The PARAFAC model

The decomposition of a three-way table X can also be defined by means of the PARAF AC model in terms of three two-way loading matrices (one for each mode) A, B, C  

In this case, the nxpxq three-way table X is decomposed into the nxr, pxr, qxr loading matrices A, B, C for the row-, column- and layer-items of X. 

In the PARAF AC model, the three loading matrices A, B, C are not necessarily orthogonal [56]. The solution of the PARAF AC model, however, is unique and does not suffer from the indeterminacy that arises in principal components and factor analysis. 

In contrast to the Tucker3 model, described above, the number of factors in each mode is identical. It is chosen to be much smaller than the original dimensions of the data table in order to achieve a considerable reduction of the data. The elements of the loading matrices A, B, C are computed such as to minimize the sum of squared residuals. 

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

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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 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.)... 

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... 

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 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... 

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. 

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. 

Fig. 7. (A) The spectral loadings calculated by PARAFAC and (B) diffusion attenuation profiles resulting from the PARAFAC model.

Direct trilinear decomposition (DTLD) is a direct non-iterative and therefore fast algorithm for solving the PARAFAC model providing a non-LS solution. If data are low-rank trilinear, the solution produced by DTLD will be close to identical to that resulting from the PARAFAC algorithm. 

In principal component analysis (PCA), a matrix is decomposed as a sum of vector products, as shown in Figure 1.6. The vertical vectors (following the object way) are called scores and the horizontal vectors (following the variable way) are called loadings. A similar decomposition is given for three-way arrays. Here, the array is decomposed as a sum of triple products of vectors as in Figure 1.7. This is the PARAFAC model. The vectors, of which there are three different types, are called loadings. 

The PARAFAC model and the CANDECOMP model are closely related and will be abbreviated as PARAFAC models. The PARAFAC model is also known under the name trilinear... 

The PARAFAC model is introduced here by generalizing the singular value decomposition. A two-way model of a matrix X (7 x J), with typical elements xy, based on a singular value decomposition truncated to R components reads in summation notation... 

For a three-way array X (7 x J x K), with elements Xyk, Equation (4.4) generalizes to the PARAFAC model... 

Across all slices Xk, the components a, and br remain the same, only their weights dk, dkR are different. Hence, all slices Xk are modeled with parallel and proportional profiles dSuaibj,..., dkR Rh R. This allows for writing the model using a simultaneous components notation. There are three fully equivalent ways of writing the PARAFAC model... 

The different expressions for the PARAFAC model of Equation (4.5) are given because in different situations some are more convenient than others. In Appendix 4.A additional ways of writing a PARAFAC model are given. 

Under mild conditions the PARAFAC model gives unique estimates (up to permutation, sign and scaling indeterminacy) that is, the calculated A, B and C cannot be changed without changing the residuals (no rotational freedom). The details of this property are treated in Section 5.2. This property is also called the intrinsic axes property because with... 

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