R: Turning Bands method

Tbm {RandomFields} | R Documentation

## Turning Bands method

### Description

The Turning Bands method is a simulation method for stationary, isotropic random fields in any dimension and defined on arbitrary points or arbitrary grids. It performs a multidimensional simulation by superposing lower-dimensional fields. In fact, the Turning Bands method is called with the Turning Bands model, see `RMtbm`.
For details see `RMtbm`.

### Usage

    RPtbm(phi, boxcox, fulldim, reduceddim, layers, lines,
          linessimufactor, linesimustep, center, points)

### Arguments

`phi` | object of class `RMmodel`; specifies the covariance function to be simulated; a univariate stationary isotropic covariance model (see `RFgetModelNames(type="positive definite", domain="single variable", isotropy="isotropic", vdim=1)`) which is valid in dimension `fulldim`.
`boxcox` | the one or two parameters of the box cox transformation. If not given, the globally defined parameters are used. see `RFboxcox` for Details.

`fulldim` | a positive integer. The dimension of the space of the random field to be simulated
`reduceddim` | a positive integer; less than `fulldim`. The dimension of the auxiliary hyperplane (most frequently a line, i.e. `reduceddim=1` used in the simulation.
`layers` | a boolean value; for space time model. If `TRUE` then the turning layers are used whenever a time component is given. If `NA` the turning layers are used only when the traditional TBM is not applicable. If `FALSE` then turning layers may never be used. Default: `TRUE`.
`lines` | Number of lines used. Default: `60`.
`linessimufactor` | `linessimufactor` or `linesimustep` must be non-negative; if `linesimustep` is positive then `linesimufactor` is ignored. If both arguments are naught then `points` is used (and must be positive). The grid on the line is `linesimufactor`-times finer than the smallest distance. See also `linesimustep`. Default: `2.0`.
`linesimustep` | If `linesimustep` is positive the grid on the line has lag `linesimustep`. See also `linesimufactor`. Default: `0.0`.
`center` | Scalar or vector. If not `NA`, the `center` is used as the center of the turning bands for `fulldim`. Otherwise the center is determined automatically such that the line length is minimal. See also `points` and the examples below. Default: `NA`.
`points` | integer. If greater than 0, `points` gives the number of points simulated on the TBM line, hence must be greater than the minimal number of points given by the size of the simulated field and the two paramters `linesimufactor` and `linesimustep`. If `points` is not positive the number of points is determined automatically. The use of `center` and `points` is highlighted in an example below. Default: `0`.

### Details

- 2-dimensional case
  It is generally difficult to use the turning bands method (`RPtbm`) directly in the 2-dimensional space. Instead, 2-dimensional random fields are frequently obtained by simulating a 3-dimensional random field (using `RPtbm`) and taking a 2-dimensional cross-section. See also the arguments `fulldim` and `reduceddim`.

- 4-dimensional case
  The turning layers can be used for the simulations with a (formal) time component. It works for all isotropic models, some special models such as `RMnsst`, and multiplicate models with that separate the time component.

### Value

`RPtbm` returns an object of class `RMmodel`

### Note

Both the precision and the simulation time depend heavily on `linesimustep` and `linesimufactor`. For covariance models with larger values of the scale parameter, `linesimufactor=2` is too small.

### Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

### References

Turning bands

- Lantuejoul, C. (2002) Geostatistical Simulation: Models and Algorithms. Springer.

- Matheron, G. (1973). The intrinsic random functions and their applications. Adv. Appl. Probab., 5, 439-468.

- Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

Turning layers

- 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.

### See Also

Gaussian, RP, `RPspectral`

### Examples

    RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
    ##                   RFoptions(seed=NA) to make them all random again

    ## isotropic example that forces the use of the turning bands method
    model <- RPtbm(RMstable(s=1, alpha=1.8))
    x <- seq(-3, 3, 0.1)
    z <- RFsimulate(model=model, x=x, y=x)
    plot(z)

    ## anisotropic example that forces the use of the turning bands method
    model <- RPtbm(RMexp(Aniso=matrix(nc=2, rep(1,4))))
    z <- RFsimulate(model=model, x=x, y=x)
    plot(z)

    ## isotropic example that uses the turning layers method
    model <- RMgneiting(orig=FALSE, scale=0.4)
    x <- seq(0, 10, 0.1)
    z <- RFsimulate(model, x=x, y=x, z=x, T=c(1,1,5))
    plot(z, MARGIN.slices=4, MARGIN.movie=3)

[Package RandomFields version 3.1.50 Index]