mcgf (version 1.1.1)

cor_fs: Calculate correlation for fully symmetric model

Description

Calculate correlation for fully symmetric model

Usage

cor_fs(nugget = 0, c, gamma = 1/2, a, alpha, beta = 0, h, u)

Value

Correlations of the same dimension as h and u.

Arguments

nugget

The nugget effect \(\in[0, 1]\).

c

Scale parameter of cor_exp, \(c>0\).

gamma

Smooth parameter of cor_exp, \(\gamma\in(0, 1/2]\).

a

Scale parameter of cor_cauchy, \(a>0\).

alpha

Smooth parameter of cor_cauchy, \(\alpha\in(0, 1]\).

beta

Interaction parameter, \(\beta\in[0, 1]\).

h

Euclidean distance matrix or array.

u

Time lag, same dimension as h.

Details

The fully symmetric correlation function with interaction parameter \(\beta\) has the form $$C(\mathbf{h}, u)=\dfrac{1}{(a|u|^{2\alpha} + 1)} \left((1-\text{nugget})\exp\left(\dfrac{-c\|\mathbf{h}\|^{2\gamma}} {(a|u|^{2\alpha}+1)^{\beta\gamma}}\right)+ \text{nugget}\cdot \delta_{\mathbf{h}=\boldsymbol 0}\right),$$ where \(\|\cdot\|\) is the Euclidean distance, and \(\delta_{x=0}\) is 1 when \(x=0\) and 0 otherwise. Here \(\mathbf{h}\in\mathbb{R}^2\) and \(u\in\mathbb{R}\). By default beta = 0 and it reduces to the separable model.

References

Gneiting, T. (2002). Nonseparable, Stationary Covariance Functions for Space–Time Data, Journal of the American Statistical Association, 97:458, 590-600.

See Also

Other correlation functions: cor_cauchy(), cor_exp(), cor_lagr_askey(), cor_lagr_exp(), cor_lagr_tri(), cor_sep(), cor_stat(), cor_stat_rs()

Examples

h <- matrix(c(0, 5, 5, 0), nrow = 2)
u <- matrix(0, nrow = 2, ncol = 2)
cor_fs(c = 0.01, gamma = 0.5, a = 1, alpha = 0.5, beta = 0.5, h = h, u = u)

h <- array(c(0, 5, 5, 0), dim = c(2, 2, 3))
u <- array(rep(0:2, each = 4), dim = c(2, 2, 3))
cor_fs(c = 0.01, gamma = 0.5, a = 1, alpha = 0.5, beta = 0.5, h = h, u = u)