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 © 2008 John Wiley & Sons, Ltd.     

Application of unfold principal component analysis and parallel factor analysis to the exploratory analysis of olive oils by means of excitation-emission matrix fluorescence spectroscopy

Discrimination between virgin olive oils and pure olive oils is of primary importance for controlling adulterations. Here, we show the potential usefulness of two multiway methods, unfold principal component analysis (U-PCA) and parallel factor analysis (PARAFAC), for the exploratory analysis of the two types of oils. We applied both methods to the excitation-emission fluorescence matrices (EEM) of olive oils and then compared the results with the ones obtained by multivariate principal component analysis (PCA) based on a fluorescence spectrum recorded at only one excitation wavelength. For U-PCA and PARAFAC, the ranges studied were λ_ex=300-400 nm, λ_em=400-695 nm and λ_ex=300-400 nm, λ_em=400-600 nm. The first range contained chlorophylls, whose peak was much more intense than those of the rest of species. The second range did not contain the chlorophylls peak but only the fluorescence spectra of the remaining compounds (oxidation products and Vitamin E). The three-component PARAFAC model on the second range was found to be the most interpretable. With this model, we could distinguish well between the two groups of oils and we could find the underlying fluorescent spectra of three families of compounds.     

Usefulness of parallel factor analysis to handle the matrix effect in the fluorescence determination of tetracycline in whey milk

The determination of tetracycline by fluorescence spectrophotometry in complex matrices has some difficulties, because the presence of other compounds in the matrix affects the analytical signal. In this work, the effect of some inorganic species that are present in whey milk on the fluorescence signal of tetracycline is studied using a D-optimal experimental design. Next, an experimental strategy is proposed in conjunction with Parallel Factor Analysis, PARAFAC, modeling that leads to suitably modeling the severe matrix effect in the determination of tetracycline in whey milk. A specific design is performed in such a way that the lack of trilinearity due to the effect of the presence of interferents on the signal is obviated. Then, ten test samples from three brands of milk, spiked with different quantities of tetracycline and measured in 2 days were analysed using this methodology (mean of the absolute value of the relative errors: 5.1%). The developed analytical method fulfils the property of trueness, the relative errors being, both in calibration and prediction, inside the interval set by Commission Decision 2002/657/EC at these concentration levels. Decision limits (CCα) at x₀ = 0 μg L⁻¹ and at x₀ = 100 μg L⁻¹ were 13.2 and 112.4 μg L⁻¹ respectively, for α = 0.05; whereas detection capabilities (CCβ) were 25.9 μg L⁻¹ and 124.4 μg L⁻¹ respectively for α = β = 0.05.     

Application of parallel factor analysis to total synchronous fluorescence spectrum of dilute multifluorophoric solutions: Addressing the issue of lack of trilinearity in total synchronous fluorescence data set

In recent years, total synchronous fluorescence (TSF) spectroscopy has become popular for the analysis of multifluorophoric systems. Application of PARAFAC, a popular deconvolution tool, requires trilinear structure in the three-way data array. The present work shows that TSF based three-way array data set of dimension sample × wavelength × Δλ does not have trilinear structure and hence it should not be subjected to PARAFAC analysis. This work also proposes that a TSF data set can be converted to an excitation–emission matrix fluorescence (EEMF) like data set which has trilinear structure, so that PARAFAC analysis can be performed on it. This also enables the retrieval of PARAFAC-separated component TSF spectra.     

CRAFT (complete reduction to amplitude frequency table) – robust and time‐efficient Bayesian approach for quantitative mixture analysis by NMR

The intrinsic quantitative nature of NMR is increasingly exploited in areas ranging from complex mixture analysis (as in metabolomics and reaction monitoring) to quality assurance/control. Complex NMR spectra are more common than not, and therefore, extraction of quantitative information generally involves significant prior knowledge and/or operator interaction to characterize resonances of interest. Moreover, in most NMR‐based metabolomic experiments, the signals from metabolites are normally present as a mixture of overlapping resonances, making quantification difficult. Time‐domain Bayesian approaches have been reported to be better than conventional frequency‐domain analysis at identifying subtle changes in signal amplitude. We discuss an approach that exploits Bayesian analysis to achieve a complete reduction to amplitude frequency table (CRAFT) in an automated and time‐efficient fashion – thus converting the time‐domain FID to a frequency‐amplitude table. CRAFT uses a two‐step approach to FID analysis. First, the FID is digitally filtered and downsampled to several sub FIDs, and secondly, these sub FIDs are then modeled as sums of decaying sinusoids using the Bayesian approach. CRAFT tables can be used for further data mining of quantitative information using fingerprint chemical shifts of compounds of interest and/or statistical analysis of modulation of chemical quantity in a biological study (metabolomics) or process study (reaction monitoring) or quality assurance/control. The basic principles behind this approach as well as results to evaluate the effectiveness of this approach in mixture analysis are presented. Copyright © 2013 John Wiley & Sons, Ltd.