Simulation Techniques
RFgetMethodNames prints and returns a list of currently
implemented methods for simulating Gaussian random fields
and max stable random fields
RFgetMethodNames()
By default, RFsimulate automatically chooses an appropriate
method for simulation. The method can also be set explicitly by the
user via RFoptions, in particular by passing
gauss.method=_a valid method string_ as an additional argument
to RFsimulate or by globally changing the options via
RFoptions(gauss.method=_a valid method string_).
The following methods are available:
(random spatial) Averages
– details soon
Boolean functions.
See marked point processes.
circulant embedding.
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.
See also cutoff embedding and intrinsic embedding for
variants of the method.
cutoff embedding.
Modified circulant embedding method so that exact simulation is guaranteed
for further covariance models, e.g. the whittle matern model.
In fact, the circulant embedding is called with the cutoff
hypermodel, see RMmodel, and A=B there.
cutoff embedding halfens the maximum number of
elements models used to define the covariance function of interest
(from 10 to 5).
Here, multiplicative models are not allowed (yet).
direct matrix decomposition.
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 8000 points, i.e. a 20x20x20 grid in three dimensions.
This implementation can use the Cholesky decomposition and
the singular value decomposition.
It allows for arbitrary points and arbitrary grids.
hyperplane method.
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.
intrinsic embedding.
Modified circulant embedding so that exact simulation is guaranteed
for further variogram models, e.g. the fractal brownian one.
Note that the simulated random field is always non-stationary.
In fact, the circulant embedding is called with the Stein
hypermodel, see RMmodel, and A=B there.
Here, multiplicative models are not allowed (yet).
Marked point processes.
Some methods are based on marked point process
P = ([x_1,m_1], [x_2,m_2], ...)
where the marks m_i
are deterministic or i.i.d. random functions on R^d.
add.MPP (Random coins).
Here the functions are elements
of the intersection (L1 cap L2)
of the Hilbert spaces L1 and L2.
A random field Z is obtained by adding the marks:
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 m_i
are all indicator functions, we obtain
a Poisson random field. If the intensity of the Poisson
process is high we obtain an approximative Gaussian random
field by the central limit theorem - this is the
add.mpp method.
max.MPP (Boolean functions).
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 max.mpp
method.
nugget.
The method allows for generating a random field of
independent Gaussian random variables.
This method is called automatically if the nugget
effect is positive except the method "circulant embedding"
or "direct" has been explicitly chosen.
The method has been extended to zonal anisotropies, see
also argument nugget.tol in RFoptions.
particular method
– details missing –
Random coins.
See marked point processes.
sequential
This method is programmed for spatio-temporal models
where the field is modelled sequentially in the time direction
conditioned on the previous k instances.
For k=5 the method has its limits for about 1000 spatial
points. It is an approximative method. The larger k the
better.
It also works for certain grids where the last dimension should
contain the highest number of grid points.
spectral TBM (Spectral turning bands).
The principle of spectral TBM
does not differ from the other
turning bands methods. However, line simulations are performed by a
spectral technique (Mantoglou and Wilson, 1982).
The standard method allows for the simulation of 2-dimensional random fields defined on arbitrary points or arbitrary grids. Here, a realisation is given as the cosine with random amplitude and random phase.
TBM2, TBM3 (Turning bands methods; turning layers).
It is generally difficult to use the turning bands method
(TBM2) directly
in the 2-dimensional space.
Instead, 2-dimensional random fields are frequently obtained
by simulating a 3-dimensional random field (using
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).TBM2 and TBM3 allow for arbitrary points, and
arbitrary grids
(arbitrary number of points in each direction, arbitrary grid length
for each direction).
Note: Both the precision and the simulation time
depend heavily on TBM*.linesimustep and
TBM*.linesimufactor
that can be set by RFoptions.
For covariance models with larger values of the scale parameter,
TBM*.linesimufactor=2 is too small.
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.
an invisible string vector of the Gaussian methods.
— details coming soon —
Most methods possess additional arguments,
see RFoptions()
that control the precision of the result. The default arguments
are chosen such that the simulations are fine for many models
and their parameters.
The example in RFvariogram()
shows a way of checking the precision.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Gneiting, T. and Schlather, M. (2004) Statistical modeling with covariance functions. In preparation.
Lantuejoul, Ch. (2002) Geostatistical simulation. New York: Springer.
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.
Original work:
Circulant embedding:
Chan, G. and Wood, A.T.A. (1997) An algorithm for simulating stationary Gaussian random fields. J. R. Stat. Soc., Ser. C 46, 171-181.
Dietrich, C.R. and Newsam, G.N. (1993) A fast and exact method for multidimensional Gaussian stochastic simulations. Water Resour. Res. 29, 2861-2869.
Dietrich, C.R. and Newsam, G.N. (1996) A fast and exact method for multidimensional Gaussian stochastic simulations: Extensions to realizations conditioned on direct and indirect measurement Water Resour. Res. 32, 1643-1652.
Wood, A.T.A. and Chan, G. (1994) Simulation of stationary Gaussian processes in [0,1]^d J. Comput. Graph. Stat. 3, 409-432.
The code used in RandomFields is based on Dietrich and Newsam (1996).
Intrinsic embedding and Cutoff embedding:
Stein, M.L. (2002) Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 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 R^2: Exploring the Limits J. Comput. Graph. Statist. Submitted.
Markov Gaussian Random Field:
Rue, H. (2001) Fast sampling of Gaussian Markov random fields. J. R. Statist. Soc., Ser. B, 63 (2), 325-338.
Rue, H., Held, L. (2005) Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability, no 104, Chapman \& Hall.
Turning bands method (TBM), turning layers:
Dietrich, C.R. (1995) A simple and efficient space domain implementation of the turning bands method. Water Resour. Res. 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. Water. Resour. Res. 18, 1379-1394.
Matheron, G. (1973) The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439-468.
Schlather, M. (2004) Turning layers: A space-time extension of turning bands. Submitted
Random coins:
Matheron, G. (1967) Elements pour une Theorie des Milieux Poreux. Paris: Masson.
RFgetMethodNames()
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