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

## Description

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

## Implemented models

 `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.

 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^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)..O(v^3*n^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^2*b^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)

## Computing demand for interpolation

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^3*n^3 + v^2*n*k) 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,...,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.)

## References

• 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`.

## Examples

 ```1 2 3 4 5 6 7``` ```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) ```

### Example output

```Loading required package: sp

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.
```

RandomFields documentation built on Feb. 6, 2020, 5:13 p.m.