Evaluation of variation matrix arrays by parallel factor analysis

Abstract

a 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 R k B R k C ‡ 2F R 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.

A second-order standard addition method based on the data treatment by calculation of variation matrix of kinetic systems analyzed by MCR-ALS

“…The involved independent reactions are as follows: (1) The extent of the reaction describes the progress of a chemical reaction, that is equal to the number of chemical transformations, defined as equation 2: [22][23][24] ( 2) where Dc j is the concentration variation vector whose elements represent the variation of component j between a given moment and a specified zero-point. e is the extent of the reaction vector whose elements designate the reaction extent at these points.…”

“…[22][23][24]31 From equation 7, it can be concluded that, for a closed system containing K independent reactions or processes (e.g., complexation reaction of a mixture of K metal cations with a ligand) and given that the ligand and all complexes involved are spectrophotometrically active, the pseudo-rank of the measurement matrix D will be ≤ K + 1, S where S is the number of absorbing species participating in the reactions. 21 Whereas the pseudo-rank of variation matrix V is ≤ min (K, S).…”

“…22,23 In this approach, variation matrix (difference spectra) is used instead of the original data matrix, which is obtained by subtracting the zero-point spectrum from each spectrum at each measurement point. [24][25][26] Variation matrix can be decomposed to reaction extent vectors and reaction spectra. Reaction extent vector designates the net variation in concentration between the i th point and a specified zero-point.…”

Hard–soft modeling parallel factor analysis to solve equilibrium processes

“…Rank overlap can be simply found where components have similar spectral profiles or analytes appearing in identical proportions throughout an experiment. It would be possible to employ the PARAFAC model for these types of data, provided that the initial conditions and restrictions are set [3,[16][17][18]. The non-negativity, unimodality and orthogonality constraints are optional ones in the PARAFAC routine belonging to N-way toolbox of Bro and Andresson that is free available from their website [19].…”

A novel second-order standard addition analytical method based on data processing with multidimensional partial least-squares and residual bilinearization

“…In certain cases, standard addition PARAFAC could not be employed because of serious profile overlapping in one of the data dimensions, in which case multivariate curve resolutionalternating least-squares (MCR-ALS) [14] was successfully applied [11]. It should be noticed that linear dependency is a general phenomenon, which is not only present in standard addition data, but also in pH-gradient [14] or kinetic-modulated spectral experiments [15,16].…”