Rotate a loading matrix orthogonally under the Crawford-Ferguson criterion using a gradient-projection optimizer along the orthogonal (Stiefel) manifold.
.rotate_cf_orth(
L,
kappa,
eps = 1e-05,
normalize = TRUE,
random_starts = 0L,
maxit = 1000L,
max_line_search = 10L,
step0 = 1,
screen_keep = 5L,
triage_maxit = 25L,
triage_improve_tol = 0
)A named list with the rotated loadings, the orthogonal rotation matrix Th
(with L %*% Th reproducing the rotated loadings), the attained criterion value, and
the convergence and validity flags. The list additionally reports the
criterion value reached at each optimized start in all_values, with a per-start convergence flag in all_converged.
Numeric matrix. The unrotated loading matrix (variables by factors).
Numeric scalar in [0, 1]. The Crawford-Ferguson parameter.
Numeric scalar. Convergence tolerance for the projected-gradient norm.
Logical scalar. If TRUE, apply Kaiser normalization before
rotation and reverse it afterwards.
Integer scalar. Number of additional random orthogonal starts.
Integer scalar. Maximum number of projected-gradient updates.
Integer scalar. Maximum number of step-halving attempts after the initial trial step in each line-search phase.
Numeric scalar. Initial step size used in the projected-gradient update.
Integer scalar. Number of screened random starts retained for triage optimization.
Integer scalar. Number of short optimization iterations used in the triage stage.
Numeric scalar. Relative improvement required for a triaged start to be promoted to full optimization.
The criterion value f and its gradient dQ/dL at the rotated loadings
L = A %*% T define the search; the engine maps the gradient to the orthogonal
transformation T, projects it onto the tangent space, performs a
non-monotone line search, and retracts back onto the orthogonal group via a
polar (singular value) projection. kappa = 0 is the quartimax criterion and
kappa = ncol(A) / (2 * nrow(A)) is the equamax criterion.
Additional random orthogonal starts may be requested. To bound runtime the solver
screens each random start by its objective, runs a short triage optimization on the
best-screened starts, and fully optimizes only those that improve on the current
incumbent by at least triage_improve_tol.
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.
Crawford, C. B., & Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35, 321-332.