Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis

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.