R: Generalized Cauchy Family Covariance Model

RMepscauchy {RandomFields} | R Documentation

## Generalized Cauchy Family Covariance Model

### Description

`RMepscauchy` is a stationary isotropic covariance model belonging to the generalized Cauchy family. In contrast to most other models it is not a correlation function. The corresponding covariance function only depends on the distance r ≥ 0 between two points and is given by

C(r) = (ε + r^α)^(-β/α)

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

### Usage

    RMepscauchy(alpha, beta, eps, 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.
`beta` | a numerical value; should be positive to provide a valid covariance function for a random field of any dimension.
`eps` | a positive value
`var,scale,Aniso,proj` | optional arguments; same meaning for any `RMmodel`. If not passed, the above covariance function remains unmodified.

### Details

This model has a smoothness parameter α and a paramater 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 `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.

### Value

`RMepscauchy` returns an object of class `RMmodel`

### Author(s)

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

### References

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

### See Also

`RMcauchy`, `RMcauchytbm`, `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 <- RMepscauchy(alpha=1.5, beta=1.5, scale=0.3, eps=0.5)
    x <- seq(0, 10, 0.02)
    plot(model)
    plot(RFsimulate(model, x=x))

[Package RandomFields version 3.1.50 Index]