RPcirculant: Circulant Embedding methods
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value References See Also Examples

Circulant embedding is a fast simulation method for stationary (possibly anisotropic) Gaussian random fields on regular grids based on Fourier transformations. It is guaranteed to be an exact method for covariance functions with finite support, e.g. the spherical model. The method is admissible for any dimension apart from memory restrictions.
The simulation is performed on a torus which represents the bended grid. To remove wrong dependencies occuring at different borders of the grid which would be close on the torus, the simulation area is multiplied by a natural number. There is also a multivariate version of circulant embedding.

Cut-off embedding is a fast simulation method for stationary, isotropic Gaussian random fields on square lattices based on the standard RPcirculant method, so that exact simulation is garantueed for further covariance models, e.g. the RMwhittle model.

In fact, the circulant embedding is called with the cutoff hypermodel, see RMcutoff. Cutoff 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).
For details see RMcutoff.

Intrinsic embedding is a fast simulation method for intrinsically stationary, isotropic Gaussian random fields on square lattices based on the standard RPcirculant method, for further variogram models, e.g. RMfbm.

Note that the simulated random field is always non-stationary. In fact, the circulant embedding is called with the Intrinsic hypermodel, see RMintrinsic.

Here, multiplicative models are not allowed (yet).
For details see RMintrinsic.

RPcirculant(phi, boxcox, force, mmin, strategy,
 maxGB, maxmem,  tolIm, tolRe, trials, useprimes, dependent,
 approx_step, approx_maxgrid)

RPcutoff(phi, boxcox, force, mmin, strategy,
 maxGB, maxmem, tolIm, tolRe, trials, useprimes, dependent,
 approx_step, approx_maxgrid, diameter, a) 
 

RPintrinsic(phi, boxcox, force, mmin, strategy,
 maxGB, maxmem, tolIm, tolRe, trials, useprimes, dependent,
 approx_step, approx_maxgrid, diameter, rawR)

phi

See RPgauss.

`boxcox`	the one or two parameters of the box cox transformation. If not given, the globally defined parameters are used. See `RFboxcox` for details.
`force`	Logical. Circulant embedding does not work if the constructed circulant matrix has negative eigenvalues. Sometimes it is convenient to replace all the negative eigenvalues by zero (`force=TRUE`) after `trials` number of trials. Default: `FALSE`.
`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`.
`strategy`	Logical. `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 `trials` is probably too small in that case. In some cases `strategy=0` works better, in other cases `strategy=1`. Just try. Clearly, if the field is isotropic and a square grid should be simulated, then `strategy=0` is the better choice. Default: `0`.
`maxGB`	Maximal memory used for the circulant matrix in units of GB. If this argument is set then `maxmem` is set to MAXINT. Default: 1.
`maxmem`	Integer. maximal number of entries in a row of the circulant matrix. The total amount of memory needed for the internal calculations is 32 (= 4 * sizeof(double)) time as large (factor 2 is needed as complex numbers must be considered for calculating the fft of the covariance matrix; another factor 2 is needed for storing the simulated result). The value of `maxmem` must be at least 2^d times as large as the number of points to be simulated. Here, d is the space dimension. In some cases even much larger. Note that `maxmem` can be used to control the automatic choice of the simulation algorithm. Namely, in case of huge circulant matrices, other simulation methods (TBM) might be faster and might be preferred by the user. If this argument is set then `maxGB` is set to `Inf`. Default: `MAXINT`.
`tolIm`	If the modulus of the imaginary part is less than `tolIm` then the eigenvalue is always considered as real (independently of the value of `force`). Default: `1E-3`.
`tolRe`	Eigenvalues between `tolRe` and 0 are always considered as 0 and set 0 (independently of the value of `force`). Default: `-1E-7`.
`trials`	Integer. A larger circulant matrix is likely to make more eigenvalues non-negative. If at least one of the thresholds `tolRe` and `tolIm` are missed then the matrix size is doubled according to `strategy`, and the matrix is checked again. This procedure is repeated up to `trials - 1` times. If there are still negative eigenvalues, the simulation method fails if `force=FALSE`. Default: `3`.
`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`.
`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 RFoptionsAdvanced for an example. See `trials` for some more information on the circulant matrix. Default: `FALSE`.
`approx_step`	Real value. It gives the grid size of the approximating grid in case circulant embedding is used although the points do not lie on a grid. If `NA` then `approx_step` is chosen such that `approx_maxgrid` is nearly reached. Default: `NA`.
`approx_maxgrid`	It defaults to `maxmem`.
`diameter`	See `RMcutoff` or `RMintrinsic`.
`a`	See `RMcutoff`.
`rawR`	See `RMintrinsic`.

Here, the algorithms by Dietrich and Newsam are implemented.

An object of class RMmodel.

Circulant Embedding

Chan, G. and Wood, A.T.A. (1997) An Algorithm for Simulating Stationary Gaussian Random Fields. Journal of the Royal Statistical Society: Series C (Applied Statistics), 46, 171–181.
Dietrich, C. R. and G. N. Newsam (1993) A fast and exact method for multidimensional gaussian stochastic simulations. Water Resour. Res. 29(8), 2861–2869.
Dietrich, C. R. and G. N. Newsam (1996) A fast and exact method for multidimensional Gaussian stochastic simulations: Extension to realizations conditioned on direct and indirect measurements Water Resour. Res. 32(6), 1643–1652.
Dietrich, C. R. and Newsam, G. N. (1997) Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix. SIAM J. Sci. Comput. 18, 1088–1107.
Wood, A. T. A. and Chan, G. (1994) Simulation of Stationary Gaussian Processes in [0, 1]^d. Journal of Computational and Graphical Statistics 3, 409–432.

Cutoff and Intrinsic

Gneiting, T., Sevecikova, H, Percival, D.B., Schlather M., Jiang Y. (2006) Fast and Exact Simulation of Large Gaussian Lattice Systems in $R^2$: Exploring the Limits. J. Comput. Graph. Stat. 15, 483–501.
Stein, M.L. (2002) Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 11, 587–599

Gaussian, RP

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

model <- RMstable(s=1, alpha=1.8)
x <- seq(-3,3,0.1)

z <- RFsimulate(model=RPcirculant(model), x=x, y=x, n=1)
plot(z)

model <- RMexp(var=10, s=2)
z <- RFsimulate(model=RPcirculant(model), 1:10)
plot(z)

model <- RMfbm(Aniso=diag(c(1,2)), alpha=1.5)
z <- RFsimulate(model, x=1:10, y=1:10)
plot(z)