Pjf. Groenen et al., Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis, PSYCHOMETRI, 65(4), 2000, pp. 511-524
An important feature of distance-based principal components analysis, is th
at the variables can be optimally transformed. For monotone spline transfor
mation, a nonnegative least-squares problem with a length constraint has to
be solved in each iteration. As an alternative algorithm to Lawson and Han
son (1974), we propose the Alternating Length-Constrained Non-Negative Leas
t-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squar
es loss function over the parameters under a length constraint, by alternat
ingly minimizing over one parameter while keeping the others fixed. Several
properties of the new algorithm are discussed. A Monte Carlo study is pres
ented which shows that for most cases in distance-based principal component
s analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or
sometimes even better in terms of the quality of the solution.