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 ≥ 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.
RMbessel(nu, var, scale, Aniso, proj)
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
|
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
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).
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
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
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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