grf: Simulation of Gaussian Random Fields

In geoR: Analysis of Geostatistical Data

Simulation of Gaussian Random Fields

Description

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

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, messages)

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". Defaults to the Cholesky decomposition. See section DETAILS below.
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 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:

coords	an n \times 2 matrix with the coordinates of the simulated data.
data	a vector (if nsim = 1) or a matrix with the simulated values. For the latter each column corresponds to one simulation.
cov.model	a string with the name of the correlation function.
nugget	the value of the nugget parameter.
cov.pars	a vector with the values of \sigma^2 and \phi, respectively.
kappa	value of the parameter \kappa.
lambda	value of the Box-Cox transformation parameter \lambda.
aniso.pars	a vector with values of the anisotropy parameters, if provided in the function call.
method	a string with the name of the simulation method used.
sim.dim	a string "1d" or "2d" indicating the spatial dimension of the simulation.
.Random.seed	the random seed by the time the function was called.
messages	messages produced by the function describing the simulation.
call	the function call.

Author(s)

Paulo Justiniano Ribeiro Jr. [e-mail],
Peter J. Diggle [e-mail].

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.

Examples

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)

geoR documentation built on May 29, 2024, 1:36 a.m.