Discretization nodes of a Shapiro-Botha variogram model

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.

Value

A vector with the discretization nodes.

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.

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)