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.
See also
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)
