Methods for Gaussian Random Fields
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^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) | |
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 |
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.)
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.
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)
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