• uniqueness;
  • rank overlap;
  • hard modeling constraint


PARAFAC model is the most famous model for analyzing three-way data. However, this method does not converge to chemically meaningful solutions when applied to three-way problems involving rank overlap profiles at least in one mode. Rank overlap can be simply found where components have similar spectral profiles or analytes appearing in identical proportions throughout an experiment. However, an appropriate selection of the initial parameters and constraints such as non-negativity and unimodality can still make PARAFAC model useful in this regard. Although such constraints reduce rotational freedom in PARAFAC solution, they are generally insufficient to wholly eliminate the rotational problem. The goal of the present paper is to incorporate hard modeling constraint in the soft-modeled PARAFAC algorithm to overcome non-uniqueness problem in the equilibrium processes involving linearly dependent factors at least in one mode. The hard constraint is introduced to force some or all of the concentration profiles to fulfill an equilibrium model that is refined at each iteration cycle of the optimization process of PARAFAC. The proposed approach is called hard–soft PARAFAC (HSPARAFAC). When the rank overlap species obeys equilibrium model in HSPARAFAC, the unique results are obtained even in the presence of non-modeled interferences. The new modification in the treatment of equilibrium data sets yields more satisfactory results than the exclusive PARAFAC algorithm. Simulated and real examples with rank overlap problem are used to confirm this statement. Copyright © 2011 John Wiley & Sons, Ltd.