *Simulation and Analysis of Random Fields*. R-News **1** (2), p. 18-20.

Further information on the package geoR can be found at: http://www.leg.ufpr.br/geoR.

##### See Also

`plot.grf`

and `image.grf`

for
graphical output,
`coords.aniso`

for anisotropy coordinates transformation
and, `chol`

,
`svd`

and `eigen`

for methods of matrix
decomposition and `GaussRF`

function
in the package RandomFields.

##### Examples

```
# NOT RUN {
sim1 <- grf(100, cov.pars = c(1, .25))
# a display of simulated locations and values
points(sim1)
# empirical and theoretical variograms
plot(sim1)
## alternative way
plot(variog(sim1, max.dist=1))
lines.variomodel(sim1)
#
# a "smallish" simulation
sim2 <- grf(441, grid = "reg", cov.pars = c(1, .25))
image(sim2)
##
## 1-D simulations using the same seed and different noise/signal ratios
##
set.seed(234)
sim11 <- grf(100, ny=1, cov.pars=c(1, 0.25), nug=0)
set.seed(234)
sim12 <- grf(100, ny=1, cov.pars=c(0.75, 0.25), nug=0.25)
set.seed(234)
sim13 <- grf(100, ny=1, cov.pars=c(0.5, 0.25), nug=0.5)
##
par.ori <- par(no.readonly = TRUE)
par(mfrow=c(3,1), mar=c(3,3,.5,.5))
yl <- range(c(sim11$data, sim12$data, sim13$data))
image(sim11, type="l", ylim=yl)
image(sim12, type="l", ylim=yl)
image(sim13, type="l", ylim=yl)
par(par.ori)
## simulating within borders
data(parana)
pr1 <- grf(100, cov.pars=c(200, 40), borders=parana$borders, mean=500)
points(pr1)
pr1 <- grf(100, grid="reg", cov.pars=c(200, 40), borders=parana$borders)
points(pr1)
pr1 <- grf(100, grid="reg", nx=10, ny=5, cov.pars=c(200, 40), borders=parana$borders)
points(pr1)
# }
```

*Documentation reproduced from package geoR, version 1.8-1, License: GPL (>= 2)*