Evaluation of variation matrix arrays by parallel factor analysis

PARAFAC is one of the most widely used algorithms for trilinear decomposition. The uniqueness properties of the PARAFAC model are very attractive regardless of whether one is interested curve resolution or not. The fact that PARAFAC provides one unique solution simplifies interpretation of the model. But in three-way data arrays the uniqueness condition can only be expected when k(A) + k(B) + k(C) >= 2F + 2, where F is the number of components and k's are the Kruskal ranks of loadings A to C. As much as second order instruments produce data of varying complexity depending upon the nature of the analytical techniques being combined, with some three-way data it is possible for patterns generated by the underlying sources of variation to have sufficient independent effects in two modes, yet nonetheless be proportional in a third mode. For example, in three-way data for spectrophotometric titrations of weak acids or bases (pH-wavelength-sample), a rank deficiency may occur in two modes, that is closure rank deficiency in the pH mode and proportionality rank deficiency in the sample direction because each analyte will have acidic and basic forms that are linear combinations in the sample mode. The goal of the present paper is to overcome the non-uniqueness problem in the second order calibration of monoprotic acids mixtures. The solution contains two steps: first each pH-absorbance matrix is pretreated by subtraction of the first spectrum from each spectrum in the data matrix. This pretreated data matrix is called the variation matrix. Second, by stacking the variation matrices, a three-way trilinear variation data array will be obtained without the proportional linear dependency problem that can be resolved uniquely by PARAFAC. It is shown, although unique results are not guaranteed by the Kruscal's condition for the original three-way data, this condition is fulfilled for pretreated three-way data. Hence, the variation array may be uniquely decomposed by the PARAFAC algorithm. Studies on simulated as well as real data array reveal the applicability of the proposed method to this kind of problem in the second order calibration of monoprotic acids. Copyright (C) 2008 John Wiley & Sons, Ltd.