Calculate correlation for fully symmetric model
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)
#> [,1] [,2]
#> [1,] 1.0000000 0.9512294
#> [2,] 0.9512294 1.0000000
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)
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1.0000000 0.9512294
#> [2,] 0.9512294 1.0000000
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.5000000 0.4794134
#> [2,] 0.4794134 0.5000000
#>
#> , , 3
#>
#> [,1] [,2]
#> [1,] 0.3333333 0.3209070
#> [2,] 0.3209070 0.3333333
#>