Resolving Factor Analysis

Resolving Factor Analysis, RFA, is an attempt to introduce the strengths of the Newton-Gauss algorithm into the model-free analysis methodology. As... 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

PLS is a factor analysis-based method that has recently demonstrated to have a high capacity to resolve complex mixtures of components with similar spectral... 

Fister, J. C., Ill, and J.M. Harris. 1996. Factor analysis of transient Raman scattering data to resolve spectra of ground- and excited-state species. In Computer assisted analytical spectroscopy. Ed. S. D. Brown. New York Wiley. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

H.B. Woodruff, P.C. Tway, and J. Cline Love, Factor analysis of mass spectra from partially resolved chromtographic peaks using simulated data, Anal. Chem., 53 81 (1981). 

Window factor analysis (WFA) was described by Malinowski and is likely the most representative and widely used noniterative resolution method [34, 35], WFA recovers the concentration profiles of all components in the data set one at a time. To do so, WFA uses the information in the complete original data set and in the subspace where the component to be resolved is absent, i.e., all rows outside of the concentration window. The original data set is projected into the subspace spanned by where the component of interest is absent, thus producing a vector that represents the spectral variation of the component of interest that is uncorrelated to all other components. This specific spectral information, combined appropriately with the original data set, yields the concentration profile of the related component. To ensure the specificity of this spectral information, all other components in the data set should be present outside of the concentration window of the component to be resolved. This means, in practice, that component peaks with embedded peak profiles under them cannot be adequately resolved. 

Most experiments have inherent in them a large number of variables. The experimenter usually ignores most of these variables, assuming on a subjective basis that they are not important. The problem has been stated by Thomas Auf der Heyde and Hans-Beat Biirgi as a twofold one First, how can we avoid subjectively carving up the data into two-dimensional subsets, i.e., how can all n variables be analyzed simultaneously in order to reveal correlations between them Second, how can the dimensionality of the problem be objectively reduced in order to interpret and visualize these correlations Two main methods are presently employed to try to resolve this problem. These methods are cluster analysis, which sorts the data into groups (clusters) so that the data can be classified, and factor analysis, which finds those variables (factors) that account most successfully for the sample variance, that is, the differences or similarities between the various sets of geometric data. 

This preliminary factor analysis leaves many questions unanswered concerning the relations among the trace elements in the total crude oils. But as pointed out previously the fact that most trace elements are associated with the asphaltene fraction of crude oils together with the facts that the contents of asphaltenes vary widely and hence that many elements were below detection limits indicates that examination of trace elements in the asphaltenes should resolve many of the uncertainties posed by the present study. Furthermore, EPR, NMR and elemental (C, H, N, O, S) analysis of the asphaltene fraction may help link the factors controlling the organic and inorganic components in these Alberta crude oils. 

A.K. Elbergali and R.G. Brereton, Influence of Noise, Peak Position and Spectral Similarities on Resolvability of Diode-Array High-Performance Liquid Chromatography by Evolutionary Factor Analysis, Chemometrics Intelligent Laboratory Systems 23 (1994), 97-106. 

In excited-state spectroscopies, including fluorescence spectroscopy, spectroscopic intensity is usually linear in functions of each of three or more independent variables, so that a three-way array of data can be fit with a trilinear model. The presence of three or more linear relationships makes algebraic methods for resolving the spectra and other properties of individual components substantially more powerful than in the case of two linear relationships. The use of a general trilinear model is sometimes known as three-way factor analysis, three-mode factor analysis, or threemode principal component analysis. For a review of the mathematics and application to spectroscopy, see our survey article. ... 

Most of our attention in this chapter has been devoted to models using three independent variables, as opposed to the two variables used in more traditional factor analysis and in most global analysis. This has the disadvantages of a requirement to identify three or more appropriate independent variables and to perform a larger number of measurements. It has the advantage of providing a richer data set, the analysis of which can yield results that are more precise than those provided by two-variable factor analysis and that are more independent of specific physical models than global analysis. In those circumstances for which a PARAFAC model is appropriate, the components can be resolved with no other information about their properties. 

Funke et al has used a factor analysis technique to investigate activity coefficients from g.l.c. In this method, the property in question (activity coefficients) is resolved into a linear sum of the minimum number of factors needed to reproduce the experimental results to the required accuracy. Using these factors, it is possible to predict activity coefficients for systems not yet measured. Funke et al. have also suggested that it should be possible to interpret the elements of the matrix into meaningful parameters but this has not yet been achieved. 

Overlapping bands can become a problem when, for example, there are two consecutive electron-transfer reactions [137]. One solution is to look at the time-or potential-resolved spectra [138], Overlapping bands are often responsible for nonlinear Nemstian plots in OTTLE studies [139]. There are only a few examples of the use of differentiating the absorbance [134], least-squares analysis [140], of the latest chemometric techniques [141]. In the latter study, evolutionary factor analysis of the spectra arising from the reduction of E. coli reductase hemoprotein (SiR-HP ) in which three species are present and the reduction of the [Cl2FeS2MoS2FeCl2] (four species present). The most challenging part of the work was the determination of the transformation matrix. 

Brereton, R. G. and Elbergali, A. K. (1994) Use of double windowing, variable selection, variable ranking and resolvability indices in window factor analysis. Journal of Chemometrics, 8, 423-37. 

Abstract This chapter introduces an application of multivariate curve resolution (MCR) technique based on a factor analysis. Not only series of IR spectra but also two-dimensional data (series of nuclear magnetic resonance (NMR), mass spectrometry (MS), and X-ray diffraction (XRD)) can deal with same manner (further more two-dimensional data generated by hyphenated techniques such as gas chromatography/mass spectrometry (GC/MS) and liquid chromatography/ultravi-olet (LC/UV) analysis, which combine two functions based on different principles, namely, chromatography, which has a separating function, and spectrometry, which provides information related to molecular structure). By using MCR techniques appropriately, the mixture data is resolved into some essential elements (chemical components, transient states and phases). The results can reveal a true chemical characteristic in your study. 

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