PrintMethodList prints the list of currently implemented methods for
simulating random fields GetMethodNames returns a list of currently implemented methods
PrintMethodList()GetMethodNames()
circulant embedding.
Introduced by Dietrich & Newsam (1993) and Wood and Chan (1994).
The way the method is implemented in this package
it only allows for quadratic grids with
common grid lengths.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 1000 points, i.e. a 10x10x10 grid in three dimensions.
This implementation can use the Cholesky decomposition and
the singular value decomposition.
It allows for arbitrary points and arbitrary grids.add.MPP(Random coins).
Here the functions are elements
of the intersection$L_1 \cap L_2$of the Hilbert spaces$L_1$and$L_2$.
A random field Z is obtained by adding the marks:$$Z(\cdot) = \sum_{[x_i,m_i] \in \Pi} m_i(\cdot - x_i)$$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 approximate Gaussian random
field by the central limit theorem -- this is theadd.mppmethod.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 themax.mppmethod.nugget.
One may specify this method (and "nugget" as covariance
function) to generate a random field of
independent Gaussian random variables. However, any other
method and any covariance function, called with zero
variance, generates also such a random field (without loss
of speed).
This method exists mainly for reasons of internal
implementation.spectral TBM(Spectral turning bands).
The principle ofspectral TBMdoes not differ from the other
turning bands methods. However, line simulations are performed by a
spectral technique (Mantoglou and Wilson, 1982); a
realisation is given as the cosine with random
amplitude and random phase.
The implementation allows the simulation of 2-dimensional random
fields defined on arbitrary points or arbitrary grids.TBM2,TBM3(Turning bands methods).
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 (usingTBM3) and taking a 2-dimensional cross-section.
TBM2andTBM3allow for arbitrary points, and
arbitrary grids
(arbitrary number of points in each direction, arbitrary grid length
for each direction)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:
Dietrich, C.R. and Newsam, G.N. (1993) A fast and exact method for multidimensional Gaussian stochastic simulations.Water Resour. Res.29, 2861--2869. Wood, A.T.A. and Chan, G. (1994) Simulation of stationary Gaussian processes in$[0,1]^d$J. Comput. Graph. Stat.3, 409--432.
Matheron, G. (1973) The intrinsic random functions and their applications.Adv. Appl. Probab.5, 439--468.
GaussRF, MaxStableRF, and
RandomFields.