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

Abstract

PARAFAC is a generalization of principal component analysis (PCA) to the situation 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 and 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 be used. Like the PARAFAC1 model, the PARAFAC2 model gives unique solutions under certain mild assumptions, whereas it is less severely constrained than PARAFAC1. It may therefore also be used for regular three-way data in situations where the PARAFAC1 model is too restricted. Usually the PARAFAC2 model 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 model 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 sets of three- and higher-way data. Furthermore, with the direct fitting approach we also gain information on the row units, in the form of ‘factor scores’. 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 becomes available, the algorithm is based on the usually much smaller cross-product matrices only. Copyright © 1999 John Wiley & Sons, Ltd.