# RMbessel

##### Bessel Family Covariance Model

`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 $\nu \ge (d-2)/2$,
$\Gamma$ 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 $(d-2)/2$ 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
$$C(r)=\sin(r)/r$$ and 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 $\nu=(d-2)/2$
(cf. Gelfand et al., 2010, pp. 21--22).

##### Value

##### 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

##### 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))
```

*Documentation reproduced from package RandomFields, version 3.1.36, License: GPL (>= 3)*