GPADF: Rotation Optimization

Description

Derivative free gradient projection rotation optimization routine used by various rotation objective.

Usage

GPForth.df(A, Tmat=diag(ncol(A)), normalize = FALSE, eps=1e-5, 
       maxit=1000, method="varimax", methodArgs=NULL)
    GPFoblq.df(A, Tmat=diag(ncol(A)), normalize = FALSE, eps=1e-5, 
       maxit=1000, method="quartimin", methodArgs=NULL)

Value

A GPArotation object which is a list with elements

loadings: The rotated loadings, one column for each factor. If randomStarts were requested then this is the rotated loadings matrix with the lowest criterion value.
Th: The rotation matrix, loadings %*% t(Th) = A.
Table: A matrix recording the iterations of the rotation optimization.
method: A string indicating the rotation objective function.
orthogonal: A logical indicating if the rotation is orthogonal.
convergence: A logical indicating if convergence was obtained.
Phi: t(Th) %*% Th. The covariance matrix of the rotated factors. This will be the identity matrix for orthogonal rotations so is omitted (NULL) for the result from GPForth.df.
G: The gradient of the objective function at the rotated loadings.

Arguments

A: initial factor loadings matrix for which the rotation criterian is to be optimized.
Tmat: initial rotation matrix.
normalize: see details.
eps: convergence is assumed when the norm of the gradient is smaller than eps.
maxit: maximum number of iterations allowed in the main loop.
method: rotation objective criterian.
methodArgs: a list ofmethodArgs arguments passed to the rotation objective

Author

Coen A. Bernaards and Robert I. Jennrich with some R modifications by Paul Gilbert.

Details

Derivative free gradient projection rotation optimization routines can be used to rotate a loadings matrix. The rotation criteria in the GPArotation package require a derivative to operate. In certain cases, the derivative is complex or non-existent. The derivative free gradient projection method provides a numerical alternative to the GPArotation package. The functions in the package GPArotateDF follow most of the functionality and logic as in the GPArotation package. Please consult the documentation in GPArotation for further details.

The argument method can be used to specify a string indicating the rotation objective. GPFoblq defaults to "quartimin" and GPForth defaults to "varimax". Available rotation objective functions include "ff.bentler", "ff.cf", "ff.cubimax", "ff.entropy", "ff.fss", "ff.geomin", "ff.infomax", "ff.oblimax", "ff.pst", "ff.quartimax","ff.quartimin", "ff.simplimax", "ff.target", and "ff.varimax". Most of the rotation criteria are avaible in the GPArotation pacakage except for cubimax and Forced Simple Structure.

The rotation criteria are in the functions prefixed by "ff." that are used in the actual function call. The ff.* function call would typically not be used directly, but are needed for rotation. Since these are illustrative of computation, these are all exported from the package namespace. New criteria for use with derivative free GP rotation do require a function of the type ff.newCriterionName that provides value for complexity f, and name of method.

Some rotation criteria (including "simplimax", "pst", "target", "cf", "fss") require one or more additional arguments. Check GPArotation documentation for details or see ff.fss.

The argument normalize gives an indication of if and how any normalization should be done before rotation, and then undone after rotation. If normalize is FALSE (the default) no normalization is done. If normalize is TRUE then Kaiser normalization is done. (So squared row entries of normalized A sum to 1.0. This is sometimes called Horst normalization.) If normalize is a vector of length equal to the number of indicators (= number of rows of A) then the colums are divided by normalize before rotation and multiplied by normalize after rotation. If normalize is a function then it should take A as an argument and return a vector which is used like the vector above.

References

Jennrich, R.I. (2004). Derivative free gradient projection algorithms for rotation. Psychometrika, 69, 475--480.

Bernaards, C.A. and Jennrich, R.I. (2005) Gradient Projection Algorithms and Software for Arbitrary Rotation Criteria in Factor Analysis. Educational and Psychological Measurement, 65, 676--696.

Examples

Run this code

  # GPRSorth and rotation name 
  data("Harman", package = "GPArotation")
  GPForth.df(Harman8, method = "quartimax")
  GPForth.df(Harman8, method = "cubimax")
  GPForth.df(Harman8, method = "varimax")
  GPFoblq.df(Harman8, method = "quartimin")

  # displaying results of factor analysis rotation output
  origdigits <- options("digits")
  Abor.unrotated <- factanal(factors = 2, covmat = ability.cov, rotation = "none")
  Abor <- GPFoblq.df(loadings(Abor.unrotated), method = "quartimin")
  Abor
  print(Abor)
  print(Abor, Table = TRUE)
  print(Abor, digits = 2)
  summary(Abor)
  options(digits = origdigits$digits)

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