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Oblique Procrustes target rotation using a k x k inner objective

Source: R/RcppExports.R
dot-oblique_procrustes.Rd

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.

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.

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.

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().

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.