Parallel factor analysis model

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 ... 

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. 

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

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. 

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]. 

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. 

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... 

Garcfa-Reiriz A, Damiani PC, OUvieri AC, Canada-Canada F, Munoz de la Pena A. Nonlinear four-way kinetic-excitation-emission fluorescence data processed by a variant of parallel factor analysis and by a neural network model achieving the second-order advantage malo-naldehyde determination in oUve oil samples. Anal Chem 2008 80 7248-56. 

Hoggard JC, Synovee RE. Parallel factor analysis (PARAFAC) of target analytes in GC x GC-TOFMS data autranated selectirm of a model with an appropriate number of factors. Anal Chem 2007 79 1611-9. 

The earliest peak nomenclature, and the one still most widely used is that of Coble et al. (1990), which denotes two peaks for humic-like fluorescence, peaks A and C, and one for tyrosine-like fluorescence, peak B. Coble (1996) introduced peaks T (tryptophan-like) and M (marine humic-like). A similar naming scheme was proposed by Parlanti et al. (2000). Since the introduction and expanding use of the multicomponent analysis technique parallel factor analysis (PARAFAC Bro, 1997 Stedmon et al 2003), peak nomenclature has evolved into a numbering scheme based on the output of the model. PARAFAC models have now been developed for diverse environments, both freshwater and marine, and the outputs have resulted in an ever-increasing number of peak designators. 

Another well-known approach for multiway data analysis is the parallel factor (PARAFAC) analysis model. For a three-way array, the PARAFAC model is... 

Following the idea of using concentration windows and the subspaces that can be derived, other noniterative methods are focused on the recovery of the response profiles (spectra). This is the case of subwindow factor analysis (SFA), proposed by Manne [38], and other derivations of this method, like parallel vector analysis (PVA) [39], Unlike WFA, SFA recovers the pure response profile of each component. The individual row response profiles are appended in a columnwise fashion, until the complete ST matrix is built. The C matrix is easily derived by least-squares according to the CR model, D = CST, as follows ... 

Jiang, J.-H., Sasic, S., Yu, R.-Q. Ozaki, Y. (2003). Resolution of two-way data from spectroscopic monitoring of reaction or process systems by parallel vector analysis (PVA) and window factor analysis (WFA) inspection of the effect of mass balance, methods and simulations. Journal of Chemometrics, Vol. 17, No. 3, pp. 186-197 Jiang, J.-H., Liang, Y. Ozaki, Y. (2004). Principles and methodologies in self-modeling curve resolution. Chemometrics and Intelligent Laboratory Systems, Vol. 71, No. 1, pp. 1-12... 

Typical nonidealities such as polydispersity in filler size and conductivity, filler waviness and entanglements, and impurities impact the measured electrical properties of polymer nanocomposites. Most analytical and simulation studies of these nonidealities have been conducted for highly simplified systems, so that the extent to which these factors can modify composite properties, particularly within the context of more dominant factors such as filler dispersion and network stmcture, is unclear. To clarify the importance of these effects, theoretical analysis or modeling of more complex systems is required. Conducting parallel experiments in model systems can enhance the efficacy of such studies. 

There are patterns of dose-response curves that preclude Schild analysis. The model of simple competitive antagonism predicts parallel shifts of agonist dose-response curves with no diminution of maxima. If this is not observed it could be because the antagonism is not of the competitive type or some other factor is obscuring the competitive nature of the antagonism. The shapes of dose-response curves can prevent measurement of response-independent... 

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