# RFMethods

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##### Simulation Techniques

PrintMethodList prints the list of currently implemented methods for simulating random fields

GetMethodNames returns a list of currently implemented methods

Keywords
spatial
##### Usage
PrintMethodList()

GetMethodNames()

##### Details

• (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.

• cutoff embedding. Modified circulant embedding method so that exact simulation is garantueed for further covariance models, e.g. the whittle matern model. In fact, the circulant embedding is called with the cutoff hypermodel, seeCovarianceFct, and$A=B$there.cutoff embeddinghalfens 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 garantueed for furthervariogrammodels, e.g. the fractal brownian one. Note that the simulated random field is alwaysnon-stationary. In fact, the circulant embedding is called with the Stein hypermodel, seeCovarianceFct, and$A=B$there.

Here multiplicative models are not allowed (yet).

• Marked point processes. Some methods are based on marked point process$\Pi=\bigcup [x_i,m_i]$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$L_1 \cap L_2$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 approximative 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.
• MarkovGaussian Random Fields. The method is due to Havard Rue and uses the property that the precision matrix (i.e. the inverse of the covariance matrix) of a Markov Gaussian random field has a small band width.Note:This method is only available on Intel Linux systems when installed as follows: 1. install GMRFLib by Havard Rue fromwww.math.ntnu.no/~hrue/GMRFsim2. install all external libraries given in GMRFLib/EXTLIBS/ (it may happen that precompiled libraries included in your distribution work better)

3. install RandomFields with option --enable-GMRF and --with-GMRF-lib=LIB_PATH and --with-GMRF-EXT-lib=LIB_PATH for the path ot GMRFLib and the EXTLIBS, respectively. E.g.,R CMD INSTALL RandomFields_* --configure-args="--enable-GMRF --with-GMRF-EXT-lib=/home/schlather/local/lib"

• nugget. The method allows for generating a random field of independent Gaussian random variables. In the isotropic case and if the simple notation of a model (withmodelandparam) is used, this method is called automatically if the nugget effect is positive except the method"circulant embedding"or"direct"have been explicitely.

• particularmethod -- details missing --
• Random coins. See marked point processes.
• sequentialThis 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 ofspectral TBMdoes 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 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 (usingTBM3) 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). TBM2andTBM3allow 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 onTBM*.linesimustepandTBM*.linesimufactorthat can be set byRFparameters. For covariance models with larger values of the scale parameter,TBM*.linesimufactor=2is 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.

##### Note

Most methods possess additional parameters, see RFparameters() that control the precision of the result. The default parameters are chosen such that the simulations are fine for many models and their parameters. The example in EmpiricalVariogram() shows a way of checking the precision.

##### Automatic selection algorithm

--- details coming soon ---

##### References

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. C46, 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 {G}aussian stochastic simulations: Extensions to realizations conditioned on direct and indirect measurementWater 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 inRandomFieldsis 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 LimitsJ. 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, no104, 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.