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.

Second- and higher-order data generation and calibration: A tutorial

“…PARAFAC2 is a variant of PARAFAC which allows for some profile changes in one 286 of the data modes [32,33]. It includes a mathematical constraint on the profiles in the 287 latter mode (namely that the cross-products of component profiles should be constant 288 across samples), but lesser chemically natural constraints in comparison with MCR-289 ALS (non-negativity, unimodality).…”

“…It includes a mathematical constraint on the profiles in the 287 latter mode (namely that the cross-products of component profiles should be constant 288 across samples), but lesser chemically natural constraints in comparison with MCR-289 ALS (non-negativity, unimodality). It is thus limited to similar changes in peak 290 positions for various constituents in different samples [25,32,34,35]. 291…”

Ad-hoc blocked design for the robustness study in the determination of dichlobenil and 2,6-dichlorobenzamide in onions by programmed temperature vaporization-gas chromatography–mass spectrometry

“…The computed design is coupled to a Parallel Factor Analysis 2 (PARAFAC2) [20,21], a 112 multiway technique which has proved to be very useful in solving common problems in GC-113 MS [22,23]. It is particularly helpful for determining compounds of interest in food 114 commodities [19,24], for solving problems as small retention time shifts, severe interferences 115 caused by unexpected derivatization artifacts or by co-eluents of the complex matrix which 116 share m/z ratios with the target compounds.…”

Speaker Adaptation Based on PARAFAC2 of Transformation Matrices for Continuous Speech Recognition

“…A representative algorithm for finding the components that minimize the squared error is the alternating least squares (ALS) [6]. The basic idea is to minimize Eq.…”

“…In the eigenvoice adaptation, training acoustic models are decomposed by PCA to obtain bases and the model for a new speaker is represented as a linearly weighted sum of bases. Our approach is closely related to the EMLLR adaptation, but bases are built from the parallel factor analysis 2 (PARAFAC2) [5], [6] of MLLR transformation matrices of training speakers. In our approach, the transformation matrix for a new speaker is expressed as a product of bases and a weight matrix.…”