RandomFields (version 3.1.12)

RMlgd: Local-Global Distinguisher Family Covariance Model

Description

RMlgd is a stationary isotropic covariance model, which is valid only for dimensions $d =1,2$. The corresponding covariance function only depends on the distance $r \ge 0$ between two points and is given by $$C(r) =1 - \beta^{-1}(\alpha + \beta)r^{\alpha} 1_{[0,1]}(r) + \alpha^{-1}(\alpha + \beta)r^{-\beta} 1_{r>1}(r)$$ where $\beta >0$ and $0 < \alpha \le (3-d)/2$, with $d$ denoting the dimension of the random field.

Usage

RMlgd(alpha, beta, var, scale, Aniso, proj)

Arguments

alpha

argument whose range depends on the dimension of the random field: $0< \alpha \le (3-d)/2$.

beta

beta > 0.

var,scale,Aniso,proj

optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

• RMlgd returns an object of class RMmodel

Details

The model is only valid for dimension $d=1,2$.

This model admits simulating random fields where fractal dimension D of the Gaussian sample and Hurst coefficient H can be chosen independently (compare also RMgencauchy.):

Here, the random field has fractal dimension $$D = d+1 - \alpha/2$$ and Hurst coefficient $$H = 1-\beta/2$$ for $0< \beta \le 1$.

References

• Gneiting, T. and Schlather, M. (2004) Stochastic models which separate fractal dimension and Hurst effect.SIAM review46, 269--282.

See Also

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
StartExample()
model <- RMlgd(alpha=0.7, beta=4, scale=0.5)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))
FinalizeExample()