GaussianFields: Methods for Gaussian Random Fields
In RandomFields: Simulation and Analysis of Random Fields

Description Implemented models Computing demand for simulations Computing demand for interpolation Computing demand for conditional simulation References See Also Examples

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

`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

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.

	grid	v	d	time	memory	comments
`RPcirculant`	yes	any	<=13	O(v^3f(n, d))	O(v^2f(n, d))
	no	any	<=13	O(v^3 f(k, d))	O(v^2f(k, d))	k ~ `approx_step`^{-d}
`RPcutoff`						see RPcirculant above
`RPcoins`	yes	1	<=4	O(k n)*	O(n)	k ~ (lattice spacing)^{-d}
	no	1	<=4	O(k n)*	O(n)	k depends on the geometry
`RPdirect`	any	any	any	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^2n^2)*	arbitrary covariance matrix (preparation)
				O(v^2n^2)*	O(v^2n^2)*	arbitrary covariance matrix (simulation)
`RPgauss`	any	any	any	O(1)..O(v^3n^3)*	O(1)..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^2b^2)*	n = S T; S and T the number of spatial and temporal locations, respectively; b =* `back_steps` (preparation)
				O(n S * b^2)*	O(n)	(simulation)
`RPspectral`	any	1	<=2	O(C(d) n)*	O(n)	C(d) : large constant increasing in d
`RPtbm`	any	1	<=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)

Assume v-variate data are given at n locations in d dimensions. To interpolate at k locations RandomFields needs

grid	v	d	time	memory	comments
any	any	any	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^3n^3 + v^2nk)*	O(v^2n^2 + vk)	arbitrary covariance matrix

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

Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.
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.
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.

RP, Other models, RMmodel, RFgetMethodNames, RFsimulateAdvanced.

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

set.seed(1)
x <- runif(90, 0, 500)
z <- RFsimulate(RMspheric(), x)
z <- RFsimulate(RMspheric(), x, max_variab=10000)

Loading required package: sp
Loading required package: RandomFieldsUtils

Attaching package: 'RandomFields'

The following object is masked from 'package:RandomFieldsUtils':

    RFoptions

The following objects are masked from 'package:base':

    abs, acosh, asin, asinh, atan, atan2, atanh, cos, cosh, exp, expm1,
    floor, gamma, lgamma, log, log1p, log2, logb, max, min, round, sin,
    sinh, sqrt, tan, tanh, trunc

NOTE: simulation is performed with fixed random seed 0.
Set 'RFoptions(seed=NA)' to make the seed arbitrary.
New output format of RFsimulate: S4 object of class 'RFsp';
for a bare, but faster array format use 'RFoptions(spConform=FALSE)'.
NOTE: simulation is performed with fixed random seed 0.
Set 'RFoptions(seed=NA)' to make the seed arbitrary.