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 vvariate 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 nonzeros  
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 vvariate 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 nonzeros  
O(v^3*n^3 + v^2*n*k)  O(v^2*n^2 + v*k)  arbitrary covariance matrix 
Assume vvariate 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 9910, Dept. of Maths and Statistics, Lancaster University.
Schlather, M. (2010) On some covariance models based on normal scale mixtures. Bernoulli, 16, 780797.
Schlather, M. (2011) Construction of covariance functions and unconditional simulation of random fields. In Porcu, E., Montero, J.M. and Schlather, M., SpaceTime 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
.
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)

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.
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