Mathematical improvements to maximum likelihood parallel factor analysis: theory and simulations

Abstract

AbstractA number of simplified algorithms for carrying out maximum likelihood parallel factor analysis (MLPARAFAC) for three‐way data affected by different error structures are described. The MLPARAFAC method was introduced to establish the theoretical basis to treat heteroscedastic and/or correlated noise affecting trilinear data. Unfortunately, the large size of the error covariance matrix employed in the general formulation of this algorithm prevents its application to solve standard three‐way problems. The algorithms developed here are based on the principle of alternating least squares, but differ from the generalized MLPARAFAC algorithm in that they do not use equivalent alternatives of the objective function to estimate the loadings for the different modes. Instead, these simplified algorithms tackle the loss of symmetry of the PARAFAC model by using only one representation of the objective function to estimate the loadings of all of the modes. In addition, a compression step is introduced to allow the use of the generalized algorithm. Simulation studies carried out under a variety of measurement error conditions were used for statistical validation of the maximum likelihood properties of the algorithms and to assess the quality of the results and computation time. The simplified MLPARAFAC methods are also shown to produce more accurate results than PARAFAC under a variety of conditions. Copyright © 2005 John Wiley & Sons, Ltd.