RandomFields (version 3.0.5)

RFsimulateAdvanced: Simulation of Random Fields -- Advanced

Description

This function simulates unconditional random fields:

It also simulates conditional random fields for

  • univariate and multivariat, spatial and spatio-temporal Gaussian random fields

For basic simulation of Gaussian random fields, see RFsimulate. See RFsimulate.more.examples and RFsimulate.sophisticated.examples for further examples.

Arguments

model
object of class RMmodel, RFformula or formula; specifies the model to be
x
matrix of coordinates, or vector of x coordinates, or object of class GridTopology or raster; if matrix, ncol(x)
y
optional vector of y coordinates, ignored if x is a matrix
z
optional vector of z coordinates, ignored if x is a matrix
T
optional vector of time coordinates, T must always be an equidistant vector or given in a gridtriple format (see argument grid); for each component of T, the random field is simulated at all location points
grid
logical; determines whether the vectors x, y, and z or the columns of x should be interpreted as a grid definition (see Details). If grid=TRUE, either x, y, and
data
matrix, data.frame or object of class RFsp; coordinates and response values of measurements in case that conditional simulation is to be performed; if a matrix or a data.frame, the first c
err.model
same as model; gives the model of the measurement errors for the measured data (which must be given in this case!), see Details, err.model=NULL (default) corresponds to error-free measurements, the most common al
distances
object of class dist representing the upper trianguar part of the matrix of Euclidean distances between the points at which the field is to be simulated; only applicable for stationary and isotropic models;
dim
integer; space or space-time dimension of the field
n
number of realizations to generate
...
further options and control arguments for the simulation that are passed to and processed by RFoptions

Value

  • By default, an object of the virtual class RFsp; result is of class RFspatialGridDataFrame if $[space-time-dimension > 1]$ and the coordinates are on a grid, result is of class RFgridDataFrame if $[space-time-dimension = 1]$ and the coordinates are on a grid, result is of class RFspatialPointsDataFrame if $[space-time-dimension > 1]$ and the coordinates are not on a grid, result is of class RFpointsDataFrame if $[space-time-dimension = 1]$ and the coordinates are not on a grid.

    The output format can be switched to the "old" array format using RFoptions, either by globally setting RFoptions(spConform=FALSE) or by passing spConform=FALSE in the call of RFsimulate. Then the object returned by RFsimulate depends on the arguments n and grid in the following way: if vdim > 1 the vdim-variate vector makes the first dimension

    if grid=TRUE an array of the dimension of the random field makes the next dimensions. Here, the dimensions are ordered in the sequence x, y, z, T (if given). Else if no time component is given, then the values are passed as a single vector. Else if the time component is given the next 2 dimensions give the space and the time, respectively. if n > 1 the repetitions make the last dimension

    Note: Conversion between the sp format and the conventional format can be done using the method RFspDataFrame2conventional and the function conventional2RFspDataFrame. InitRFsimulate returns 0 if no error has occurred and a positive value if failed.

Details

RFsimulate simulates different classes of random fields, controlled by the wrapping model.

If the wrapping function of the model argument is a covariance or variogram model (i.e., one of list obtained by RFgetModelNames(type="negative definite", group.by="type"), by default, a Gaussian field with the corresponding covariance structure is simulated. By default, the simulation method is chosen automatically through internal algorithms. The simulation method can be set explicitly by enclosing the covariance function with a method specification.

If other than Gaussian fields are to be simulated, the model argument must be enclosed by a function specifying the type of the random field.

There are different possibilities of passing the locations at which the field is to be simulated. If grid=FALSE, all coordinate vectors (except for the time component $T$) must have the same length and the field is only simulated at the locations given by the rows of $x$ or of cbind(x, y, z). If $T$ is not missing, the field is simulated for all combinations $(x[i, ], T[k])$ or $(x[i], y[i], z[i], T[k])$, $i=1, ...,$nrow(x), $k=1, ...,$length(T), even if model is not explicitly a space-time model. If grid=TRUE, the vectors x, y, z and T or the columns of x and T are interpreted as a grid definition, i.e. the field is simulated at all locations $(x_i, y_j, z_k, T_l)$, as given by expand.grid(x, y, z, T). Here, grid means equidistant in each direction, i.e. all vectors must be equidistant and in ascending order. In case of more than 3 space dimensions, the coordinates must be given in matrix notations. To enable different grid lengths for each direction in combination with the matrix notation, the gridtriple notation c(from, stepsize, len) is used: If x, y, z, T or the columns of x are of length 3, they are internally replaced by seq(from=from, to=from+(len-1)*stepsize, by=stepsize) , i.e. the field is simulated at all locations expand.grid(seq(x$from, length.out=x$len, by=x$stepsize), seq(y$from, length.out=y$len, by=y$stepsize), seq(z$from, length.out=z$len, by=z$stepsize), seq(T$from, length.out=T$len, by=T$stepsize))

If data is passed, conditional simulation is performed.

  • if of classRFsp,ncol(data@coords)must equal the dimension of the index space. Ifdata@datacontains only a single variable, variable names are optional. Ifdata@datacontains more than one variable, variables must be named andmodelmust be given in the tilde notationresp ~ ...(seeRFformula) and"resp"must be contained innames(data@data).
  • % Beschreibung hier stimmt nicht so ganz mit Examples unten ueberein Ifdatais a matrix or a data.frame, eitherncol(data)equals$(dimension of index space + 1)$and the order of the columns is (x, y, z, T, response) or, ifdatacontains more than one response variable (i.e.ncol(data) > (dimension of index space + 1)),colnames(data)must containcolnames(x)or those of"x", "y", "z", "T"that are not missing. The response variable name is matched withmodel, which must be given in the tilde notation. If"x", "y", "z", "T"are missing anddatacontainsNAs,colnames(data)must contain an element which starts withdata; the corresponding column and those behind it are interpreted as the given data and those before the corresponding column are interpreted as the coordinates.
  • ifxis missing,RFsimulatesearches forNAs in the data and performs a conditional simulation for them.

Specification of err.model: In geostatistics we have two different interpretations of a nugget effect: small scale variability and measurement error. The result of conditional simulation usually does not include the measurement error. Hence the measurement error err.model must be given separately. For sake of generality, any model (and not only the nugget effect) is allowed. Consequently, err.model is ignored when unconditional simulation is performed.

References

General
  • Lantuejoul, Ch. (2002)Geostatistical simulation.New York:Springer.
  • 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.

Original work:

  • Circulant embedding: Chan, G. and Wood, A.T.A. (1997) An algorithm for simulating stationary Gaussian random fields.J. R. Stat. Soc., Ser. C46, 171-181. Dietrich, C.R. and Newsam, G.N. (1993) A fast and exact method for multidimensional Gaussian stochastic simulations.Water Resour. Res.29, 2861-2869. Dietrich, C.R. and Newsam, G.N. (1996) A fast and exact method for multidimensional {G}aussian stochastic simulations: Extensions to realizations conditioned on direct and indirect measurementWater Resour. Res.32, 1643-1652.

Wood, A.T.A. and Chan, G. (1994) Simulation of stationary Gaussian processes in$[0,1]^d$J. Comput. Graph. Stat.3, 409-432.

The code used inRandomFieldsis based on Dietrich and Newsam (1996).

  • Intrinsic embedding and Cutoff embedding: Stein, M.L. (2002) Fast and exact simulation of fractional Brownian surfaces.J. Comput. Graph. Statist.11, 587--599. Gneiting, T., Sevcikova, H., Percival, D.B., Schlather, M. and Jiang, Y. (2005) Fast and Exact Simulation of Large Gaussian Lattice Systems in$R^2$: Exploring the LimitsJ. Comput. Graph. Statist.Submitted.
  • Markov Gaussian Random Field: Rue, H. (2001) Fast sampling of Gaussian Markov random fields.J. R. Statist. Soc., Ser. B,63(2), 325-338. Rue, H., Held, L. (2005)Gaussian Markov Random Fields: Theory and Applications.Monographs on Statistics and Applied Probability, no104, Chapman \& Hall.
  • Turning bands method (TBM), turning layers: Dietrich, C.R. (1995) A simple and efficient space domain implementation of the turning bands method.Water Resour. Res.31, 147-156. Mantoglou, A. and Wilson, J.L. (1982) The turning bands method for simulation of random fields using line generation by a spectral method.Water. Resour. Res.18, 1379-1394.
  • Matheron, G. (1973) The intrinsic random functions and their applications.Adv. Appl. Probab.5, 439-468.

    Schlather, M. (2004) Turning layers: A space-time extension of turning bands.Submitted

  • Random coins: Matheron, G. (1967)Elements pour une Theorie des Milieux Poreux. Paris: Masson.
  • See Also

    RFoptions, RMmodel, RFgui, methods for simulating Gaussian random fields, RFfit, RFempiricalvariogram, RFsimulate.more.examples, RFsimulate.sophisticated.examples, RPgauss,

    Examples

    Run this code
    RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
    ##                   RFoptions(seed=NA) to make them all random again
    
    \dontrun{
    
    #############################################################
    ##                                                         ##
    ## Example 1: Specification of simulation method           ##
    ##                                                         ##
    #############################################################
    
     ## usage of a specific method
     ## -- the complete list is obtained by RFgetMethodNames()
     model <- RMstable(alpha=1.5)
     x <- runif(100, max=20) 
     y <- runif(100, max=20) # 100 points in 2 dimensional space
     simulated <- RFsimulate(model = RPdirect(model), x=x, y=y, 
     cPrint=3) # direct matrix decomposition
     plot(simulated)
    
    
    
    #############################################################
    ##                                                         ##
    ## Example 2: Turnung band with different number of lines  ##
    ##                                                         ##
    #############################################################
    model <- RMstable(alpha=1.5)
    x <- seq(0, 10, if (interactive()) 0.01, 1)
    z <- RFsimulate(model = RPtbm(model), x=x, y=x)
    plot(z)
    
    
    
    #############################################################
    ##                                                         ##
    ## Example 3: Shot noise fields (random coins)             ##
    ##                                                         ##
    #############################################################
    
     x <- GridTopology(0, .1, 500)
    
     z <- RFsimulate(model=RPpoisson(RMgauss()), x=x, mpp.intensity = 100)
    
     plot(z)
     par(mfcol=c(2,1))
     plot(z@data[,1:min(length(z@data), 1000)], type="l")
     hist(z@data[,1])
     
     
     z <- RFsimulate(x=x, model=RPpoisson(RMball()), mpp.intensity = 0.1)
     
     plot(z)
     par(mfcol=c(2,1))
     plot(z@data[,1:min(length(z@data), 1000)], type="l")
     hist(z@data[,1])
     
    
    
    
     #############################################################
     ##                                                         ##
     ## Example 4: a 2d random field based on                   ##
     ## covariance functions valid in 1d only                   ##
     ##                                                         ##
     #############################################################
    
    x <- seq(0, 2, 0.1)
    model <- RMfbm(alpha=0.5, Aniso=matrix(nrow=1, c(1, 0))) + 
             RMfbm(alpha=0.9, Aniso=matrix(nrow=1, c(0, 1)))
    z <- RFsimulate(x, x, model=model)
    plot(z)
    
    
    
    #############################################################
    ##                                                         ##
    ## Example 5 : Brownian sheet                              ##
    ## (using Stein's method)                                  ##
    ##                                                         ##
    #############################################################
     
    # 2d
    step <- 0.3 ## nicer, but also time consuming if step = 0.1
    x <- seq(0, 5, step)
    alpha <- 1 # in [0,2)
    z <- RFsimulate(x=x, y=x, model=RMfbm(alpha=alpha))
    plot(z)
    
    
    # 3d
    z <- RFsimulate(x=x, y=x, z=x,
     model=RMfbm(alpha=alpha))
    }
    FinalizeExample()

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