PARAFAC models

The PARAFAC model can also be defined by means of an extended matrix notation ... 

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

FIGURE 3.22 PARAFAC model for three-way data X. R components are used to approximate A by a trilinear model as defined in Equation 3.35. 

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. 

FIGURE 2.21 Spectral profiles recovered by the PARAFAC model at different pH values based on the deconvolution of UV-Vis absorption spectra, featuring the various anthocyanin secondary monomeric forms. (Reprinted from Levi, M.A.B. et al., Talanta, 62, 299, 2004. Copyright 2004 Elsevier Science B.V. With permission.)... 

FIGURE 12.1 Construction and decomposition of a three-way array via the trilinear PARAFAC model. 

With the exception of the Tucker3 model, the models discussed here are intrinsically linear models, and a straightforward application thus assumes linear interactions and behavior of the samples. While many of the systems of interest to chemists contain nonlinearities that violate the assumptions of the models, the PARAFAC model forms an excellent starting point from which many subsidiary methods are... 

Alternating least squares (ALS) methods are both slower, due to their numeric intensity, and more flexible than eigenvalue-eigenvector problem-based methods for solving Equation 12.1a and Equation 12.1b. The basic PARAFAC model of Equation... 

Although the PARAFAC model is a trilinear model that assumes linear additivity of effects between species, the model can be successfully employed when there is a nonlinear dependence between analyte concentration and signal intensity. Provided that the spectral profiles in the X- and Y-ways are not concentration dependent, the resolved Z-way profiles will be a nonlinear function of analyte... 

The PARAFAC model is often applicable for calibration when a finite number of factors cannot fully model the data set. In these traditionally termed nonbilinear applications, the additional terms in the PARAFAC model successively approximate the variance in the data set. This approximation is analogous to employing additional factors in a PLS or PCR model [5], Nonbilinear rank annihilation (NBRA) exploits the property that, in many cases when the PARAFAC model is applied to a set consisting of a pure analyte spectrum and mixture spectrum, some factors will be unique to the analyte, some will be unique to the interferent, and some factors will describe both analyte and interferent information [40], Accurate calibration and prediction can be accomplished with the factors that are unique to the analyte. If these factors can be found by mathematically multiplying the pure spectrum by a, then the estimated relative concentrations that decrease by 1/a are unique to the analyte [41], In Reference [41] the necessary conditions required to enable accurate prediction with nonbilinear data are discussed. 

Visual inspection of the estimated factors is not to be trusted in the presence of degenerate factors, which occur when two or more factors are collinear in one or more of the three ways. When this is the case, in the concentration way or Z-way, the PARAFAC model is still valid, but the rotational uniqueness of the X-way and Y-way protiles of the degenerate factors is lost. This often results in estimated protiles that are hard to interpret. If the collinearity occurs in the X-way or Y-way, the PARAFAC model may not be appropriate, and the constrained Tucker3 model should be used instead. Collinearity in the X-way or Y-way can be checked by successively performing PCA on data unfolded to an / x (J K) matrix, and then to a. / x (l K) matrix. If there are no collinearities in the X- or Y-ways, the optimal number of factors determined by both unfoldings will be the same. 

Harchman, R.A. and Lundy, M.E., The PARAFAC model for three-way factor analysis and multidimensional scaling, in Research Methods for Multimode Data Analysis, Law, H.G. et al., Eds., Praeger, New York, 1984. 

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]. 

In this study, we propose an approach based on unique optical configuration, efficient acquisition of a multidimensional data set, and decomposition of unknown fluorescent components by using the PARAFAC model. Further, we demonstrate that our approach is powerful and effective enough to track complicated responses in living cells by analyzing the autofluorescence of native molecules. 

We obtained y-Em maps by scanning the x-position (1 tm step, 640 positions in total). Typically, two frames of different delay times (x = 0.0 and 3.0 ns) were obtained at each x-position. As a result, a fluorescence data set consisted of 640 (x) x 480 (y) x 640 (Em) x 2 (x). For the PARAFAC calculations, this data set was binned with 25-nm steps along the fluorescence wavelength dimension to reduce data size. In addition, the spatial dimensions of 640 (x) x 480 (y) were reshaped to the one-dimensional array (size of 307 200), and then reshaped again to the spatial dimensions of 640 (x) x 480 (y) after calculations. Therefore, the data set, which consisted of 10 (Em) x 2 (x) x 307 200 (xy), was fitted by the PARAFAC model. 

There are several methods for decomposing the 3 D data set X with I xj x K. The two major methods are PARAFAC and Tucker3. Since the PARAFAC model can be considered a constrained version ofthe Tucker3 model, we first describe the Tucker3 model and then give a description of the PARAFAC model. 

The 3 D PARAFAC model with F components is depicted in Figure 32.4 and can be formulated as follows ... 

The number of components F was determined by the core consistency diagnostic, as proposed by Bro and Kiers [29]. This is based on evaluating the appropriateness of the PARAFAC model by comparing the core arrays of the Tucker3 and PARAFAC models, because the PARAFAC model is a constrained version of the Tucker3 model. Therefore, the core consistency can be defined as... 

In PARAFAC modeling, non-negativity constraints were applied to all three dimensions. All analyses were performed with the N-way toolbox for MATLAB [30], which is a set of MATLAB routines designed to perform multi-way data analysis. 

First, we examined a number of fluorescent components from the 3D data for the mixture. Based on the formulation given by Eq. (32.4), we calculated the consistency of the PARAFAC model with various numbers of components. The consistency for the 1- and 2-component models was almost 100% in both cases, while that for the 3-component model decreased to 70% (Figure 32.8). In 4- or more component models, the consistencies were almost 0%, indicating that the PARAFAC model was no longer adequate in these cases. Therefore, we determined that 3 components was an appropriate number. 

PARAFAC model, and then obtained the vectors of ojwith elements of a,y(l = 1. 

Figure 32.8 The consistency (%) calculated for the PARAFAC model with different numbers of components, using Eq. (32.4).

Figure 32.9 The three components extracted from the 3D fluorescence data ofthe mixture in Figure 32.7 using the PARAFAC model. The upper panels are Ex-Em maps constructed by Oyig> bf, while the bottom panels are x-Em maps constructed by Cj-<2> bj ...

Analysis and decomposition of a multidimensional data set by using the PARAFAC model. 

Different schemes have been developed to allow the validation of three-way models. One method used for PARAFAC models is split-half analysis, where the data set is split into two parts and individual modelling is performed on the two halves. If the two models show similar spectral loadings, the variation expressed in the two halves is comparable and it can be assumed that an appropriate number of components have been chosen. As for PCA, visual interpretability is also important and usually substantially easier for PARAFAC, because the components directly represent chemically meaningful phenomena. Other tools that may aid in deciding the correct number of components can be found in the literature.15... 

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