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The second type of resolution methods can be regarded as evolutionary. Examples are evolving factor analysis (EFA) , window factor analysis (WFA) , heuristic evolving latent projections (HELP) [12-14], and more recently, subwindow factor analysis . Their common feature is the use
resolution methods for two-dimensional data
of the informative ‘‘windows,’’ such as the selective information regions, zero-concentration regions, and also the regions of (A - 1) components. Thus, the correct estimation of the abovementioned windows based on the local rankmap is crucial in order to obtain the correct resolution results. The resolution condition for “hyphenated” two-way data was comprehensively reviewed by Manne . Three so-called resolution theorems are presented here.
Theorem 3.1. If all interfering compounds that appear inside the concentration window of a given analyte also appear outside this window, it is possible to calculate the concentration profile of the analyte.
Theorem 3.2. If for every interferent the concentration window of the analyte has a subwindow where the interferent is absent, then it is possible to calculate the spectrum of the analyte.
Theorem 3.3. For a resolution based on rank information in the chromatographic direction, the condition of Theorems 3.1 and 3.2 are not only sufficient but also necessary.
As stated before, selectivity is the cornerstone for the resolution of twoway data. But, from Theorems 3.1-3.3, it does not seem necessary to have selective information for complete resolution without ambiguities. Yet, this is not true because the rationale behind these three theorems is that the component that elutes out first should also be the one devolving first in the concentration profiles. Thus, there must be some kind of selective information available in the data. If one cannot mine this out, complete resolution for all the components involved with no ambiguities is impossible. Of course, it is not necessary that every component have its own selective information. But for systems of two components or more, two selective information regions are necessary for complete resolution without ambiguities. The simplest example is the embedded peak with only two components in the chromatogram where only one item of selective information is available. The complete resolution of embedded peaks without additional modeling assumption, to our knowledge, seems impossible for the two-way data so far. Trilinear data may help us obtain the complete resolution desired. The more recent progress for multivariate resolution methodology is based mainly on evolutionary methods. Thus, evolutionary methods are the main focus of this book.
two-dimensional signal processing techniques in chemistry
3.6.2.I. Evolving Factor Analysis (EFA)
The first attempt to efficiently use the separation ability in the chromatographic direction for estimating the chemical rankmap might be the evolving factor analysis developed by Gampp et al. , which was proposed primarily to deal with the titration data. This technique was later extended to the analysis of chromatographic two-way data. Its most attractive feature is application of the evolving information in the elution direction for titration, chromatography, and other chemical procedures. This opens a new door for chemometricians to work with two-way data. The methodology from EFA seems simple. It embraces the spectra to be factor-analyzed in an incremental fashion and then collects the eigenvalues and plots them against the retention timepoints (see also Figs. 3.9 and 3.10). This evolving factor-analyzing procedure can also be conducted in the reverse direction. Finally, the points of appearance and disappearance of every chemical component can be determined in this manner. The only assumption involved is that the component that first appears will disappear also first in an evolving manner.
In order to deal with the peak purity problem for two-way chromatography, Keller and Massart developed a method termed fixed-size moving-window factor analysis (FSMWFA)  (see also Figs. 3.11 and 3.12). Instead of factor-analyzing the data matrix in an incremental fashion, the method factor-analyzes the spectra in a fixed-sized window and moves the window along the chromatographic direction. The eigenvalues thus obtained are also plotted against the retention time. This method has been
Figure 3.9. Illustration diagram of the evolving factor analysis (EFA) algorithm. The algorithm can be conducted in both the forward and backward directions. It embraces the spectra to be factor-analyzed in a stepwise increasing way and then collects the eigenvalues to be plotted against the retention timepoints.
resolution methods for two-dimensional data
Figure 3.10. The resulting plot of evolving factor analysis of a three-component system. With the help of the results obtained, the points for every chemical component appearing and disappearing can be determined in this way. The only assumption of the method is that the component that first appears will first disappear in an evolving pattern.