RandomFields (version 3.1.16)

RMstable: Stable Family / Powered Exponential Model

Description

RMstable is a stationary isotropic covariance model belonging to the so called stable family. The corresponding covariance function only depends on the distance $r \ge 0$ between two points and is given by $$C(r) = e^{-r^\alpha}$$ where $0 < \alpha \le 2$.

Usage

RMstable(alpha, var, scale, Aniso, proj)
RMpoweredexp(alpha, var, scale, Aniso, proj)

Arguments

alpha

a numerical value; should be in the interval (0,2] to provide a valid covariance function for a random field of any dimension.

var,scale,Aniso,proj

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

Value

RMstable returns an object of class RMmodel

Details

The parameter $\alpha$ determines the fractal dimension $D$ of the Gaussian sample paths: $$ D = d + 1 - \frac{\alpha}{2}$$ where $d$ is the dimension of the random field. For $\alpha < 2$ the Gaussian sample paths are not differentiable (cf. Gelfand et al., 2010, p. 25).

Each covariance function of the stable family is a normal scale mixture. The stable family includes the exponential model (see RMexp) $\alpha = 1$ and the Gaussian model (see RMgauss) for $\alpha = 2$.

The model is called stable, because in the 1-dimensional case the covariance is the characteristic function of a stable random variable (cf. Chiles, J.-P. and Delfiner, P. (1999), p. 90).

References

Covariance function

• Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.
• Diggle, P. J., Tawn, J. A. and Moyeed, R. A. (1998) Model-based geostatistics (with discussion). Applied Statistics 47, 299--350.
• Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.

Tail correlation function (for $0 < \alpha \le 1$)

• Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

See Also

RMexp, RMgauss, 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 <- RMstable(alpha=1.9, scale=0.4)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))