RandomFields (version 3.0.5)

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

parameter in $(0,2]$ refering to the RFfractaldimension of the process

var,scale,Aniso,proj

optional parameters; same meaning for any RMmodel. If not passed, the above variogram remains unmodified.

Value

• RMfbm returns an object of class RMmodel.

Details

The parameter $\alpha$ must satisfy $\alpha \in (0,2]$.

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 RFfractaldimension $D$ 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

set.seed(0)
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))