Evolving factor analysis

Keller, H. R. and Massart, D. L., Artefacts in Evolving Factor Analysis-Based Methods for Purity Control in Liquid Chromatography with Diode-Array Detection, Ana/yt/ca Chimica Acta 263, 1992, 21-28. 

Fixed-size window evolving factor analysis (FSWEFA)... 

M. Maeder, Evolving factor analysis for the resolution of overlapping chromatographic peaks. Anal. Chem., 59 (1987) 527-530. 

M. Maeder and A.D. Zuberbuhler, The resolution of overlapping chromatographic peaks by evolving factor analysis. Anal. Chim. Acta, 181 (1986) 287-291. 

H.R. Keller and D.L. Massart, Artifacts in evolving factor analysis-based methods for peak purity control in liquid-chromatography with diode array detection. Anal. Chim. Acta, 263 (1992) 21-28. 

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. 

The Singular Value Decomposition of a matrix Y into the product USV is full of rich and powerful information. The model-free analyses we discussed so far are based on the examination of the matrices of eigenvectors U and V. Evolving Factor Analysis, EFA, is primarily based on the analysis of the matrix S of singular values. 

Evolving factor analysis can, and should be performed in both forward and backward directions. The forward plot, calculated above and shown in Figure 5-34, indicates the appearance of new components. The backward plots of Figure 5-36 are calculated similarly by determination of the singular values of the set of the last 1, 2, 3,. .. spectra in Y, as seen in the schematic of Figure 5-35. These plots indicate the disappearance of the components. 

Local Rank and Resolution Evolving Factor Analysis... 

LOCAL RANK AND RESOLUTION EVOLVING FACTOR ANALYSIS AND RELATED TECHNIQUES... 

Some of the local-rank analysis methods, such as evolving-factor analysis (EFA) [27-29], are more process oriented and rely on the sequential evolution of the components as a function of time or any other variable in the data set, while others, such as fixed-size moving-window-evolving-factor analysis (FSMW-EFA) [30, 31], can be applied to processes and mixtures. EFA and FSMW-EFA are the two pioneering local-rank analysis methods and can still be considered the most representative and widely used. 

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. 

Keller, H.R., Massart, D.L., Liang, Y.Z., and Kvalheim, O.M., Evolving factor analysis in the presence of heteroscedastic noise, Anal. Chim. Acta, 263, 29-36, 1992. 

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. 

FIGURE 9.7. Scheme of the evolving factor analysis procednre. (a) Forward analysis (b) Backward analysis and (c) Reconstmcted abstract profiles. 

Fixed-size moving-window-evolving factor analysis (FSMW-EFA). This technique, also called window factor analysis (WFA), is based on a window of a predefined number of rows or spectra, typically from three to five, which... 

FSMW-EFA—fixed-size moving-window-evolving factor analysis FT—Fourier transform... 

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