RMgencauchy: Generalized Cauchy Family Covariance Model
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value References See Also Examples

RMgencauchy is a stationary isotropic covariance model belonging to the generalized Cauchy family. The corresponding covariance function only depends on the distance r ≥ 0 between two points and is given by

C(r) = (1 + r^α)^(-β/α)

where 0 < α ≤ 2 and β > 0. See also RMcauchy.

1	RMgencauchy(alpha, beta, var, scale, Aniso, proj)

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

This model has a smoothness parameter α and a parameter β which determines the asymptotic power law. More precisely, this model admits simulating random fields where fractal dimension D of the Gaussian sample and Hurst coefficient H can be chosen independently (compare also with RMlgd): Here, we have

D = d + 1 - α/2, 0 < α ≤ 2

and

H = 1 - β/2, β > 0.

I. e. the smaller β, the longer the long-range dependence.

The covariance function is very regular near the origin, because its Taylor expansion only contains even terms and reaches its sill slowly.

Each covariance function of the Cauchy family is a normal scale mixture.

Note that the Cauchy Family (see RMcauchy) is included in this family for the choice α = 2 and β = 2 γ.

RMgencauchy returns an object of class RMmodel.

Covariance function

Gneiting, T. and Schlather, M. (2004) Stochastic models which separate fractal dimension and Hurst effect. SIAM review 46, 269–282.

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.

RMcauchy, RMcauchytbm, RMmodel, RFsimulate, RFfit.

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

model <- RMgencauchy(alpha=1.5, beta=1.5, scale=0.3)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))