R: Variogram Model of Fractal Brownian Motion

RMfbm {RandomFields} | R Documentation

## Variogram Model of Fractal Brownian Motion

### Description

`RMfbm` is an intrinsically stationary isotropic variogram model. The corresponding centered semi-variogram only depends on the distance r ≥ 0 between two points and is given by

γ(r) = r^α

where 0 < α ≤ 2.
By now, the model is implemented for dimensions up to 3.
For a generalized model see also `RMgenfbm`.

### Usage

    RMfbm(alpha, var, scale, Aniso, proj)

### Arguments

`alpha` | numeric in (0,2]; refers to the fractal dimension of the process
`var,scale,Aniso,proj` | optional arguments; same meaning for any `RMmodel`. If not passed, the above variogram remains unmodified.

### Details

The variogram is unbounded and belongs to a non-stationary process with stationary increments. For α=1 and `scale=2` we get a variogram corresponding to a standard Brownian Motion.

For 0 < α < 2 the quantity H=α/2 is called Hurst index and determines the fractal dimension D of the corresponding Gaussian sample paths

D = d + 1 - H

where d is the dimension of the random field (see Chiles and Delfiner, 1999, p. 89).

### Value

`RMfbm` returns an object of class `RMmodel`.

### Author(s)

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

### References

- Chiles, J.-P. and P. Delfiner (1999) Geostatistics. Modeling Spatial Uncertainty. New York, Chichester: John Wiley & Sons.

- Stein, M.L. (2002) Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 11, 587–599

### See Also

`RMgenfbm`, `RMmodel`, `RFsimulate`, `RFfit`.

### Examples

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

    model <- RMfbm(alpha=1)
    x <- seq(0, 10, 0.02)
    plot(model)
    plot(RFsimulate(model, x=x))

[Package RandomFields version 3.1.50 Index]