R: Stable Family / Powered Exponential Model

RMstable {RandomFields} | R Documentation

## 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 ≥ 0 between two points and is given by

C(r)=e^{-r^α}

where 0 < α ≤ 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.

### Details

The parameter α determines the fractal dimension D of the Gaussian sample paths:

D = d + 1 - α/2

where d is the dimension of the random field. For α < 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`) α = 1 and the Gaussian model (see `RMgauss`) for α = 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).

### Value

`RMstable` returns an object of class `RMmodel`

### Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

### 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 < α ≤ 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

`RMbistable`, `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))

[Package RandomFields version 3.1.50 Index]