Generalized Fisher Transformation of Correlation Matrices

Forward and inverse generalized Fisher transformation (GFT) of correlation matrices, gamma = vecl(log C), which maps the positive definite correlation matrices one-to-one onto the Euclidean space of dimension n(n-1)/2, see Archakov and Hansen (2021) . The inverse is computed from a variational characterization by the GFT-FP+N algorithm: matrix-free inexact Newton steps for the log-diagonal residual, solved by preconditioned conjugate gradients, with fixed-point safeguards that guarantee global convergence. Sequential inversion with a tangent predictor, a certified quadrature preconditioner, and comparison solvers (the plain fixed point, Broyden's method, full Newton, Anderson acceleration, and limited-memory BFGS) are included. Uses base R only.

GFT (R package)

Forward and inverse generalized Fisher transformation (GFT) of correlation matrices in base R.

The GFT maps a non-singular n x n correlation matrix C to the unrestricted vector gamma = vecl(log C) in R^d, d = n(n-1)/2, and is a bijection onto R^d (Archakov and Hansen, 2021, Econometrica). The inverse is computed by GFT-FP+N (Archakov and Hansen, 2026, arXiv:2609.19028): matrix-free inexact Newton steps for the log-diagonal residual, solved by preconditioned conjugate gradients, with fixed-point safeguards that guarantee global convergence. Version 1.2.0 of the package matches release v1.2.0 of the Julia implementation.

library(GFT)
C <- 0.9^abs(outer(1:5, 1:5, "-"))
z <- gft(C)          # forward: correlation matrix -> R^10
r <- inv_gft(z)      # inverse: R^10 -> correlation matrix
max(abs(r$C - C))    # ~1e-15

Solvers: inv_gft (GFT-FP+N, recommended; with the optional certified quadrature preconditioner), inv_gft_path (sequential inversion with warm starts and the tangent predictor gft_predict), and the comparison solvers inv_gft_fp (fixed point), inv_gft_broyden (Chen, Fei and Yu, 2025), inv_gft_newton (full Newton), inv_gft_anderson (Anderson acceleration, plain or guarded), and inv_gft_lbfgs (limited-memory BFGS). All report eigendecomposition counts and convergence diagnostics.

Installation

# from CRAN (once accepted)
install.packages("GFT")

# development version
remotes::install_github("reinhardhansen/GFT", subdir = "r")

Correspondence with the Julia package

R/GFT.R is a line-faithful port of the Julia reference implementation (julia/src/GFT.jl, same repository). The test suite includes golden values generated by an independent NumPy implementation and shared with the Julia tests, so a passing run is a cross-language verification. dev/ contains a development harness (webR) and a script comparing solver iteration counts against the serialized draws used in the paper's supplement; it is excluded from the built package.

References

Archakov, I. and Hansen, P. R. (2021). A new parametrization of correlation matrices. Econometrica, 89(4), 1699-1715. doi:10.3982/ECTA16910
Archakov, I. and Hansen, P. R. (2026). A variational approach to the generalized Fisher transformation of correlation matrices. Working paper.
Chen, H., Fei, Y. and Yu, J. (2025). Multivariate stochastic volatility models based on generalized Fisher transformation. Journal of Econometrics, 251, 106041.