Method to simulate the Nugget effect
Method to simulate the Nugget effect. (Only for advanced users)
RPnugget(phi, boxcox, tol, vdim)
phi |
object of class |
boxcox |
the one or two parameters of the box cox transformation.
If not given, the globally defined parameters are used.
See |
tol |
points at a distance less than or equal to Default: |
vdim |
positive integer; the model is treated
Default is that it takes the value of the submodel |
This method only allows RMnugget as a submodel.
The method also allows for zonal nugget effects. Only there the
argument tol becomes important.
For the zonal nugget effect, the anisotropy matrix Aniso
should be given in RMnugget. There, only the
kernel of the
matrix is important.
The
locations at a distance less than or equal to the RFoptions
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 nugget.tol=0.0. However, if the anisotropy matrix
does not have full rank and nugget.tol=0.0, then the
simulations are likely to be odd.
The value of nugget.tol
should be of order 1e-15.
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
allow_duplicated_locations in RFoptions.
In case several measurement have been taken in single space-time
locations,
measurement errors can be separated from spatial noise.
In this case 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 RMnugget(Ansio=something) and
RMnugget(proj=something) model the spatial nugget effect
(with and without zonal anisotropy in case Aniso has low and
full rank respectively).
RPnugget
Even for advanced users, there is no need to call
RPnugget directly, as this is done internally when
the RMnugget is involved in the covariance model.
RPnugget returns an object of class
RMmodel.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Schlather, M. (1999) 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.
Gaussian,
RP,
RPcoins,
RPhyperplane,
RPspectral,
RPtbm.
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))
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