gft: Generalized Fisher Transformation

In GFT: Generalized Fisher Transformation of Correlation Matrices

Generalized Fisher Transformation

Description

Computes the generalized Fisher transformation \gamma = \mathrm{vecl}(\log C): the below-diagonal elements, stacked column by column, of the matrix logarithm of a positive definite correlation matrix C.

Usage

gft(C)

Arguments

C	a positive definite correlation matrix (square, symmetric, numeric). The symmetric part (C + C')/2 is used.

Details

The matrix logarithm is computed from the eigendecomposition of C. For n = 2 the transformation reduces to Fisher's classical z-transformation z = \mathrm{atanh}(\rho). The map is a bijection between the positive definite correlation matrices and R^{n(n-1)/2}; its inverse is computed by inv_gft.

Value

A numeric vector of length n(n-1)/2 containing \mathrm{vecl}(\log C).

References

Archakov, I. and Hansen, P. R. (2021). A new parametrization of correlation matrices. Econometrica, 89(4), 1699–1715. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3982/ECTA16910")}

See Also

inv_gft, vecl, unvecl.

Examples

# n = 2: reduces to Fisher's z-transformation
C <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
all.equal(gft(C), atanh(0.5))

C <- 0.9^abs(outer(1:5, 1:5, "-"))
z <- gft(C)
max(abs(inv_gft(z)$C - C))

GFT documentation built on Aug. 23, 2026, 5:10 p.m.