RandomFields (version 3.1.12)

GaussianFields: Methods for Gaussian Random Fields

Description

Here, all the methods (models) for simulating Gaussian random fields are listed

Implemented models

ll{ RPcirculant simulation by circulant embedding RPcutoff simulation by a variant of circulant embedding RPcoins simulation by random coin / shot noise RPdirect through the square root of the covariance matrix RPgauss generic model that chooses automatically among the specific methods RPhyperplane simulation by hyperplane tessellation RPintrinsic simulation by a variant of circulant embedding RPnugget simulation of (anisotropic) nugget effects RPsequential sequential method RPspecific model specific methods (very advanced) RPspectral spectral method RPtbm turning bands }

Computing demand for simulations

Assume at $n$ locations in $d$ dimensions a $v$-variate field has to be simulated. Let $$f(n, d) = 2^d n \log(n)$$ The following table gives in particular the time and memory needed for the specific simulation method. lllllll{ grid $v$ $d$ time memory comments RPcirculant yes any $\le 13$ $O(v^3f(n, d))$ $O(v^2f(n, d))$ no any $\le 13$ $O(v^3 f(k, d))$ $O(v^2f(k, d))$ $k \sim$approx_step${}^{-d}$ RPcutoff see RPcirculant above RPcoins yes $1$ $\le 4$ $O(k n)$ $O(n)$ $k \sim$$(lattice spacing)^{-d}$ no $1$ $\le 4$ $O(k n)$ $O(n)$ $k$ depends on the geometry RPdirect any any any $O(1)..O(v^2 n^2)$ $O(v^2 n^2)$ effort to investigate the covariance matrix, if matrix_methods is not specified (default) $O(v n)$ $O(v n)$ covariance matrix is diagonal see spam $O(z + v n)$ covariance matrix is sparse matrix with $z$ non-zeros $O(v^3 n^3)$ $O(v^2 n^2)$ arbitrary covariance matrix (preparation) $O(v^2 n^2)$ $O(v^2 n^2)$ arbitrary covariance matrix (simulation) RPgauss any any any $O(1) \ldots O(v^3n^3)$ $O(1)\ldots O(n^2)$ only the selection process; $O(1)$ if first method tried is successful RPhyperplane any $1$ $2$ $O(n / s^d)$ $O(n / s^d)$ $s =$scale RPintrinsic see RPcirculant above RPnugget any any any $O(v n)$ $O(v n)$ RPsequential any $1$ any $O(S^3 b^3)$ $O(S^2 b^2)$ $n=ST$; $S$ and $T$ the number of spatial and temporal locations, respectively; $b =$back_steps (preparation) $O(n S b^2)$ $O(S^2 b^2) + O(n)$ (simulation) RPspectral any $1$ $\le 2$ $O(C(d) n)$ $O(n)$ $C(d)$ : large constant increasing in $d$ RPtbm any $1$ $\le 4$ $O(C(d) (n + L)$ $O(n + L)$ $C(d)$ : large constant increasing in $d$; $L$ is the effort needed to simulate on a line (or plane) RPspecific only the specific part * * RMplus any any any O(v n) O(v n) * * RMS any any any O(1) O(v n) * * RMmult any any any O(v n) O(v n) }

Computing demand for interpolation

Assume $v$-variate data are given at $n$ locations in $d$ dimensions. To interpolate at $k$ locations RandomFields needs lllllll{ grid $v$ $d$ time memory comments any any any $O(1)..O(v^2 n^2)$ $O(v^2 n^2)$ effort to investigate the covariance matrix, if matrix_methods is not specified (default) $O(v ^2 n k)$ $O(v (n + k))$ covariance matrix is diagonal see spam+ O(v^2nk) $O(z + v (n + k))$ covariance matrix is sparse matrix with $z$ non-zeros $O(v^3 n^3 + v^2nk)$ $O(v^2 n^2 + v*k)$ arbitrary covariance matrix }

Computing demand for conditional simulation

Assume $v$-variate data are given at $n$ locations $x_1,\ldots, x_n$ in $d$ dimensions. To conditionally simulated at $k$ location $y_1,\ldots, y_k$, the computing demand equals the sum of the demand for interpolating and the demand for simulating on the $k+n$ location. (Grid algorithms for simulating will apply if the $k$ locations $y_1,\ldots, y_k$ are defined by a grid and the $n$ locations $x_1,\ldots, x_n$ are a subset of $y_1,\ldots, y_k$, a situation typical in image analysis.)

References

• Chiles, J.-P. and Delfiner, P. (1999)Geostatistics. Modeling Spatial Uncertainty.New York: Wiley. % \item Gneiting, T. and Schlather, M. (2004) % Statistical modeling with covariance functions. % \emph{In preparation.}
• 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.
• Schlather, M. (2010) On some covariance models based on normal scale mixtures.Bernoulli,16, 780-797.
• Schlather, M. (2011) Construction of covariance functions and unconditional simulation of random fields. In Porcu, E., Montero, J.M. and Schlather, M.,Space-Time Processes and Challenges Related to Environmental Problems.New York: Springer. % \item Schlather, M. (2002) Models for stationary max-stable % random fields. \emph{Extremes} \bold{5}, 33-44.
• Yaglom, A.M. (1987)Correlation Theory of Stationary and Related Random Functions I, Basic Results.New York: Springer.
• Wackernagel, H. (2003)Multivariate Geostatistics.Berlin: Springer, 3nd edition.

See Also

RP, Other models, RMmodel, RFgetMethodNames RFsimulateAdvanced

Examples

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

library(RandomFields, lib="~/TMP")
RFoptions(print = 3)
set.seed(1)

x <- runif(9000, 0, 500)
z <- RFsimulate(RMspheric(), x)
z <- RFsimulate(RMspheric(), x, max_variab=10000)FinalizeExample()