Evolving factor analysis data modeling

An important group of methods relies on the inherent order of the data, typically time in kinetics or chromatography. These methods are often based on Evolving Factor Analysis and its derivatives. Another well known family of model-free methods is based on the Alternating Least-Squares algorithm that solely relies on restrictions such as positive spectra and concentrations. 

There are many chemometric methods to build initial estimates some are particularly suitable when the data consists of the evolutionary profiles of a process, such as evolving factor analysis (see Figure 11.4b in Section 11.3) [27, 28, 51], whereas some others mathematically select the purest rows or the purest columns of the data matrix as initial profiles. Of the latter approach, key-set factor analysis (KSFA) [52] works in the FA abstract domain, and other procedures, such as the simple-to-use interactive self-modeling analysis (SIMPLISMA) [53] and the orthogonal projection approach (OPA) [54], work with the real variables in the data set to select rows of purest variables or columns of purest spectra, that are most dissimilar to each other. In these latter two methods, the profiles are selected sequentially so that any new profile included in the estimate is the most uncorrelated to all of the previously selected ones. 

Gampp, H., Maeder, M., Meyer, C J., and Zuberbiihler, A., Calculation of equilibrium constants from multiwavelength spectroscopic data, IV model free least squares refinement by use of evolving factor analysis, Talanta, 33, 943-951, 1986. 

It is possible to analyse these titration curves quantitatively. The number of absorbing species in solution may be determined from the raw data by factor analysis, with evolving factor analysis often giving a good indication of the stoichiometry of the species [16]. If we have a model for the complexes formed, together with an estimate for the stability constant of each species, then we can calculate a speciation of the particles at any point in the titration from the known total concentrations of metal and ligand. This speciation may be used to obtain estimates of the molar extinction coefficients of the particles (if they are unknown), and calculate the absorbance [17] a non-linear least-squares refinement of the stability constants is then carried out to give the best fit of calculated and observed absorbance. 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

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