graphicalExtremes (version 0.2.0)

Gamma2Sigma: Transformation of \(\Gamma\) matrix to \(\Sigma\) or \(\Sigma^k\) matrix

Description

Transforms the Gamma matrix from the definition of a Huesler--Reiss distribution to the corresponding \(\Sigma\) or \(\Sigma^k\) matrix.

Usage

Gamma2Sigma(Gamma, k = NULL, full = FALSE)

Value

Numeric \(\Sigma^k\) matrix of size \((d-1) \times (d-1)\) if full = FALSE, and \(\Sigma\) of size \(d \times d\) if full = TRUE.

Arguments

Gamma

Numeric \(d \times d\) variogram matrix.

k

NULL (default) or an integer between 1 and d. Indicates which matrix \(\Sigma\), or \(\Sigma^k\) should be produced.

full

Logical. If true, then the kth row and column in \(\Sigma^k\) are included and the function returns a \(d \times d\) matrix. By default, full = FALSE.

Details

Every \(d \times d\) Gamma matrix in the definition of a Huesler--Reiss distribution can be transformed into a \((d-1) \times (d-1)\) \(\Sigma^k\) matrix, for any k from 1 to d. The inverse of \(\Sigma^k\) contains the graph structure corresponding to the Huesler--Reiss distribution with parameter matrix Gamma. If full = TRUE, then \(\Sigma^k\) is returned as a \(d \times d\) matrix with additional kth row and column that contain zeros. For details see eng2019;textualgraphicalExtremes and hen2022;textualgraphicalExtremes.

References

See Also

Other MatrixTransformations: Gamma2Theta(), Gamma2graph(), Sigma2Gamma(), Theta2Gamma()

Examples

Gamma <- cbind(
  c(0, 1.5, 1.5, 2),
  c(1.5, 0, 2, 1.5),
  c(1.5, 2, 0, 1.5),
  c(2, 1.5, 1.5, 0)
)
Gamma2Sigma(Gamma, k = 1, full = FALSE)