Configuration: To Configuration (varimax)...

A command that rotates the selected Configuration object to a new Configuration object whose coordinates have maximum squared variance.

Settings

Normalize rows
when selected, the distances of all points to the origin will be made equal before iteration starts. We remember these scale factors and restore the original distances after the iteration process has stopped.
Quartimax
when selected, the sum of fourth powers, normalized or raw, will be maximized.
Maximum number of iterations
sets a limit to the number of iterations. One iteration consists of numberOfDimensions· (numberOfDimensions–1)/2 planar rotations of all pairs of dimensions.
Tolerance
also determines when the iteration stops. This happens if |vivi+1| < tolerance · vi, where vi is the squared variance for the ith iteration.

The iteration process stops when either the maximum number of iterations is reached or the tolerance criterion is met, which ever one is first.

Algorithm

The Varimax rotation procedure was first proposed by Kaiser (1958). Given a numberOfPoints × numberOfDimensions configuration A, the procedure tries to find an orthonormal rotation matrix T such that the sum of variances of the columns of B*B is a maximum, where B = AT and * is the element wise (Hadamard) product of matrices. A direct solution for the optimal T is not available, except for the case when numberOfDimensions equals two. Kaiser suggested an iterative algorithm based on planar rotations, i.e., alternate rotations of all pairs of columns of A.

However, this procedure is not without problems: the varimax function may have stationary points that are not even local maxima. We have incorporated an algorithm of Ten Berge (1995) that prevents this unpleasant situation from happening.


© djmw, April 7, 2004