RandomFields (version 3.0.32)

RMfbm: 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.

Value

• RMfbm returns an object of class RMmodel.

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).

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, if (interactive()) 0.02 else 1) 
plot(model, ylim=c(0,1))
plot(RFsimulate(model, x=x))
FinalizeExample()