Variogram Model of Fractal Brownian Motion
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.
RMfbm(alpha, var, scale, Aniso, proj)
alpha |
numeric in (0,2]; refers to the fractal dimension of the process |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
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).
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
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.
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))
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.