PARAFAC analysis

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

Early visualization in chemical three-way analysis was used in curve resolution methods, and especially GRAM. The extraction and visualization of unknown spectra (curve resolution) was considered very important. Tucker and PARAFAC analysis were used much in the exploratory sense and PARAFAC also for curve resolution. The first publication of Ho et al. [1978] has figures of eigenvalues expressed as a function of a parameter /S, see Figure 8.1. The second publication of Ho et al. [1980] also contains contour plots for the excitation-emission spectra of polyaromatic compounds (see Figures 8.2 and 8.3). They also show excitation and emission spectra as line plots (see Figures 8.4 and 8.5). The plots are only used to show the raw data and show no results after analysis. An early publication of de Ligny et al. [1984] has all the results shown as numbers in tables. 

Kiers HAL, A three-step algorithm for CANDECOMP/PARAFAC analysis of large data sets with multicollinearity, Journal of Chemometrics, 1998, 12, 155-171. 

PARAFAC Analysis for Temperature-Dependent NMR Spectra of Poly(Lactic Acid) Nanocomposite... 

This chapter provides a tutorial on the fundamental concept of Parallel factor (PARAFAC) analysis and a practical example of its application. PARAFAC, which attains clarity and simplicity in sorting out convoluted information of highly complex chemical systems, is a powerful and versatile tool for the detailed analysis of multi-way data, which is a dataset represented as a multidimensional array. Its intriguing idea to condense the essence of the information present in the multi-way data into a very compact matrix representation referred to as scores and loadings has gained considerable popularity among scientists in many different areas of research activities. 

In the second part of this chapter, an illustrative example of PARAFAC analysis for three-way data obtained in an actual laboratory experiment is presented to show how PARAFAC trilinear model can be constructed and analyzed to derive in-depth understanding of the system from the data. Thermal deformation of several types of poly lactic add (PLA) nanocomposites xmdergoing grass-to-rubber transition is probed by cross-polarization magic-angle (CP-MAS) NMR spectroscopy. Namely, sets of temperature-dependent NMR spectra are measured under varying clay content in the PLA nanocomposite samples. While temperature strongly affects molecular dynamics of PLA, the clay content in the samples also influences the molecular mobility. Thus, NMR spectra in this study become a three-way... 

In PARAFAC analysis, the set of matrices A, B and C are usually obtained by iteratively solving alternating least-squares (ALS) problems min - A(C B) over A for fixed... 

A pertinent example for PARAFAC analysis based on NMR spectra of PLA nanocomposites is provided here to show how certain useful information can be effectively extracted from an actual laboratory experiment. 

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. 

The emission weights, in particular, present maxima and minima at wavelengths which correspond quite well with the position of the signals associated with the components identified by the PARAFAC analysis of the same data array. The same information can be visualized as landscapes, by plotting, for each factor, the product wf refolded into a matrix (see Figure 19). 

Schmidt B, Jaroszewski JW, Bro R, Witt M, Staerk D. Combining PARAFAC analysis of HPLC-PDA profdes and structural characterization using HPLC-PDA-SPE-NMR-MS experiments commercial preparations of St. John s wort. Anal Chem 2008 80 1978-87. 

PCA and parallel factor (PARAFAC) analysis were used to investigate the role of different technological parameters/cooking strategies adopted by different producers on the quality of final cooked musts. 

Taking into account the trilinear structure of the data (Figure 10), characterised by three distinct sources of systematic variability, that is cooking time, monitored variables, and different cooking processes, PARAFAC analysis... 

Before PARAFAC analysis, the data array was centred across the first mode (cooking processes), in order to remove the offsets, and scaled to unit variance within the secrnid mode. 

Stedmon, C.A. and Markager, S.S. (2005a). Resolving the variability in dissolved organic matter flnorescence in a temperate estuary and its catchment using PARAFAC analysis. Limnol. Oceanogr., 50(2), 686-697. 

In our example, there are 5 /3 2 = 10 pairs of components that can be plotted in order to explore the relationships between the five PARAFAC components from the Horsens catchment model. Four of these are shown in Figure 10.8 (note log-log scales). Similar to the PCA in Figure 10.4, the PARAFAC analysis detects three clusters of samples. However, due to the additional information provided by the PARAFAC spectra, it is possible to obtain a more direct chemical interpretation of the differences between sites than previously. Component 3 appears to represent a terrestrially derived fraction that is more or less absent in the WTP organic matter (Figure 10.8, first and last panels). Closer analysis shows that this signal was present in the water in conjunction with periods of intensive rain where soil organic matter is also expected to be a more significant fraction of the municipal... 

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