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

Description Usage Arguments Details Value References See Also Examples

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.

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

`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.

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.

RMcauchy returns an object of class RMmodel.

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.

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