Parallel factor analysis

Parachlorobenzotrifluoride, 6 1341 Paracortex, in wool fibers, 22 173 Para-crystalline lattice model, 24 464 Paracyclophane synthesis, 24 38 PARAFAC (PARAllel FACtor analysis),... 

Multiway methods For analyzer data where a single sample generates a second order array (ex. GC/MS, LC/UV, excitation/emission fluorescence), multiway chemometric modehng methods, such as PARAFAC (parallel factor analysis) [121,122], can be used to exploit the second order advantage to perform effective calibration transfer and instrument standardization. 

Absorption spectra have also been used in the reexamination of pH-dependent color and structural transformations in aqueous solutions of some nonacylated anthocyanins and synthetic flavylium salts." ° In a recent study, the UV-Vis spectra of flower extracts of Hibiscus rosasinensis have been measured between 240 and 748 nm at pH values ranging from 1.1 to 13.0." Deconvolution of these spectra using the parallel factor analysis (PARAFAC) model permitted the study of anthocyanin systems without isolation and purification of the individual species (Figure 2.21). The model allowed identification of seven anthocyanin equilibrium forms, namely the flavylium cation, carbinol, quinoidal base, and E- and Z-chalcone and their ionized forms, as well as their relative concentrations as a function of pH. The spectral profiles recovered were in agreement with previous models of equilibrium forms reported in literature, based on studies of pure pigments. 

Olivieri, A.C. and Faber, N.M., Standard error of prediction in parallel factor analysis of three-way data, Chemom. Intell. Lab. Syst., 70, 75-82, 2004. 

Equation 11.17 is the fundamental expression of the PARAFAC (parallel factor analysis) model [77], which is used to describe the decomposition of trilinear data sets. For nontrilinear systems, the core C is no longer a regular cube (ncr x ncc x net), and the non-null elements are spread out in different manners, depending on each particular data set. The variables ncr, ncc, and net represent the rank in the row-wise, columnwise, and tubewise augmented data matrices, respectively. Each element in the original data set can now be obtained as shown in Equation 11.18 ... 

Three-way calibrations methods, such as the generalized rank annihilation method (GRAM) and parallel factor analysis (PARAFAC), are becoming increasingly prevalent tools to solve analytical challenges. The main advantage of three-way calibration is estimation of analyte concentrations in the presence of unknown, uncalibrated... 

PARAFAC (parallel factor analysis) differs from the Tucker3 models in diat each of the diree dimensions contains the same number of components. Hence die model can be represented as die sum of contributions due to g components, just as in normal PCA, as illustrated in Figure 4.41 and represented algebraically by... 

The parallel factor analysis (PARAFAC) model [18-20] is based on a multilinear model, and is one of several decomposition methods for a multidimensional data set. A major advantage of this model is that data can be uniquely decomposed into individual contributions. Because of this, the PARAFAC model has been widely applied to 3D and also higher dimensional data in the field of chemometrics. It is known that fluorescence data is one example that corresponds well with the PARAFAC model [21]. 

Shirakawa, H. and Miyazaki, H.S. (2004) Blind spectral decomposition of single-cell fluorescence by parallel factor analysis. Biophys.J., 86, 1739-1752. 

Harshman, R.A. and Lundy, M.E. (1994) PARAFAC Parallel factor analysis. Comp. Stat. Data Anal., 18, 39-72. 

Do T, McIntyre NS, Application of parallel factor analysis and X-ray photoelectron spectroscopy to the initial stages in oxidation of aluminium, Surface Science, 1999, 435, 136-141. 

Moreda-Pineiro A, Marcos A, Fisher A, Hill SJ, Parallel factor analysis for the study of systematic error in inductively coupled plasma atomic emission spectrometry and mass spectrometry, Journal of Analytical Atomic Spectrometry, 2001, 16, 360-369. 

Paatero P, The multilinear engine - a table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model, Journal of Computational and Graphical Statistics, 1999, 8, 854-888. 

Wise BM, Gallagher NB, Butler SW, White DD, Barna GG, A comparison of principal component analysis, multiway principal component analysis, trihnear decomposition and parallel factor analysis for fault detection in a semiconductor etch process, Journal of Chemometrics, 1999, 13,... 

Yongnian N, Shaojing S, Serge K. Spectrofluorimetric studies on the binding of salicylic acid to bovine serum albumin using warfarin and ibuprofen as site markers with the aid of parallel factor analysis. Anal Chim Acta. 2006 2 206-15. 

Ebrahimi, D., Kennedy, D. F., Messerle, B. A. Hibbert, D. B. (2008). High throughput screening arrays of rhodium and iridium complexes as catalysts for intramolecular hydroamination using parallel factor analysis. Analyst, Vol. 133,... 

Parallel factor analysis (PARAFAC) (Harshman, 1970 Bro, 1997 Amigo et al., 2010) is a technique that is ideally suited for interpreting multivariate separations data. PARAFAC is a decomposition model for multivariate data which provides three matrices. A, B and C which contain the scores and loadings for each component. The residuals, E, and the number of factors, r, are also extracted. The PARAFAC decomposition finds the best... 

Direct analysis of a three-way data array is feasible by parallel factor analysis (PAR AFAC) or by Tucker models. 

Parallel Factor Analysis The PARAFAC model for an element of the three-way arrayX(/ xJxK) in Figure 5.15 is as follows ... 

Figure 5.15 The parallel factor analysis (PARAFAC) model for F factors.

Methods for simultaneous Af-way regression can be based on the decomposition of the X array by multiway methods introduced in Section 5.2 (parallel factor analysis (PARAFAC) or Tucker models) and regressing the dependent variable on the components of those models. A drawback with this approach is that the separately estimated components are not necessarily predictive for Y. This caused the development of improved algorithms for multiway regression analysis of that kind. 

Guimet et al. used two potential multiway methods for the discrimination between virgin olive oils and pure olive oils the unfold principal component analysis (U-PCA) and parallel factor analysis (PARAFAC), for the exploratory analysis of these two types of oils. Both methods were applied to the excitation-emission fluorescence matrices (EEM) of olive oils and followed the comparison of the results with the ones obtained with multivariate principal component analysis (PCA) based on a fluorescence spectrum recorded at only one excitation wavelength. 

Guimet, R, Boque, R., and Ferre, J. 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. Analytica Chimica Acta, 515, 75-85. 2004. 

Mixtures of acetylsalicylic acid and ascorbic acid have been studied by using parallel factor analysis and partial least-squares. The former is used for spectral deconvolution, and pK estimation for both acids. The simultaneous determination of fosinopril and hydrochlorothiazide in pharmaceutical formulations consists of extracting both compounds in an aqueous solution, measuring by multiwavelength UV spectrophotometry hydrochlorothiazide acts as an internal standard to verify the accuracy of the analysis. 

Ohno, T., A. Amirbahman, and R. Bro. 2008. Parallel factor analysis of excitation-emission matrix fluorescence spectra of water soluble soil organic matter as basis for the determination of conditional metal binding parameters. Environmental Science Technology 42, no. 1 186-192. doi 10.1021/es071855f. 

Santfn, C., Y. Yamashita, X. Otero, M. Alvarez, and R. Jaffe. 2009. Characterizing humic substances from estuarine soils and sediments by excitation-emission matrix spectroscopy and parallel factor analysis. Biogeochemistry 96, no. 1 131-147. doi 10.1007/ S10533-009-9349-1. 

Stedmon, C. A., and R. Bro. 2008. Characterizing dissolved organic matter fluorescence with parallel factor analysis A tutorial. Limnology and Oceanography Methods 6, no. Nov 572-579. 

Such arrays raise the question of more generalizations of the table-oriented techniques presented in Chapters 3.9 to 3.11. The most prominent representatives of factorial methods are the so-called Tucker3 [21] and PARAFAC (parallel factor analysis) [22] models. For three-way arrays, the Tucker3 model is expressed as... 

Ghasemi, J., and Abbasi, B., 2005. Simultaneous spectrophotometric determination of group B vitamins using parallel factor analysis. Journal of the Chinese Chemical Society. 52 1123-1129. 

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