R: Cauchy Family Covariance Model

RMcauchy {RandomFields} | R Documentation

## Cauchy Family Covariance Model

### Description

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

C(r) = (1 + r^2)^(-γ)

where γ > 0. See also `RMgencauchy`.

### Usage

    RMcauchy(gamma, var, scale, Aniso, proj)

### Arguments

`gamma` | 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.

### Details

The paramater γ determines the asymptotic power law. The smaller γ, 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.

The generalized Cauchy Family (see `RMgencauchy`) includes this family for the choice α = 2 and β = 2 γ. The generalized Hyperbolic Family (see `RMhyperbolic`) includes this family for the choice ξ = 0 and γ = -ν/2; in this case scale=δ.

### Value

`RMcauchy` 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.

- Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.

### See Also

`RMcauchytbm`, `RMgencauchy`, `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 <- RMcauchy(gamma=1)
    x <- seq(0, 10, 0.02)
    plot(model, xlim=c(-3, 3))
    plot(RFsimulate(model, x=x, n=4))

[Package RandomFields version 3.1.50 Index]