A New Formulation of the Nonmetric Strain Problem in Multidimensional Scaling

Abstract

A natural extension of classical metric multidimensional scaling is proposed. The result is a new formulation of nonmetric multidimensional scaling in which the strain criterion is minimized subject to order constraints on the disparity variables. Innovative features of the new formulation include: the parametrization of the p-dimensional distance matrices by the positive semidefinite matrices of rank ≤p; optimization of the (squared) disparity variables, rather than the configuration coordinate variables; and a new nondegeneracy constraint, which restricts the set of (squared) disparities rather than the set of distances. Solutions are obtained using an easily implemented gradient projection method for numerical optimization. The method is applied to two published data sets.