RFparameters {RandomFields} | R Documentation |
RFparameters
sets and returns control parameters for the simulation
of random fields
RFparameters(..., no.readonly=FALSE)
... |
arguments in |
no.readonly |
If |
The possible parameters are
General options
PracticalRange
logical or .
If not FALSE
the range of the primitive
covariance functions is
adjusted so that cov(1) is zero for models with finite range.
The value of
cov(1) is about 0.05 (for scale=1
)
for models without range.
See CovarianceFct
or type
PrintModelList()
for the list of
primitive models.
FALSE
: the practical range ajustment is not used.
TRUE
: PracticalRange
is applicable only if
the value is known exactly, or, at least, can be approximated by
a closed formula.
2
: if the practical range is not known exactly it
is approximated numerically.
Default: FALSE
[init].
PrintLevel
If PrintLevel
<=0
there is not any output on the screen. The
higher the number the more tracing information is given.
Default: 1 [init, do].
1 : error messages
2 : messages about partial failures of the algorithm
>2 : additional informations
Note that PrintLevel
is also used in other packages
as a default, for example in SoPhy
(risk.index
and create.roots
). The changing of
PrintLevel
here may cause some unexpected effects in these
functions. See the documentation there.
General options for simulating
pch
Character or empty string.
The character is printed after each
performed simulation if more than one simulation is performed at
once. If pch='!'
then an absolute
counter is shown instead of the character.
If pch='%'
then a
counter of percentages is shown instead of the character.
Note that also '^H's are printed in
the last two cases,
which may have undesirable interactions with some few other R
functions, e.g. Sweave
.
Default: '*'
[do].
Storing
Logical.
If FALSE
then the intermediate results are
destroyed after the simulation of the random field(s)
or if an error had occured.
On the other hand, if Storing=TRUE
, then
several simulations performed with DoSimulateRF
for the same
model parameters are performed faster.
See alse CE.several
, TBMCE.several
and
local.several
for related parameters.
Default: FALSE
[do].
skipchecks
logical.
If TRUE
, the check whether the given parameter values
and the dimension are within the allowed range is skipped.
Do not change the value of this variable except you really
know what you do.
Default: FALSE
[init].
stationary.only
Logical or NA. Used for the automatic choice of methods. See also RFMethods.
TRUE
: the simulation of non-stationary random
fields is refused. In particular, the intrinsic
embedding method is excluded and
the simulation of Brownian motion is rejected.
FALSE
: intrinsic embedding is always allowed,
actually it's the first one considered in the automatic
selection algorithm.
NA
: the simulation of the Brownian motion allowed,
but intrinsic embedding is not used for stationary random fields.
Default: NA
[init].
exactness
logical or NA.
TRUE
: add.MPP, hyperplanes and
all turning bands methods are excluded.
If the circulant embedding method is considered as badly
behaved, then the matrix decomposition methods are preferred.
FALSE
: if the circulant embedding method is
considered as badly behaved or the number of points to be
simulated is large, the turning bands methods are
rather preferred.
NA
: approximative non-exact methods are excluded,
i.e. TBM2 if the Abel transform of the covariance function
cannot be given explicitely.
Default: NA
[init].
every
integer.
if greater than zero, then every every
th iteration is
printed if simulated by TBM or random coin method.
Default: 0
[do].
aniso
logical.
If TRUE
missing anisotropy parameter
or scale parameter in model $
are
intepreted as identity matrix. If FALSE
missing anisotropy or scale parameter is intepreted
as scale parameter of value 1.0. The latter coding tends to faster
simulation, but also to more error messages. The parameter should
be changed only by advanced users.
Default: TRUE
[init].
Options for simulating with the standard circulant embedding method
CE.force
Logical. Circulant embedding does not work if a
certain circulant
matrix has negative eigenvalues. Sometimes it is convenient
to replace all the negative eigenvalues by zero
(CE.force=TRUE
) after CE.trials
number of trials.
Default: FALSE
[init].
CE.mmin
Scalar or vector, integer if positive.
CE.mmin
determines the initial size of the circulant
matrix. If CE.mmin=0
the minimal starting size is
determined automatically according to the
dimensions of the grid.
If CE.mmin>0
then the absolute starting size is given.
If CE.mmin<0
then the automatically determined
matrix size is multiplied
by |\code{CE.mmin}|; here CE.mmin
must be smaller
than -1; the
value -1 takes over the minimal starting size.
Note: in any cases, the initial size might be increased according
to CE.useprimes
.
Default: 0
[init].
CE.strategy
0 : if the circulant
matrix has negative eigenvalues then the
size in each direction is doubled;
1 : the size is enhanced only in
one direction, namely that one where the covariance function has the
largest value at the end point of the grid — note that
the default value of CE.trials
is probably too small
in that case.
In some cases CE.strategy=0
works better, in other cases
CE.strategy=1
. Just try.
Default: 0
[init].
CE.maxmem
maximal total size of the circulant matrix.
The total amount of memory needed for the internal calculations
is about 16 (=2 * sizeof(double))
times as large as CE.maxmem
if RFparameters
()$Storing=FALSE
and 32 (=4 * sizeof(double)) time as large if Storing=TRUE
.
Note that CE.maxmem
can be used to control the automatic
choice of the simulation algorithm. Namely, in case of huge
circulant matrices, other simulation
methods (TBM) are faster and might be preferred be the user.
Default: 4096^2 = 16777216
[init].
CE.tolIm
If the modulus of the imaginary part is less than
CE.tolIm
then the eigenvalue is considered as real.
Default: 1E-3
[init].
CE.tolRe
Eigenvalues between CE.tolRe
and 0 are considered as
0 and set 0. Default: -1E-7
[init].
CE.trials
A larger circulant matrix is likely to make more eigenvalues
non-negative. If at least one of the thresholds CE.tolRe
and
CE.tolIm
are missed then the matrix size is doubled
according to CE.strategy
,
and the matrix is checked again. This procedure is repeated
up to CE.trials-1
times. If there are still negative
eigenvalues, the simulation method fails if CE.force=FALSE
.
Default: 3
[init].
CE.several
logical.
If FALSE
only half the memory is need, but
only a single independent realisation can created.
Default: TRUE
[init].
CE.useprimes
Logical. If FALSE
the columns of the circulant matrix
have length 2^k for some k. Otherwise the algorithm
tries to find a nicely factorizable number close to the size of the
given matrix. Default: TRUE
[init].
CE.dependent
Logical. If FALSE
then independent random fields are created. If TRUE
then at least 4 non-overlapping rectangles are taken out of the
the expanded grid defined by the circulant matrix.
These simulations are dependent.
See below for an example.
See CE.trials
for some
more information on the circulant matrix.
Default: FALSE
[init].
CE.method
0 or 1. Decomposition of the covariance matrix. This parameter is only relevant if multivariate random fields are simulated.
Default (CE.method=0
):
If .Random.seed
is fixed, Cholesky decomposition allows for
fixing the random field of the first component whilst
parameters for the second field are changed.
If CE.method=0
and approximate circulant embedding is
allowed, i.e. CE.force=TRUE
,
SVD is tried for the last failed attempt of Cholesky
decomposition. (SVD, in contrast to Cholesky decomposition, allows
for approximate circulant embedding.)
If CE.method=1
, Cholesky
decomposition will not be attempted, but singular value
decomposition used instead in all attempts. (SVD is slower, but
more precise.)
Default: 0
[init].
Options for simulating by simple matrix decomposition
direct.bestvariables
integer.
When searching for an appropriate simuation method
the matrix decomposition method (see ‘direct method’ below)
is preferred if the number of variables is less than or equal to
direct.bestvariables
.
Default: 800
[init]
direct.maxvariables
If the number of variables to generate is
greater than direct.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=NULL
in
GaussRF
).
Default: 4096
[init]
direct.method
Decomposition of the covariance matrix.
If direct.method=1
, Cholesky
decomposition will not be attempted, but singular value
decomposition
used instead if direct.svdtolerance
positive.
In case of a multivariate random field, direct.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 direct.method = 0
or 1
it starts with the first component of all locations, then the
second components follow, etc.
Default: 0
[init].
direct.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 direct.svdtolerance
. No check is performed if
direct.svdtolerance
is negative.
Default: 1e-12
[init].
Options for simulating hyperplane tessellations
hyper.superpos
integer.
number of superposed hyperplane tessellations.
Default: 300
[do].
hyper.maxlines
integer.
Maximum number of allowed lines.
Default: 1000
[init].
hyper.mar.distr
integer. code for the marginal distribution used in the simulation:
0
uniform distribution
1
Frechet distribution with form parameter
hyper.mar.param
2
Bernoulli distribution (Binomial with n=1) with
parameter hyper.mar.param
The parameter should not be changed yet.
Default: 0
[do].
hyper.mar.param
Parameter used for the marginal
distribution. The parameter should not be changed yet.
Default: 0
[do].
Options for simulating with the local ce
methods (cutoff, intrinsic)
local.force
see CE.force above.
Default: FALSE
[init].
local.mmin
see CE.mmin above.
Difference: if local.mmin=0
the automatic determination
of the initial size of the circulant matrix takes into
account an expansion factor. This expansion factor is
intended to make the circulant matrix positive definite
and is either theoretically or numerically known, or guessed.
If the usual strategy of circulant embedding
(doubling the grid sizes) should be taken over then
local.mmin
must
be set to -1
.
Default: 0
[init].
local.maxmem
see CE.maxmem
above.
Default: 20000000
[init].
local.tolIm
see CE.tolIm
above.
Default: 1E-7
[init].
local.tolRe
see CE.tolRe
above.
Default: -1E-9
[init].
local.several
see CE.several
above.
Default: 1
[init].
local.useprimes
see CE.useprimes
above.
Default: TRUE
[init].
local.dependent
see CE.dependent
above.
Default: FALSE
[init].
Options for simulating a Gaussian Markov random fields
markov.neighbours
2 or 3.
Number of neighbours in each direction.
Default: 2
[init].
markov.precision
See also the GMRF manual.
Default: 1.0
[init].
markov.cyclic
logical.
simulation on a torus; should not be changed
except the user knows what he/she does. (The covariance function
will be (slightly?!) changed then without further notice!)
Default: FALSE
[init].
markov.maxmem
integer.
maximum numer of points of the grid.
Default: 250000
[init].
Options for simulating by random coins
mpp.locations
integer. Way of defining the point process.
-1 : automatic choice
0 : grid
1 : Poisson point process
2 : uniform distribution
3 : single point follows d dimensional Gaussian distribution
Default: -1
[init].
mpp.intensity
real.
Number of superposed
realisations (to approximate the normal distribution; total number
for all (additive) components with same anisotropy);
if mpp.intensity<=0.0
then only a single
value is simulated (for checking).
Default: 100
[do].
mpp.plus
In order avoid edge effects, the simulation area is enlarged by
a constant r so that all marks have their
(supposed) support in the ball with radius r centred at
the origin; see also mpp.approxzero
.
If mpp.plus>0
the true radius r is replaced by
mpp.radius
.
Default: 0.0
[init].
mpp.relRadius
is added to the effectiveRadius.
The latter gives the approximate radius for the support of the
grain.
Default: 2.5
[init].
mpp.approxzero
Functions that
do not have
compact support are set to zero outside the ball outside for which the
function has absolute values less than MPP.approxzero
.
Default: 0.001
[init].
mpp.samplingdist
mpp.samplingdist
gives the grid distance
to numerically approximate the volume of the grain.
[-mpp.samplingr,mpp.sampling]^d
gives the
domain over which the approximation is calculated.
Default: 0.01
[init].
mpp.samplingr
see mpp.samplingdist
Default: 5
[init].
Options for simulating nugget effects
Simulating a nugget effect seems trivial. It gets complicated
and best methods (including direct
and circulant
embedding
!) fail if zonal anisotropies are considered,
where sets of points have to be identified that belong to the
same subspace of eigenvalue 0 of the anisotropy matrix.
nugget.tol
points at a distance less than or equal to 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.
Default: 0.0
[init].
nugget.meth
logical. If TRUE
any method given the user for the
simulation of the nugget effect is replaced by the method
‘nugget’ whenever appropriate (zonal nugget or a method
different form circulant embedding, direct and sequential).
Options for using the sequential method
The method works only for spatio-temporal settings (and
grids).
sequ.max
integer.
maximum number of allowed variables on which
the conditional distribution is conditionned.
Default: 5000
[init].
sequ.back
integer.
number of previous instances on which
the algorithm should condition.
If less than one then the number of previous instances
equals sequ.max
/ (number of spatial points).
Default: 5
[init].
sequ.initial
First, N=(number of spatial points) * sequ.back
number of points are simulated. Then, sequentially,
all spatial points for the next time instance
are simulated at once, based on the previous sequ.back
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 sequ.initial
is non-negative, then sequ.initial
first steps are performed.
If sequ.initial
is negative, then
sequ.back
- sequ.initial
initial steps are performed. The latter ensures that
none of the very first N variables are returned.
Default: -10
[init].
Options for special simulation methods
Special methods exist for the following covariance functions
this covariance function is simulated by a certain moving average of a spatially independent, but temporally dependent random field Y.
spec.MA.r
gives the radius beyond which the bivariate
standard normal density is considered as being zero
Default: 2.5
[init].
spec.MA.dist
the random field Y is
approximated by a grid; the grid length is given by
spec.MA.dist
.
Default: 0.1
[init].
Options for simulating with a turning bands method
Currently, there are 3 variants of the turning bands method
implemented:
spectral
The spectral turning bands method is implemented for 2 (and 1) dimensions only.
TBM2
It is based on the two dimensional turning bands operator and is applicable for 1 and 2 dimensions. As an additional dimension the time dimension can be added.
TBM3
It is based on the three dimensional turning bands operator and is applicable for 1,2,3 dimensions. As an additional dimension the time dimension can be added.
The following parameters are used.
spectral.grid
Logical.
The angle of the lines is random if
spectral.grid=FALSE
,
and k*pi/spectral.lines
for k in 1:spectral.lines
,
otherwise. This parameter is only considered
if the spectral measure, not the density is used.
Default: TRUE
[do].
spectral.ergodic
In case of an additive model and spectral.ergodic=FALSE
,
the additive component are chosen proportional to their
variance. In total spectral.lines
are simulated. If
spectral.ergodic=TRUE
, the components are simulated
separately and then added.
Default: FALSE
[do].
spectral.lines
Number of lines used (in total for all additive components of the
covariance function). Default: 500
[do].
spectral.metro
Logical. If TRUE
then preference is given
to the (slower) metropolis sampling algorithm of the spectral
density.
Default: FALSE
[init].
spectral.nmetro
integer. Considered if the Metropolis
algorithm is used. It gives the number of metropolis steps
before returning the next draw from the spectral density.
If spectral.nmetro
is not positive thenRandomFields
tries to find a good
choice for
spectral.nmetro
by itself.
Default: 0
[init].
spectral.sigma
real. Considered if the Metropolis
algorithm is used. It gives the standard deviation of the
multivariate normal distribution of the proposing
distribution.
If spectral.sigma
is not positive thenRandomFields
tries to find a good
choice for
spectral.sigma
itself.
Default: 0
[init].
TBM.method
character.
The preferred method to simulate on the line for TBM2
and
TBM3
;
If ‘Nothing’ then automatic choice.
Default: "Nothing"
[init].
TBM.center
Scalar or vector.
If not NA
, the TBM.center
is used as the center of
the turning bands for TBM2
and TBM3
.
Otherwise the center is determined
automatically such that the line length is minimal.
See also TBM.points
and the examples below.
Default: NA
[init].
TBM.points
integer. If greater than 0,
TBM.points
gives the number of points simulated on the TBM
line, hence
must be greater than the minimal number of points given by
the size of the simulated field and the two paramters
TBMx.linesimufactor
and TBMx.linesimustep
.
If TBM.points
is not positive the number of points is
determined automatically.
The use of TBM.center
and TBM.points
is highlighted
in an example below.
Default: 0
[init].
TBM2.every
If TBM2.every>0
then every
TBM2.every
th iteration is announced.
Default: 0
[do].
TBM2.lines
Number of lines used.
Default: 60
[do].
TBM2.linesimufactor
TBM2.linesimufactor
or
TBM2.linesimustep
must be non-negative; if
TBM2.linesimustep
is positive then TBM2.linesimufactor
is ignored.
If both
parameters are naught then TBM.points
is used (and must be
positive).
The grid on the line is TBM2.linesimufactor
-times
finer than the smallest distance.
See also TBM2.linesimustep
.
Default: 2.0
[init].
TBM2.linesimustep
If TBM2.linesimustep
is positive the grid on the line has lag
TBM2.linesimustep
.
See also TBM2.linesimufactor
.
Default: 0.0
[init].
TBM2.layers
Logical or integer. If TRUE
then the turning layers are used whenever
a time component is given.
If FALSE
the turning layers are used only when the
traditional TBM is not applicable.
If negative then turning layers may never be used.
If greater than 1 then only turning layers may be used.
Default: FALSE
[init].
TBM3.every
If TBM3.every>0
then every
TBM3.every
th iteration is announced.
Default: 0
[do].
TBM3.lines
Number of lines used.
Default: 500
[do].
TBM3.linesimufactor
See TBM2.linesimufactor
for
the meaning.
Default: 2.0
[init].
TBM3.linesimustep
See
TBM2.linesimustep
for the meaning.
Default: 0.0
[init].
TBM3.layers
See
TBM2.layers
for the meaning.
Default: FALSE
[init].
TBMCE.force
see TBM.method
and CE.force
Default: FALSE
[init].
TBMCE.mmin
see TBM.method
and
CE.mmin
. Default: 0
[init].
TBMCE.strategy
see TBM.method
and
CE.strategy
. Default: 0
[init].
TBMCE.maxmem
see TBM.method
and
CE.maxmem
. Default: 10000000
[init].
TBMCE.tolIm
see TBM.method
and
CE.tolIm
. Default: 1E-3
[init].
TBMCE.tolRe
see TBM.method
and
CE.tolRe
. Default: -1E-7
[init].
TBMCE.trials
see TBM.method
and
CE.trials
. Default: 3
[init].
TBMCE.useprimes
see TBM.method
and
CE.useprimes
. Default: TRUE
[init].
TBMCE.dependent
see TBM.method
and CE.dependent
.
Default: FALSE
[init].
Options specific to simulating max-stable random fields
maxstable.maxGauss
Max-stable random fields.
The simulation of the max-stable process based on random fields uses
a stopping rule that necessarily needs a finite upper endpoint
of the marginal distribution of the random field.
In the case of extremal Gaussian random fields,
see MaxStableRF
, the upper endpoint is
approximated by maxstable.maxGauss
.
Default: 3.0
[init].
General comments on the options
The following refers to the simulation of Gaussian random fields
(InitGaussRF
, GaussRF
), but most
parts also apply
for the simulation of max-stable random fields
(InitMaxStableRF
, MaxStableRF
).
Some of the global parameters determine the basic settings of a
simulation, e.g. direct.method
(which chooses a square
root of a positive definite matrix). The values of
such parameters are read by
InitGaussRF
and stored in an internal register.
Changing
such a parameter between calling InitGaussRF
and calling
DoSimulateRF
or between subsequent calls of
GaussRF
will not have any effect. These parameters have
the flag "[init]".
Parameters like TBM2.lines
(which determines the number of
i.i.d. processes to be simulated on the line)
are only relevant when generating
random numbers. These parameters are read by DoSimulateRF
(or by the second part of GaussRF
), and
are marked by "[do]".
Storing
has an influence on both, InitGaussRF
and
DoSimulateRF
. InitGaussRF
may reserve
more memory if Storing=TRUE
. DoSimulateRF
will
free the register
if Storing=FALSE
, whatever the value of Storing
was
when InitGaussRF
was called.
The distinction between [init] and [do] is also relevant if
GaussRF
is used and called a second time
with the same parameters for the random field and if
RFparameters()$Storing=TRUE
.
Then GaussRF
realises that the second call has the
same random field parameters, and
takes over the stored intermediate results (that have been calculated
with the RFparameters()
at that time). To prevent the use of
stored intermediate results or to take into account intermediate
changes of RFparameters
set RFparameters(Storing=FALSE)
or use
DeleteRegister()
between calls of GaussRF
.
A programme that checks whether the parameters are well
adapted to a specific simulation problem is given as an example of
EmpiricalVariogram()
.
For further details on the implemented methods, see RFMethods.
If any parameter has been given
RFparameters
returns an invisible list of
the given parameters in full name.
Otherwise the complete list of parameters is returned. Further the
values of the following internal readonly variables are returned:
* covmaxchar
: max. name length for variogram/covariance
models
* covnr
: number of currently implemented
variogram/covariance models
* distrmaxchar
: max. name length for a distribution
* distrnr
: number of currently implemented
distributions
* maxdim
: maximum number of dimensions for a
random field
* maxmodels
: maximum number of elementary models in
definition of a complex covariance model
* methodmaxchar
: max. name length for methods
* methodnr
: number of currently implemented simulation methods
Martin Schlather, martin.schlather@math.uni-goettingen.de http://www.stochastik.math.uni-goettingen.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.
GaussRF
,
GetPracticalRange
,
MaxStableRF
,
RandomFields
,
and RFMethods
.
RFparameters(Storing=TRUE) str(RFparameters()) ############################################################ ## ## ## use of TBM.points and TBM.center ## ## ## ############################################################ ## The following example shows that the same realisation ## can be obtained on different grid geometries (or point ## configurations, i.e. grid, non-grid) using TBM3 (or TBM2) x1 <- seq(-150,150,1) y1 <- seq(-15, 15, 1) x2 <- seq(-50, 50, 1) model <- "exponential" param <- c(0, 1, 0, 10) meth <- "TBM3" ###### simulation of a random field on long thing stripe runif(1) rs <- get(".Random.seed", envir=.GlobalEnv, inherits = FALSE) z1 <- GaussRF(x1, y1, model=model, param=param, grid=TRUE, method=meth, TBM.center=0, Storing=TRUE) do.call(getOption("device"), list(height=1.55, width=12)) par(mar=c(2.2, 2.2, 0.1, 0.1)) image(x1, y1, z1, col=rainbow(100)) polygon(range(x2)[c(1,2,2,1)], range(y1)[c(1,1,2,2)], border="red", lwd=3) ###### definition of a random field on a square of shorter diagonal assign(".Random.seed", rs, envir=.GlobalEnv) tbm.points <- length(GetRegisterInfo(meth="TBM")$S$line) z2 <- GaussRF(x2, x2, model=model, param=param, grid=TRUE, register=1, method=meth, TBM.center=0, TBM.points=tbm.points) do.call(getOption("device"), list(height=4.3, width=4.3)) par(mar=c(2.2, 2.2, 0.1, 0.1)) image(x2, x2, z2, zlim=range(z1), col=rainbow(100)) polygon(range(x2)[c(1,2,2,1)], range(y1)[c(1,1,2,2)], border="red", lwd=3) ############################################################ ## ## ## use of exactness ## ## ## ############################################################ x <- seq(0, 1, 1/30) model <- list("+", list("stable", alpha=1.0), list(model="gencauchy", alpha=1.0, beta=2.0) ) for (exactness in c(NA, FALSE, TRUE)) { readline(paste("\n\nexactness: `", exactness, "'; press return")) z <- GaussRF(x, x, grid=TRUE, gridtriple=FALSE, model=model, exactness=exactness, stationary.only=NA, Print=1, n=1, TBM2.linesimustep=1, Storing=TRUE) Print(GetRegisterInfo()$meth$name) } ############################################################# ## The following gives a tiny example on the advantage of ## ## local.dependent=TRUE (and CE.dependent=TRUE) if in a ## ## study most of the time is spent with simulating the ## ## Gaussian random fields. Here, the covariance at a pair ## ## of points is estimated for n independentent repetitions ## ## and 2*n locally dependent dependent repetitions . ## ## To get the precision, the procedure is repeated m times.## ############################################################# # In the example below, local.dependent speeds up the simulation # by about factor 16 at the price of an increased variance of # factor 1.5 x <- seq(0, 1, len=10) y <- seq(0, 1, len=10) grid.size <- c(length(x), length(y)) model <- list("$", var=1.1, aniso=matrix(nc=2, c(2,1,0.5,1)), list(model="exp")) (CovarianceFct(matrix(c(1, -1), ncol=2), model=model)) ## true value m <- if (interactive()) 100 else 2 n <- if (interactive()) 100 else 10 # using local.dependent=FALSE (which is the default) c1 <- numeric(m) unix.time( for (i in 1:m) { cat("", i) z <- GaussRF(x, y, model=model, grid=TRUE, method="cu", n=n, local.dependent=FALSE, pch="") c1[i] <- cov(z[1,length(y), ], z[length(x), 1, ]) }) # many times slower than with local.dependent=TRUE mean(c1) sd(c1) # using local.dependent=TRUE... c2 <- numeric(m) unix.time( for (i in 1:m) { cat("", i) z <- GaussRF(x, y, model=model, grid=TRUE, method="cu", n=2 * n, local.dependent=TRUE, pch="") c2[i] <- cov(z[1,length(y),], z[length(x), 1 , ]) }) mean(c2) sd(c2) # the sd is samller (using more locally dependent realisations) ## but it is (much) faster! For n=n2 instead of n=2 * n, the ## value of sd(c2) would be larger due to the local dependencies ## in the realisations.