Coefficients of an extended Shapiro-Botha variogram model

Computes the coefficients of an extended Shapiro-Botha variogram model.

kappasb(x, dk = 0)

Arguments

x: numeric vector (on which the kappa function will be evaluated).
dk: dimension of the kappa function.

Value

A vector with the coefficients of an extended Shapiro-Botha variogram model.

Details

If dk >= 1, the coefficients are computed as: $$\kappa_d(x) = (2/x)^{(d-2)/2} \Gamma(d/2) J_{(d-2)/2}(x)$$ where $J_p$ is the Bessel function of order $p$.
If dk == 0, the coefficients are computed as: $$\kappa _\infty(x) = e^{-x^2}$$ (corresponding to a model valid in any spatial dimension).
NOTE: some authors denote these functions as $\Omega_d$.

References

Shapiro, A. and Botha, J.D. (1991) Variogram fitting with a general class of conditionally non-negative definite functions. Computational Statistics and Data Analysis, 11, 87-96.

Examples

kappasb(seq(0, 6*pi, len = 10), 2)
#>  [1]  1.00000000  0.16979382 -0.37808962  0.22027691  0.07521839 -0.23876759
#>  [7]  0.15750739  0.05562746 -0.18861103  0.12906352
  
curve(kappasb(x/5, 0), xlim = c(0, 6*pi), ylim = c(-1, 1), lty = 2)
for (i in 1:10) curve(kappasb(x, i), col = gray((i-1)/10), add = TRUE)
abline(h = 0, lty = 3)