Revision

**c103e17c562b32dcb179c0853306f28a0e456d50**authored by Martin Schlather on**11 April 2005, 00:00:00 UTC**, committed by Gabor Csardi on**11 April 2005, 00:00:00 UTC****1 parent**e7b694a

Methods.Rd

```
\name{RFMethods}
\alias{RFMethods}
\alias{PrintMethodList}
\alias{GetMethodNames}
\title{Simulation Techniques}
\description{
\code{PrintMethodList} prints the list of currently implemented methods for
simulating random fields
\code{GetMethodNames} returns a list of currently implemented methods
}
\usage{
PrintMethodList()
GetMethodNames()
}
\details{
\itemize{
\item Boolean functions.\cr
See marked point processes.
\item \code{circulant embedding}. \cr
Introduced by Dietrich & Newsam (1993) and Wood and Chan (1994).
Circulant embedding is a fast simulation method based on
Fourier transformations. It is garantueed to be an exact method
for covariance functions with finite support, e.g. the spherical
model.
\item \code{cutoff embedding}. \cr
Modified circulant embedding method so that exact simulation is garantueed
for further covariance models, e.g. the whittle matern model.
Here multiplicative models are not allowed (yet).
\item \code{direct matrix decomposition}.\cr
This method 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 1000 points, i.e. a 10x10x10 grid in three dimensions.
This implementation can use the Cholesky decomposition and
the singular value decomposition.
It allows for arbitrary points and arbitrary grids.
\item \code{hyperplane method}.\cr
The method is based on a tessellation of the space by
hyperplanes. Each cell takes a spatially constant value
of an i.i.d. random variables. The superposition of several
such random fields yields approximatively a Gaussian random field.
\item \code{intrinsic embedding}. \cr
Modified circulant embedding so that exact simulation is garantueed
for further \emph{variogram} models, e.g. the fractal brownian one.
Note that the simulated random field is always non-stationary.
Here multiplicative models are not allowed (yet).
\item Marked point processes.\cr
Some methods are based on marked point process
\eqn{\Pi=\bigcup [x_i,m_i]}{P = ([x_1,m_1], [x_2,m_2], ...)}
where the marks \eqn{m_i}{m_i}
are deterministic or i.i.d. random functions on \eqn{R^d}{R^d}.
\itemize{
\item \code{add.MPP} (Random coins).\cr
Here the functions are elements
of the intersection \eqn{L_1 \cap L_2}{(L1 cap L2)}
of the Hilbert spaces \eqn{L_1}{L1} and \eqn{L_2}{L2}.
A random field Z is obtained by adding the marks:
\deqn{ Z(\cdot) = \sum_{[x_i,m_i] \in \Pi} m_i(\cdot - x_i)}{
Z(.) = sum_i m_i( . - x_i)}
In this package, only stationary Poisson point fields
are allowed
as underlying unmarked point processes.
Thus, if the marks \eqn{m_i}{m_i}
are all indicator functions, we obtain
a Poisson random field. If the intensity of the Poisson
process is high we obtain an approximate Gaussian random
field by the central limit theorem - this is the
\code{add.mpp} method.
\item \code{max.MPP} (Boolean functions).\cr
If the random functions are multiplied by suitable,
independent random values, and then the maximum is
taken, a max-stable random field with unit Frechet margins
is obtained - this is the \code{max.mpp}
method.
}
\item \code{nugget}.\cr
One may specify this method (and "nugget" as covariance
function) to generate a random field of
independent Gaussian random variables. However, any other
method and any covariance function, called with zero
variance, generates also such a random field (without loss
of speed).
This method exists mainly for reasons of internal
implementation.
\item Random coins.\cr
See marked point processes.
\item \code{spectral TBM} (Spectral turning bands).\cr
The principle of \code{spectral TBM}
does not differ from the other
turning bands methods. However, line simulations are performed by a
spectral technique (Mantoglou and Wilson, 1982); a
realisation is given as the cosine with random
amplitude and random phase.
The implementation allows the simulation of 2-dimensional random
fields defined on arbitrary points or arbitrary grids.
\item \code{TBM2}, \code{TBM3} (Turning bands methods; turning layers).\cr
It is generally difficult to use the turning bands method
(\code{TBM2}) directly
in the 2-dimensional space.
Instead, 2-dimensional random fields are frequently obtained
by simulating a 3-dimensional random field (using
\code{TBM3}) and taking a 2-dimensional cross-section.
TBM3 allows for multiplicative models; in case of anisotropy the
anisotropy matrices must be multiples of the first matrix or the
anisotropy matrix consists of a time component only (i.e. all
components are zero except the very last one).\cr
\code{TBM2} and \code{TBM3} allow for arbitrary points, and
arbitrary grids
(arbitrary number of points in each direction, arbitrary grid length
for each direction).
The turning layers are used for the simulations with time component.
Here,
if the model is a
multiplicative covariance function then the
product may contain matrices with pure time component. All
the other matrices must be equal up to a factor and the temporal
part of the anisotropy matrix (right column) may contain only
zeros, except the very last entry.
}
}
\note{Most methods possess additional parameters,
see \code{\link{RFparameters}()}
that control the precision of the result. The default parameters
are chosen such that the simulations are fine for many models
and their parameters.
The example in \code{\link{EmpiricalVariogram}()}
shows a way of checking the precision.
}
\references{
Gneiting, T. and Schlather, M. (2004)
Statistical modeling with covariance functions.
\emph{In preparation.}
Lantuejoul, Ch. (2002) \emph{Geostatistical simulation.}
\bold{New York:} Springer.
Schlather, M. (1999) \emph{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:
\itemize{
\item Circulant embedding:
Chan, G. and Wood, A.T.A. (1997)
An algorithm for simulating stationary Gaussian random fields.
\emph{J. R. Stat. Soc., Ser. C} \bold{46}, 171-181.
Dietrich, C.R. and Newsam, G.N. (1993)
A fast and exact method for multidimensional Gaussian
stochastic simulations.
\emph{Water Resour. Res.} \bold{29}, 2861-2869.
Wood, A.T.A. and Chan, G. (1994)
Simulation of stationary Gaussian processes in \eqn{[0,1]^d}{[0,1]^d}
\emph{J. Comput. Graph. Stat.} \bold{3}, 409-432.
\item Intrinsic embedding and Cutoff embedding:
Stein, M.L. (2002)
Fast and exact simulation of fractional Brownian surfaces.
\emph{J. Comput. Graph. Statist.} \bold{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
\eqn{R^2}: Exploring the Limits
\emph{J. Comput. Graph. Statist.} Submitted.
\item Turning bands method (TBM), turning layers:
Dietrich, C.R. (1995) A simple and efficient space domain implementation
of the turning bands method. \emph{Water Resour. Res.} \bold{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. \emph{Water. Resour. Res.} \bold{18}, 1379-1394.
Matheron, G. (1973)
The intrinsic random functions and their applications.
\emph{Adv. Appl. Probab.} \bold{5}, 439-468.
Schlather, M. (2004)
Turning layers: A space-time extension of turning bands.
\emph{Submitted}
\item Random coins:
Matheron, G. (1967) \emph{Elements pour une Theorie des Milieux
Poreux}. Paris: Masson.
}
}
\section{Automatic selection algorithm}{
--- more details coming soon ---
}
\author{Martin Schlather, \email{schlath@hsu-hh.de}
\url{http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html}
Yindeng Jiang \email{jiangyindeng@gmail.com} (circulant embedding
methods \sQuote{cutoff} and \sQuote{intrinsic})
}
\seealso{\code{\link{GaussRF}}, \code{\link{MaxStableRF}},
\code{\link{PrintModelList}},
\code{\link{RandomFields}}.}
\examples{
PrintMethodList()
}
\keyword{spatial}
```

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