R: Bessel Family Covariance Model

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 ≥ 0 between two points and is given by

C(r) = 2^ν Γ(ν+1) r^{-ν} J_ν(r)

where ν ≥ (d-2)/2, Γ denotes the gamma function and J_ν 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 (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 ν=-0.5 which gives the covariance function

C(r)=cos(r)

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

A second important case is ν=0.5 with covariance function

C(r)=sin(r)/r

and which is valid for d ≤ 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 ν=(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

### 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))

[Package RandomFields version 3.1.50 Index]