PARAFAC2 - Part I. A direct fitting algorithm for the PARAFAC2 model

PARAFAC is a generalization of principal component analysis (PCA) to the si tuation where a set of data matrices is to be analysed. If each data matrix has the same row and column units, the resulting data are three-way data a nd can be modelled by the PARAFAC1 model. If each data matrix has the same column units but different (numbers of) row units, the PARAFAC2 model can b e used. Like the PARAFACI model, the PARAFAC2 model gives unique solutions under certain mild assumptions, whereas it is less severely constrained tha n PARAFAC 1. It may therefore also be used for regular three-way data in si tuations where the PARAFAC1 model is too restricted. Usually the PARAFAC2 m odel is fitted to a set of matrices with cross-products between the column units. However, this model-fitting procedure is computationally complex and inefficient. In the present paper a procedure for fitting the PARAFAC2 mod el directly to the set of data matrices is proposed. It is shown that this algorithm is more efficient than the indirect fitting algorithm. Moreover, it is more easily adjusted so as to allow for constraints on the parameter matrices, to handle missing data, as well as to handle generalizations to s ets of three- and higher-way data. Furthermore, with the direct fitting app roach we also gain information on the row units, in the form of 'factor sco res'. As will be shown, this elaboration of the model in no way limits the feasibility of the method. Even though full information on the row units be comes available, the algorithm is based on the usually much smaller cross-p roduct matrices only. Copyright (C) 1999 john Wiley & Sons, Ltd.