PARAFAC Decomposition

Fig. 31.20. PARAFAC decomposition of an nxpxq three-way table X. The core matrix I is an rxrxr three-way identity matrix. A, B and C represent the nxr, pxr and qxr loading matrices of the row-, column- and layer-items of X, respectively.

We have noted that three-way resolution methods generally work with the unfolded matrices. Depending on the algorithm used, all three types of unfolded matrices may be used, or only some of them. In the PARAFAC decomposition of a trilinear data set, all three types of unfolded data matrices are used, whereas in the resolution of a nontrilinear data set by the MCR-ALS method, only one type of unfolded matrix is used. 

PARAFAC Decomposition Without any Prior Knowledge of Constituents... 

The concept of fc-rank was introduced by Kruskal [1976], see also Harshman and Lundy [1984], It is a useful concept in deciding whether a PARAFAC decomposition of a given array is unique (see Chapters 4 and 6). 

The principle of parallel proportional profiles has a natural chemical interpretation for curve resolution. Similar ideas evolved independently in chemistry initiated by Ho et al. in the late 1970s and early 1980s [Ho et al. 1978, Ho et al. 1980, Ho et al. 1981], They developed the method of rank annihilation, which is close to the idea of a PARAFAC decomposition. 

Near infrared spectra are known to be difficult to interpret and to exhibit nonlinear behaviour. For example, the spectra of pure analytes can change in mixtures. This invalidates Beer s law. For the interpretation of the results, spectra of pure chemicals and known mixtures are also collected. The goal of the example is to check whether a PARAFAC decomposition of the three-way array allows the interpretation of the loadings as a function of what happens in the reaction. 

Fig. 7, 8 and 9 show results obtained from A, B and C matrices derived from PARAFAC analysis of the three-way NMR spectral data collected imder varying temperature and day content, respectively. Two major factors are indicated here, reflecting the fact that there are two species present in the system One of the important benefits derived from PARAFAC decomposition of the multi-way data is the ability to rationally clarify the effect of the applied perturbations. For example, the matrix A represents abstract information on the temperature-induced behavior of the PLA imder the influence of the day content. In contrast, the matrix C holds essential information on the spectral intensity variation induced by the addition of the clay under the influence of the temperature. The matrix B contains loading vectors which provides chemical or physical interpretation to the p>attems observed in the score matrices A and C. 

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... 

Estimation of the parameters A, B, and C is usually carried out by the alternating least squares (ALS) algorithm. As an example for PARAFAC decompositions, we consider the evaluation of the folding states of a protein by means of mass spectrometry... 

Figure 5.18 Folding states of the protein myoglobin derived from PARAFAC decomposition of charge states observed in ESI-MS spectra in dependence on pH values.

Morales R, Ortiz MC, Sarabia LA. Usefulness of a PARAFAC decomposition in the fiber selection procedure to determine chlorophenois by means SPME-GC-MS. Anal Bioanal Chem 2012 403 1095-107. 

Figure 10.6. PARAFAC decomposition of the Horsens catchment fluorescence dataset. Spectra are shown above contour plots, with spectra for two independently modeled halves of the data set overlaid.

Although the decomposition of a data table yields the elution profiles of the individual compounds, a calibration step is still required to transform peak areas into concentrations. Essentially we can follow two approaches. The first one is to start with a decomposition of the peak cluster by one of the techniques described before, followed by the integration of the peak of the analyte. By comparing the peak area with those obtained for a number of standards we obtain the amount. One should realize that the decomposition step is necessary because the interfering compound is unknown. The second approach is to directly calibrate the method by RAFA, RBL or GRAFA or to decompose the three-way table by Parafac. A serious problem with these methods is that the data sets measured for the sample and for the standard solution should be perfectly synchronized. 

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 ... 

Equation 11.18 defines the decomposition of nontrilinear data sets and is the underlying expression of the Tucker3 model [78], Detailed descriptions of the PARAFAC and Tucker3 models are given in Chapter 12, Section 12.4. 

MCR-ALS solutions can be additionally constrained to fulfill a trilinear model [82], When this trilinearity constraint is applied, the profiles in the three different modes (Useo, utemP, and VT) are directly recovered and can be compared with the profiles obtained using PARAFAC- or Tucker-based model decompositions. MCR-ALS results have already been compared with Tucker3-ALS and PARAFAC-ALS results in the resolution of different chemical systems [81],... 

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

PARAFAC refers both to the parallel factorization of the data set R by Equation 12.1a and Equation 12.lb and to an alternating least-squares algorithm for determining X, Y, and Z in the two equations. The ALS algorithm is known as PARAFAC, emanating from the work by Kroonenberg [31], and as CANDECOMP, for canonical decomposition, based on the work of Harshman [32], In either case, the two basic algorithms are practically identical. 

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. 

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

Fig. 5. Schematic representation of the decomposition of the three-way array X into scores and loadings and a residual performed by PARAFAC.

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. 

Finally, two important decomposition methods for two-way analysis (PCA) and three-way analysis (PARAFAC) are introduced briefly, because these methods are needed in the following chapter. 

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... 

In order to improve the calibration results, an analyte concentration in the mixture close to the analyte concentration in the standard would be favorable. This is not a practical solution, but an alternative is to use multiple standards. All five standards are used now. The calibration equations become slightly more complicated (i.e. a combination of cases (ii) and (iii) from above, see Appendix 10.C), but end up in a relatively straightforward three-way model. Generalized rank annihilation cannot be used anymore, since there are more than two slices in the three-way array. The authors used direct trilinear decomposition to solve this problem but an ordinary PARAFAC-ALS algorithm can also be used. The results are presented in Table 10.5 and show improvements at low concentrations. 

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