A solution to the weighted procrustes problem in which the transformation is in agreement with the loss function

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45737/1/11336_2005_Article_BF02296976.pd

Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm

In connection with a least-squares solution for fitting one matrix, A , to another, B , under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure, e , and a refined similarity measure, e s , which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix, B , e.g. , e ( A , kB ) ≠ e ( A , B ) for k ≠ 1. Therefore, both measures are useless for answering questions of the type: “Does A fit B better than A fits C ?”. In this note two new measures of fit are suggested which do not depend upon the norms of A and B , which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45731/1/11336_2005_Article_BF02291666.pd