Calculate Cauchy correlation

Calculate Cauchy correlation

Usage

cor_cauchy(x, a, alpha, nu = 1, nugget = 0, is.dist = FALSE)

Arguments

x

A numeric vector, matrix, or array.

a

Smooth parameter, \(a>0\).

alpha

Scale parameter, \(\alpha\in(0, 1]\).

nu

Power parameter, \(\nu>0\). Default is 1.

nugget

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

is.dist

Logical; if TRUE, x is a distance matrix or an array of distance matrices.

Value

Correlations of the same dimension as x.

Details

The Cauchy correlation function with scale parameter \(a\) and smooth parameter \(\alpha\) has the form $$C(x)=(1-\text{nugget})(a|x|^{2\alpha} + 1)^{-\nu}+\text{nugget}\cdot \delta_{x=0},$$ where \(\delta_{x=0}\) is 1 when \(x=0\) and 0 otherwise.

References

Gneiting, T., and Schlather, M. (2004). Stochastic Models That Separate Fractal Dimension and the Hurst Effect. SIAM Review, 46(2), 269–282.

See also

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

Examples

x <- matrix(c(0, 5, 5, 0), nrow = 2)
cor_cauchy(x, a = 1, alpha = 0.5)
#>           [,1]      [,2]
#> [1,] 1.0000000 0.1666667
#> [2,] 0.1666667 1.0000000

x <- matrix(c(0, 5, 5, 0), nrow = 2)
cor_cauchy(x, a = 1, alpha = 0.5, nugget = 0.3, is.dist = TRUE)
#>           [,1]      [,2]
#> [1,] 1.0000000 0.1166667
#> [2,] 0.1166667 1.0000000