# disc.sb: Discretization nodes of a Shapiro-Botha variogram model In npsp: Nonparametric Spatial Statistics

## Description

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

## Usage

 `1` ```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,…, 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,…, 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.

`kappasb`, `fitsvar.sb.iso`.
 ```1 2 3 4 5 6 7 8``` ```disc.sb( 12, 1, 1.0) 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) ```