R: Methods for Gaussian Random Fields

GaussianFields {RandomFields} | R Documentation

## Methods for Gaussian 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 simulated at k location y_1,...,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,...,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.)

### Author(s)

Martin Schlather, schlather@math.uni-mannheim.de http://ms.math.uni-mannheim.de/de/publications/software

### 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.

### 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

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

[Package RandomFields version 3.1.50 Index]