| GaussianFields {RandomFields} | R Documentation |
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 | \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) | |
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(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 |
Assume v-variate data are given at
n locations x_1,\ldots, x_n in d dimensions.
To conditionally simulate at k locations
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 locations.
(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.)
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
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)