RMbessel {RandomFields}

R Documentation

Bessel Family Covariance Model

Description

RMbessel is a stationary isotropic covariance model belonging to the Bessel family. The corresponding covariance function only depends on the distance r \ge 0 between two points and is given by

C(r) = 2^\nu \Gamma(\nu+1) r^{-\nu} J_\nu(r)

where \nu \ge \frac{d-2}2, \Gamma denotes the gamma function and J_\nu is a Bessel function of first kind.

Usage

RMbessel(nu, var, scale, Aniso, proj)

Arguments

nu	a numerical value; should be equal to or greater than \frac{d-2}2 to provide a valid covariance function for a random field of dimension d.
var, scale, Aniso, proj	optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Details

This covariance models a hole effect (cf. Chiles, J.-P. and Delfiner, P. (1999), p. 92, cf. Gelfand et al. (2010), p. 26).

An important case is \nu=-0.5 which gives the covariance function

C(r)=\cos(r)

and which is only valid for d=1. This equals RMdampedcos for \lambda = 0, there.

A second important case is \nu=0.5 with covariance function

C(r)=\sin(r)/r

which is valid for d \le 3. This coincides with RMwave.

Note that all valid continuous stationary isotropic covariance functions for d-dimensional random fields can be written as scale mixtures of a Bessel type covariance function with \nu=\frac{d-2}2 (cf. Gelfand et al., 2010, pp. 21–22).

Value

RMbessel returns an object of class RMmodel.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/

References

• Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.

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

• http://homepage.tudelft.nl/11r49/documents/wi4006/bessel.pdf

See Also

RMdampedcos, RMwave, 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 <- RMbessel(nu=1, scale=0.1)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))

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