https://github.com/cran/RandomFields
Tip revision: e994a4415e67fa60cbfd3f208aaab20872521c0b authored by Martin Schlather on 14 February 2019, 21:02:19 UTC
version 3.3
version 3.3
Tip revision: e994a44
RFsimulateAdvanced.Rd
\name{RFsimulateAdvanced}
\alias{RFsimulateAdvanced}
\title{Simulation of Random Fields -- Advanced}
\description{
This function simulates \bold{unconditional} random fields:
\itemize{
\item univariate and multivariate,
spatial and spatio-temporal Gaussian random fields
\item \link[=RPpoisson]{stationary Poisson fields}
\item \link[=RPchi2]{Chi2 fields}
\item \link[=RPt]{t fields}
\item \link[=RPbernoulli]{Binary fields}
\item \link[=RPmaxstable]{stationary max-stable random fields}.
}
It also simulates \bold{conditional} random fields for
\itemize{
\item univariate and multivariate,
spatial and spatio-temporal Gaussian random fields.
}
For basic simulation of Gaussian random fields, see \link{RFsimulate}.
See \link{RFsimulate.more.examples} and
\link{RFsimulate.sophisticated.examples} for further examples.
}
\arguments{
\item{model}{object of class \code{\link[=RMmodel-class]{RMmodel}},
\command{\link{RFformula}} or \command{\link[stats]{formula}};
specifies the model to be simulated
\itemize{
\item if of class \code{\link[=RMmodel-class]{RMmodel}}, \code{model}
specifies
\itemize{
\item the type of random field by using \code{RP}functions,
e.g.,
\itemize{
\item \command{\link{RPgauss}}: Gaussian random field (default
if none of the functions in the list is given)
% \item \command{\link{RPbernoulli}}:
% indicator field, based on a Gaussian field
% \item \command{\link{RPpoisson}}: shot noise model
% \item \command{\link{RPbrownresnick}}: Brown-Resnick processes
% \item \command{\link{RPopitz}}: Extremal t process
\item \command{\link{RPsmith}}: Smith model
% \item \command{\link{RPschlather}}: Extremal Gaussian process
% (also called Schlather model)
See\code{ \link{RP}} for an overview.
}
\item the covariance or variogram model in case of a Gaussian random
field (\command{\link{RPgauss}}) and for fields based on Gaussian fields
(e.g. \command{\link{RPbernoulli}});
type \code{\link{RFgetModelNames}(type="variogram")}
for a list of available models; see also \command{\link{RMmodel}}.
\item the shape function in case of a shot noise process; type
\code{\link{RFgetModelNames}(type='shape')} for a list of
available models.
% \item the main structural element of a max-stable field
% (\command{\link{RPbrownresnick}}, \command{\link{RPopitz}},
% \command{\link{RPsmith}} and \command{\link{RPschlather}})
}
\item if of class \code{\link{RFformula}} or \command{\link[stats]{formula}},
\code{submodel} specifies a linear mixed model where random
effects can be modelled by Gaussian random fields;
see \command{\link{RFformula}} for details on model
specification.
}
}
\item{x}{matrix of coordinates, or vector of x coordinates, or object
of class \code{\link[sp:GridTopology-class]{GridTopology}} or
\code{\link[raster]{raster}};
if matrix, \code{ncol(x)} is the dimension
of the index space; matrix notation is required in case of more than 3 space
dimensions; in this case, if \code{grid=FALSE}, \code{x_ij} is the
i-th coordinate in the j-th dimension; otherwise, if
\code{grid=TRUE}, the columns of \code{x} are interpreted as
gridtriples (see \code{grid}); if of class \code{\link[sp:GridTopology-class]{GridTopology}},
\code{x} is interpreted as grid definition and \code{grid}
is automatically set to \code{TRUE}.
}
\item{y}{optional vector of y coordinates, ignored if \code{x}
is a matrix}
\item{z}{optional vector of z coordinates, ignored if \code{x}
is a matrix}
\item{T}{optional vector of time coordinates,
\code{T} must always be an equidistant vector or given in a
gridtriple format (see argument \code{grid}); for each component
of \code{T}, the random field is simulated at all location points.
}
\item{grid}{logical; determines whether the vectors \code{x},
\code{y}, and \code{z} or the columns of \code{x} should be
interpreted as a grid definition (see Details). If \code{grid=TRUE},
either \code{x}, \code{y}, and \code{z} must
be equidistant vectors in ascending order or the columns of \code{x}
must be given in the gridtriple format:
\code{c(from, stepsize, len)}.
Note: If \code{grid} is not given, \command{\link{RFsimulate}} tries to
guess what is meant.
\code{c(from, stepsize, len)} (see Details)
}
\item{data}{matrix, data.frame or object of class \command{\link[=RFsp-class]{RFsp}};
coordinates and response values of
measurements in case that conditional simulation is to
be performed;
if a matrix or a data.frame, the first columns are
interpreted as coordinate vectors, and the last column(s) as
(multiple) measurement(s) of the field; if \code{x} is missing,
\code{data} may contain \code{NA}s, which are then replaced by
conditionally simulated values; if \code{data} is missing, unconditional
simulation is performed;
for details on matching of variable names see Details; if of class
\command{\link[=RFsp-class]{RFsp}}
}
\item{err.model}{same as \code{model}; gives the model of the
measurement errors for the measured \code{data} (which must be given
in this case!), see Details.
\code{err.model=NULL} (default) corresponds to error-free
measurements, the most common alternative is
\code{err.model=\link{RMnugget}()};
ignored if \code{data} is missing.
}
\item{distances}{object of class \code{\link{dist}} representing
the upper triangular part of the matrix of Euclidean distances
between the points at which the field is to be simulated; only
applicable for stationary and isotropic models; if not \code{NULL},
\code{dim} must be given and \code{x}, \code{y}, \code{z} and
\code{T} must be missing or \code{NULL}.
If \code{distances} are given, the current value of \code{spConform}, see
\command{\link{RFoptions}}, is ignored and instead
\code{spConform=FALSE} is used. (This fact may change in future.) % To Do
}
\item{dim}{
integer; space or space-time dimension of the field
}
\item{n}{number of realizations to generate}
\item{...}{further options and control arguments for the simulation
that are passed to and processed by \command{\link{RFoptions}}
}
}
\details{
\command{\link{RFsimulate}} simulates different classes of random fields,
controlled by the wrapping model.
If the wrapping function of the \code{model} argument is a covariance
or variogram model, i.e., one of the list obtained by
\code{\link{RFgetModelNames}(type="variogram",
group.by="type")}, by default, a Gaussian field
with the corresponding covariance structure is simulated.
By default, the simulation method is chosen automatically through internal algorithms.
The simulation method can be set explicitly by enclosing the
covariance function with a \link[=Gaussian]{method specification}.
If other than Gaussian fields are to be simulated, the \code{model}
argument must be enclosed by a function specifying the type of the
random field.
There are different possibilities of passing the locations at which
the field is to be simulated. If \code{grid=FALSE}, all coordinate
vectors (except for the time component \eqn{T}) must have the same
length and the field is only simulated at the locations given by the
rows of \eqn{x} or of \code{cbind(x, y, z)}. If \eqn{T} is not
missing, the field is simulated for all combinations
\eqn{(x[i, ], T[k])} or \eqn{(x[i], y[i], z[i], T[k])},
\eqn{i=1, ..., }\code{nrow(x)}, \eqn{k=1, ..., }\code{length(T)},
even if \code{model} is not explicitly a space-time model.
\cr
If \code{grid=TRUE}, the vectors \code{x},
\code{y}, \code{z} and \code{T} or the columns of \code{x} and
\code{T} are
interpreted as a grid definition, i.e. the field is simulated at all
locations \eqn{(x_i, y_j, z_k, T_l)}, as given by
\code{expand.grid(x, y, z, T)}.
Here, \dQuote{grid} means \dQuote{equidistant in each direction}, i.e. all
vectors must be equidistant and in ascending order.
In case of more than 3 space dimensions, the coordinates must be
given in matrix notation. To enable different grid lengths for each
direction in combination with the matrix notation, the
\dQuote{gridtriple} notation \code{c(from, stepsize, len)} is used:
If \code{x}, \code{y},
\code{z}, \code{T} or the columns of \code{x} are of length 3, they
are internally replaced by \code{seq(from=from,
to=from+(len-1)*stepsize, by=stepsize)} , i.e. the field
is simulated at all locations\cr
\code{expand.grid(seq(x$from, length.out=x$len, by=x$stepsize),
seq(y$from, length.out=y$len, by=y$stepsize),
seq(z$from, length.out=z$len, by=z$stepsize),
seq(T$from, length.out=T$len, by=T$stepsize))}
If \code{data} is passed, conditional simulation is performed.
\itemize{
\item
If of class \code{\link[=RFsp-class]{RFsp}},
\code{ncol(data@coords)} must equal the dimension of the index
space. If \code{data@data} contains only a single variable,
variable names are optional. If \code{data@data} contains
more than one variable, variables must be named and \code{model}
must be given in the tilde notation \code{resp ~ ... } (see
\command{\link{RFformula}}) and \code{"resp"} must be contained
in \code{names(data@data)}.
\item
% Beschreibung hier stimmt nicht so ganz mit Examples unten ueberein
If \code{data} is a matrix or a data.frame, either \code{ncol(data)}
equals \eqn{(dimension of index space + 1)} and the order of the
columns is (x, y, z, T, response) or, if \code{data} contains
more than one
response variable (i.e. \code{ncol(data) > (dimension of index
space + 1)}), \code{colnames(data)} must contain
\code{colnames(x)} or those of \code{"x", "y", "z", "T"} that
are not missing. The response variable name is matched with
\code{model}, which must be given in the tilde notation. If
\code{"x", "y", "z", "T"} are missing and \code{data} contains
\code{NA}s, \code{colnames(data)} must contain an element which starts
with \sQuote{data}; the corresponding column and those behind it are
interpreted as the given data and those before the corresponding
column are interpreted as the coordinates.
\item
If \code{x} is missing, \command{\link{RFsimulate}} searches for
\code{NA}s in the data and performs a conditional simulation
for them.
}
Specification of \code{err.model}:
In geostatistics we have two different interpretations of a nugget
effect: small scale variability and measurement error.
The result of conditional simulation usually does not include the
measurement error. Hence the measurement error \code{err.model}
must be given separately. For sake of generality, any model (and not
only the nugget effect) is allowed.
Consequently, \code{err.model} is ignored
when unconditional simulation is performed.
}
\value{By default,
an object of the virtual class
\command{\link[=RFsp-class]{RFsp}};
result is of class
\command{\link[=RFspatialGridDataFrame]{RFspatialGridDataFrame}}
if \eqn{[space-time-dimension > 1]} and the coordinates are on a grid,
result is of class
\command{\link[=RFgridDataFrame]{RFgridDataFrame}}
if \eqn{[space-time-dimension = 1]} and the coordinates are on a grid,
result is of class
\command{\link[=RFspatialPointsDataFrame]{RFspatialPointsDataFrame}}
if \eqn{[space-time-dimension > 1]} and the coordinates are not on a grid,
result is of class
\command{\link[=RFpointsDataFrame]{RFpointsDataFrame}}
if \eqn{[space-time-dimension = 1]} and the coordinates are not on a
grid.
The output format can be switched to the "old" array format using
\code{\link{RFoptions}}, either by globally setting
\code{\link{RFoptions}(spConform=FALSE)} or by passing \code{spConform=FALSE}
in the call of \command{\link{RFsimulate}}.
Then the object returned by \command{\link{RFsimulate}}
depends on the arguments \code{n} and \code{grid} in the following way:\cr
If \code{vdim > 1} the \code{vdim}-variate vector makes the first dimension.
If \code{grid=TRUE} an array of the dimension of the
random field makes the next dimensions. Here, the dimensions
are ordered in the sequence \code{x}, \code{y}, \code{z}, \code{T}
(if given).
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 the space and the time, respectively.
If \code{n > 1} the repetitions make the last dimension.
Note: Conversion between the \pkg{sp}
format and the conventional format can be
done using the method \command{RFspDataFrame2conventional} and the
function \command{conventional2RFspDataFrame}.
\code{InitRFsimulate} returns 0 if no error has occurred and a positive value
if failed.\cr
}
\note{Advanced options are
\itemize{
\item \code{spConform} (suppressed return of S4 objects)
\item \code{practicalrange} (forces range of covariances to be one)
\item \code{exactness} (chooses the simulation method by precision)
\item \code{seed} (sets \code{\link[base]{.Random.seed}} locally or
globally)
}
See \command{\link{RFoptions}} for further options.
}
\references{
% Gneiting, T. and Schlather, M. (2004)
% Statistical modeling with covariance functions.
% \emph{In preparation.}
General
\itemize{
\item
Lantuejoul, Ch. (2002) \emph{Geostatistical simulation.}
\bold{New York:} Springer.
\item
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.
Dietrich, C.R. and Newsam, G.N. (1996)
A fast and exact method for multidimensional {G}aussian stochastic
simulations: Extensions to realizations conditioned on direct and
indirect measurement
\emph{Water Resour. Res.} \bold{32}, 1643-1652.
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.
The code used in \cite{RandomFields} is based on
Dietrich and Newsam (1996).
\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 Markov Gaussian Random Field:
Rue, H. (2001) Fast sampling of Gaussian Markov random fields.
\emph{J. R. Statist. Soc., Ser. B}, \bold{63} (2), 325-338.
Rue, H., Held, L. (2005) \emph{Gaussian Markov Random Fields:
Theory and Applications.}
Monographs on Statistics and Applied Probability, no \bold{104},
Chapman \& Hall.
\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.
}
}
\me
\seealso{
\command{\link{RFoptions}},
\command{\link{RMmodel}},
\command{\link{RFgui}},
\command{\link[=Gaussian]{methods for simulating Gaussian random fields}},
\command{\link{RFfit}},
\command{\link{RFvariogram}},
\link{RFsimulate.more.examples},
\link{RFsimulate.sophisticated.examples},
\command{\link{RPgauss}}.
}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
\dontshow{\dontrun{
#############################################################
## ##
## Example 1: Specification of simulation method ##
## ##
#############################################################
## usage of a specific method
## -- the complete list is obtained by RFgetMethodNames()
model <- RMstable(alpha=1.5)
x <- runif(100, max=20)
y <- runif(100, max=20) # 100 points in 2 dimensional space
simulated <- RFsimulate(model = RPdirect(model), x=x, y=y) # cholesky
plot(simulated)
#############################################################
## ##
## Example 2: Turnung band with different number of lines ##
## ##
#############################################################
model <- RMstable(alpha=1.5)
x <- seq(0, 10, 0.01)
z <- RFsimulate(model = RPtbm(model), x=x, y=x)
plot(z)
#############################################################
## ##
## Example 3: Shot noise fields (random coins) ##
## ##
#############################################################
x <- GridTopology(0, .1, 500)
z <- RFsimulate(model=RPpoisson(RMgauss()), x=x, mpp.intensity = 100)
plot(z)
par(mfcol=c(2,1))
plot(z@data[,1:min(length(z@data), 1000)], type="l")
hist(z@data[,1])
z <- RFsimulate(x=x, model=RPpoisson(RMball()), mpp.intensity = 0.1)
plot(z)
par(mfcol=c(2,1))
plot(z@data[,1:min(length(z@data), 1000)], type="l")
hist(z@data[,1])
#############################################################
## ##
## Example 4: a 2d random field based on ##
## covariance functions valid in 1d only ##
## ##
#############################################################
x <- seq(0, 2, 0.1)
model <- RMfbm(alpha=0.5, Aniso=matrix(nrow=1, c(1, 0))) +
RMfbm(alpha=0.9, Aniso=matrix(nrow=1, c(0, 1)))
z <- RFsimulate(x, x, model=model)
plot(z)
#############################################################
## ##
## Example 5 : Brownian sheet ##
## (using Stein's method) ##
## ##
#############################################################
# 2d
step <- 0.3 ## nicer, but also time consuming if step = 0.1
x <- seq(0, 5, step)
alpha <- 1 # in [0,2)
z <- RFsimulate(x=x, y=x, model=RMfbm(alpha=alpha))
plot(z)
# 3d
z <- RFsimulate(x=x, y=x, z=x,
model=RMfbm(alpha=alpha))
#############################################################
## ##
## Example 5 : Non-Geometric anisotropy ##
## ##
#############################################################
x <- seq(0.1, 6, 0.12)
Aniso <- R.c(R.p(1)^2, R.p(2)^1.5)
z <- RFsimulate(RMexp(Aniso = Aniso) + 10, x, x)
plot(z)
}}
\dontshow{FinalizeExample()}}
\keyword{spatial}