Parallel factor analysis data array

Multiway methods For analyzer data where a single sample generates a second order array (ex. GC/MS, LC/UV, excitation/emission fluorescence), multiway chemometric modehng methods, such as PARAFAC (parallel factor analysis) [121,122], can be used to exploit the second order advantage to perform effective calibration transfer and instrument standardization. 

Direct analysis of a three-way data array is feasible by parallel factor analysis (PAR AFAC) or by Tucker models. 

Finally, an example should be provided about a class of methods, which have also explorative purposes, which will be discussed with more detail and theoretical depth in Chapter 7, that is multiway analysis methods [81]. These methods, among which parallel factor analysis (PARAFAC) will be shown here in action compared to PCA, are to some extent referred to as the conceptual (and mathematical) extension of PCA to arrays of order higher than two. They show their potentiality when the variability of a data set is related to different sources, or conditions, at which a full set of properties for each sample is measured [17-21]. An example, quite common in the food science analysis, is the excitation-emission fluorescence landscape, where, for each sample, an emission spectrum is recorded at each wavelength of the excitation signal. 

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

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. 

Combined X-ray and electron diffraction analysis led to an orthorhombic unit-cell, with a = 2.468 mn, 1) = 1.152 nm, and c = 1.054 nm. The space group is P2,2,21. Two parallel chains are related, pairwise, by a two-fold screw-axis parallel to the chain axis, and pairs of chains pack in an antiparallel array. The (110) growth planes ol the crystal are parallel to the direction of highest atomic densities. The transformation CTA II cellulose II was discussed. The R factor is 30% with the X-ray diffraction data, and 26% with the electron diffraction data. 

Some of the first ideas on multi-way analysis were published by Raymond Cattell [1944,1952], Thurstone s principle of parsimony states that a simple structure should be found to describe a data matrix or its correlation matrix with the help of factors [Thurstone 1935], For the simultaneous analysis of several matrices together, Cattell proposed to use the principle of parallel proportional profiles [Cattell 1944], The principle of parallel proportional profiles states that a set of common factors should be found that can be fitted with different dimension weights to many data matrices at the same time. This is the same as finding a common set of factors for a stack of matrices, a three-way array. To quote Cattell ... 

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