Computes the discretization nodes of a `nonparametric' extended Shapiro-Botha variogram model, following Gorsich and Genton (2004), as the scaled roots of Bessel functions.

disc.sb(nx, dk = 0, rmax = 1)

Arguments

nx

number of discretization nodes.

dk

dimension of the kappa function (dk >= 1, see Details below).

rmax

maximum lag considered.

Details

If dk >= 1, the nodes are computed as: $$x_i = q_i/rmax; i = 1,\ldots, nx,$$ where \(q_i\) are the first \(n\) roots of \(J_{(d-2)/2}\), \(J_p\) is the Bessel function of order \(p\) and \(rmax\) is the maximum lag considered. The computation of the zeros of the Bessel function is done using the efficient algorithm developed by Ball (2000).

If dk == 0 (corresponding to a model valid in any spatial dimension), the nodes are computed so the gaussian variogram models involved have practical ranges: $$r_i = 2 ( 1 + (i-1) ) rmax/nx; i = 1,\ldots, nx.$$

References

Ball, J.S. (2000) Automatic computation of zeros of Bessel functions and other special functions. SIAM Journal on Scientific Computing, 21, 1458-1464.

Gorsich, D.J. and Genton, M.G. (2004) On the discretization of nonparametric covariogram estimators. Statistics and Computing, 14, 99-108.

See also

Examples

disc.sb( 12, 1, 1.0)
#> [1] 1.570796 4.712389 7.853982 10.995574 14.137167 17.278760 20.420352 #> [8] 23.561945 26.703538 29.845130 32.986723 36.128316
nx <- 1 dk <- 0 x <- disc.sb(nx, dk, 1.0) h <- seq(0, 1, length = 100) plot(h, kappasb(x * h, 0), type="l", ylim = c(0, 1))
abline(h = 0.05, lty = 2)