Uniqueness Proof for a Family of Models Sharing Features of Tucker's Three-Mode Factor Analysis and PARAFAC/Candecomp

Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, X k =A A D k H B D k B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where A D k = B D k ). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.