1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174
|
\name{Independent Variables}
\alias{RPnugget}
\alias{Nugget}
\title{Method to simulate the Nugget effect}
\description{
Method to simulate the Nugget effect. (Only for advanced users)
}
\usage{
RPnugget(phi, boxcox, tol, vdim)
}
\arguments{
\item{phi}{object of class \code{\link[=RMmodel-class]{RMmodel}};
specifies the covariance model to be simulated. The only possible
model for \code{phi} is \command{\link{RMnugget}}.}
\item{boxcox}{the one or two parameters of the box cox transformation.
If not given, the globally defined parameters are used.
See \command{\link{RFboxcox}} for details.
}
\item{tol}{
points at a distance less than or equal to \code{nugget.tol}
are considered as being identical. This strategy applies to
the simulation method and the covariance function itself.
Hence, the covariance function is only positive definite
if \code{nugget.tol=0.0}. However, if the anisotropy matrix
does not have full rank and \code{nugget.tol=0.0}, then
the simulations are likely to be odd.
The value of \code{nugget.tol}
should be of order \eqn{10^{-15}}{1e-15}.
Default: \code{0.0}
}
\item{vdim}{positive integer; the model is treated
\code{vdim}-variate, \code{vdim=1} (default) corresponds to a
univariate random field.
Mostly, the value of \code{vdim} is set automatically.
Default is that it takes the value of the submodel \code{phi}.}
}
\details{
\describe{
\item{General}{
This method only allows \command{\link{RMnugget}} as a submodel.
}
\item{Anisotropy}{
The method also allows for zonal nugget effects. Only there the
argument \code{tol} becomes important.
For the zonal nugget effect, the anisotropy matrix \code{Aniso}
should be given in \command{\link{RMnugget}}. There, only the
kernel of the
matrix is important.
}
\item{Points close together}{
The
locations at a distance less than or equal to the \link{RFoptions}
\code{nugget.tol}
are considered as being identical. This strategy applies to
the simulation method and the covariance function itself.
Hence, the covariance function is only positive definite
if \code{nugget.tol=0.0}. However, if the anisotropy matrix
does not have full rank and \code{nugget.tol=0.0}, then the
simulations are likely to be odd.
The value of \code{nugget.tol}
should be of order \eqn{10^{-15}}{1e-15}.
}
\item{Repeated measurements}{
Measurement errors are mathematically not distinguishable from spatial
nugget effects as long as measurements are not repeated at the very
same space-time
location. So there is no need to distinguish the spatial nugget
effect from a measurement error.
This is the default, see
\code{allow_duplicated_locations} in \link{RFoptions}.
In case several measurement have been taken in single space-time
locations,
measurement errors can be separated from spatial noise.
In this case \code{RMnugget()} models the measurement error (which
corresponds to a non-stationary model in an abstract space) by
default and the
measurement error model cannot be extended beyond the given
locations.
On the other hand \code{RMnugget(Ansio=something)} and
\code{RMnugget(proj=something)} model the spatial nugget effect
(with and without zonal anisotropy in case \code{Aniso} has low and
full rank respectively).
}
\item{Role of \command{RPnugget}}{
Even for advanced users, there is no need to call
\command{RPnugget} directly, as this is done internally when
the \link{RMnugget} is involved in the covariance model.
}
}
}
\value{
\code{RPnugget} returns an object of class
\code{\link[=RMmodel-class]{RMmodel}}.
}
\references{
\itemize{
\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.
}
}
\me
\seealso{ \link{Gaussian},
\link{RP},
\command{\link{RPcoins}},
\command{\link{RPhyperplane}},
\command{\link{RPspectral}},
\command{\link{RPtbm}}.
}
\keyword{methods}
\examples{\dontshow{StartExample()}
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
x <- y <- 1:2
xy <- as.matrix(expand.grid(x, y)) ## we get 4 locations
## Standard use of the nugget effect
model <- RMnugget(var = 100)
RFcovmatrix(model, x=xy)
as.vector(RFsimulate(model, x=x, y=x, tol=1e-10))
## zonal nugget effect, which is not along the axes
model <- RMnugget(Aniso=matrix(1, nr=2, nc=2))
RFcovmatrix(model, x=xy)
as.vector(RFsimulate(model, x=x, y=x, tol=1e-10))
## All the following examples refer to repeated measurements
RFoptions(allow_duplicated_locations = TRUE)
(xy <- rbind(xy, xy)) ## now, the 4 locations are repeated twice
## standard situation: the nugget is interpreted as measurement error:
model <- RMnugget()
RFcovmatrix(model, x=xy)
as.matrix(RFsimulate(model, x=xy))
## any anisotropy matrix with full rank: spatial nugget effect
model <- RMnugget(Aniso=diag(2))
RFcovmatrix(model, x=xy)
as.matrix(RFsimulate(model, x=xy))
## anisotropy matrix with lower rank: zonal nugget effect
model <- RMnugget(Aniso=matrix(c(1, 0, 0, 0), nc=2))
RFcovmatrix(model, x=xy)
as.matrix(RFsimulate(model, x=xy))
## same as before: zonal nugget effect
model <- RMnugget(Aniso=t(c(1,0)))
RFcovmatrix(model, x=xy)
as.matrix(RFsimulate(model, x=xy))
\dontshow{FinalizeExample(); RFoptions(allow_duplicated_locations = FALSE) }}
|