cor_fs {mcgf}

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)

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.

Value

Correlations of the same dimension as h and u.

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)