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 \ge 0 between two points and is given by

\gamma(r) = r^\alpha

where \alpha \in (0,2].
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 \alpha=1 and scale=2 we get a variogram corresponding to a standard Brownian Motion.

For \alpha \in (0,2) the quantity H = \frac{\alpha} 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, https://www.wim.uni-mannheim.de/schlather/

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))

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