dot-oblique_procrustes: Oblique Procrustes target rotation using a k x k inner...

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Oblique Procrustes target rotation using a k x k inner objective

Description

Compute an oblique target rotation for a loading matrix using a targetQ-compatible parameterization and a ⁠k x k⁠ objective.

Usage

.oblique_procrustes(
  A,
  B,
  S_r = NULL,
  T_init_r = NULL,
  eps = 1e-05,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1,
  normalize = FALSE,
  random_starts = 0L,
  screen_keep = 2L,
  triage_maxit = 25L,
  triage_improve_tol = 0
)

Arguments

A	Numeric matrix. Loading matrix to be rotated.
B	Numeric matrix. Target loading matrix with the same dimensions as A.
S_r	Optional numeric ⁠k x k⁠ matrix containing crossprod(A). Supplying this is useful when the same A is rotated repeatedly. Ignored when normalize = TRUE because the normalized cross-product is different.
T_init_r	Optional numeric ⁠k x k⁠ starting transformation matrix. If NULL, the identity matrix is used for the primary start.
eps	Numeric scalar. Convergence tolerance for the projected-gradient norm.
maxit	Integer scalar. Maximum number of full projected-gradient updates.
max_line_search	Integer scalar. Maximum number of step-halving attempts after the initial trial step in each line-search phase.
step0	Numeric scalar. Initial step size used in the projected-gradient update.
normalize	Logical scalar. If TRUE, apply Kaiser normalization to the loadings (only) before rotation and reverse it afterwards; the target is left unnormalized, matching GPArotation::targetQ(normalize = TRUE).
random_starts	Integer scalar. Number of additional random starts.
screen_keep	Integer scalar. Number of screened random starts retained for triage optimization.
triage_maxit	Integer scalar. Number of short optimization iterations used in the triage stage.
triage_improve_tol	Numeric scalar. Relative improvement required for a triaged start to be promoted to full optimization.

Details

The rotated loading matrix is defined as L = A %*% solve(t(T)), and the corresponding factor correlation matrix is Phi = t(T) %*% T. The optimization is carried out over the transformation matrix T under the oblique normalization constraint diag(t(T) %*% T) = 1.

Non-invertible candidate transformations are rejected rather than evaluated through a pseudo-inverse.

Additional random starts may be requested. To reduce runtime, the solver uses a two-stage strategy for extra starts: cheap objective screening, followed by short triage optimization, followed by full optimization only for starts that improve on the current incumbent by at least triage_improve_tol.

The routine is intended for repeated oblique target rotations in workflows such as bootstrap alignment or consensus alignment of exploratory factor solutions across multiply imputed datasets. It follows the same oblique transformation convention as GPArotation::targetQ().

Value

A named list containing the rotated loadings, transformation matrix, factor correlation matrix, target criterion value, convergence diagnostics, line-search diagnostics, and multi-start summaries.

References

Bernaards, C. A., & Jennrich, R. I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65, 676-696.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

EFAtools documentation built on Aug. 21, 2026, 5:16 p.m.