Square roots {RandomFields}R Documentation

Methods relying on square roots of the covariance matrix

Description

Methods relying on square roots of the covariance matrix

Usage

RPdirect(phi, root_method, svdtolerance, max_variab) 

RPsequential(phi, max_variables, back_steps, initial)

Arguments

phi

object of class RMmodel; specifies the covariance model to be simulated.

root_method

Decomposition of the covariance matrix. If root_method=1 or 3, Cholesky decomposition will not be attempted, but singular value decomposition performed instead. In case of a multivariate random field, root_method = 2 or 3 orders the covariance such that first all components are considered for the first variable, then all components for the second one, and so on. If root_method = 0 or 1 it starts with the first component of all locations, then the second components follow, etc.

Default: 0 .

svdtolerance

If SVD decomposition is used for calculating the square root of the covariance matrix then the absolute componentwise difference between the covariance matrix and square of the square root must be less than svdtolerance. No check is performed if svdtolerance is negative.

Default: 1e-12 .

max_variab

If the number of variables to generate is greater than maxvariables, then any matrix decomposition method is rejected. It is important that this option is set conveniently to avoid great losses of time during the automatic search of a simulation method (method="any method").

Default: 8192

max_variables

The maximum size of the conditional covariance matrix (default to 5000)

back_steps

Number of previous instances on which the algorithm should condition. If less than one then the number of previous instances equals max / (number of spatial points).

Default: 10 .

initial

First, N=(number of spatial points) * back_steps number of points are simulated. Then, sequentially, all spatial points for the next time instance are simulated at once, based on the previous back_steps instances. The distribution of the first N points is the correct distribution, but differs, in general, from the distribution of the sequentially simulated variables. We prefer here to have the same distribution all over (although only approximatively the correct one), hence do some initial sequential steps first. If initial is non-negative, then initial first steps are performed. If initial is negative, then back_steps - initial initial steps are performed. The latter ensures that none of the very first N variables are returned.

Default: -10 .

Details

RPdirect is based on the well-known method for simulating any multivariate Gaussian distribution, using the square root of the covariance matrix. The method is pretty slow and limited to about 8000 points, i.e. a 20x20x20 grid in three dimensions. This implementation can use the Cholesky decomposition and the singular value decomposition. It allows for arbitrary points and arbitrary grids.

RPsequential is programmed for spatio-temporal models where the field is modelled sequentially in the time direction conditioned on the previous k instances. For k=5 the method has its limits for about 1000 spatial points. It is an approximative method. The larger k the better. It also works for certain grids where the last dimension should contain the highest number of grid points.

Value

RPsequential returns an object of class RMmodel

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

References

See Also

RP, RPcoins, RPhyperplane, RPspectral, RPtbm.

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again
model <- RMgauss(var=10, s=10) + RMnugget(var=0.01)
plot(model, xlim=c(-25, 25))

z <- RFsimulate(model=RPdirect(model), 0:10, 0:10, n=4)
plot(z)
[Package RandomFields version 3.0.62 Index]