Band factor analysis

Bulmer, J.T., et. al. "Factor Analysis as a Complement to Band Resolution Techniques. I. The Method and its Application to Self-Association of Acetic Acid",./. Phys. Chem. 1973, (77) 256-262. 

In Section 34.2 we explained that factor analysis consists of a rotation of the principal components of the data matrix under certain constraints. When the objects in the data matrix are ordered, i.e. the compounds are present in certain row-windows, then the rotation matrix can be calculated in a straightforward way. For non-ordered spectra with three or less components, solution bands for the pure factors are obtained by curve resolution, which starts with looking for the purest spectra (i.e. rows) in the data matrix. In this section we discuss the VARDIA [27,28] technique which yields clusters of pure variables (columns), for a certain pure factor. 

Our band shape methods have made use of the principal component method of factor analysis (Pancoska etal., 1979 Malinowski, 1991) to characterize the protein spectra in terms of a relatively small number of coefficients (loadings) (Pancoska et al., 1994 1995 Baumruk et al., 1996). This approach is similar, in its initial stages, to various methods (Selcon, Variselect, etc.) that have been used for determining protein secondary structure from ECD data (Hennessey and Johnson, 1981 Provencher and Glockner, 1981 Johnson, 1988 Pancoska and Keiderling, 1991 Sreerama and Woody, 1993, 1994 Venyaminov and Yang, 1996). At this point, one can say these traditional quantitative methods have had little impact upon structural studies of denatured proteins. 

Fig. 14. Factor analysis loadings (first and second spectral components) for thermal unfolding of RNase A as monitored with amide F FTIR and far-UV ECD. In each case a pretransition is evident in the curves before the main transition at 55°C. This full band shape analysis can sense smaller variations and can be partitioned to give added insight. Since the main ECD change could be shown to be loss of intensity, the major structural change was unfolding of a helix. The frequency dispersion of the FTIR change showed that some /3-sheet loss accompanied this pretransitional helix unfolding, but that most sheet loss was in the main transition.

Fig. 15.6. Results of factor analysis in the 600-800 cm 1 region in human skin and a cultured skin model (Epiderm ). Data from human skin (8 x 12 pixels) and cultured skin (7 x 12 pixels) have been concatenated. Pixels marked with x s were excluded from the analysis, a The cholesterol (700cm-1) and Phe vibrations (620 cm-1) are marked, b An image of the 700/620 cm-1 band intensity ratios...

In many practical cases, the factors f i are very close to unity and can be omitted. The parameters So, and mo,- are then equal to their gas-phase values oto, and mo,. Equation [100] then gives the polarizability change in terms of spectroscopic moments and gas-phase solute dipoles. Experimental measurement and theoretical calculation of Aao = aoi - ocoi is still challenging. Perhaps the most accurate way to measure Akq presently available is that by Stark spectroscopy,which also gives Awq. Equation [100] can therefore be used as an independent source of Aao, provided all other parameters are available, or as a consistency test for the band shape analysis. 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

We thank a referee for pointing out that instead of measuring a simple difference spectrum, a detailed factor analysis should have been performed. Although this may give more accurate values for the wavelength of the new absorption maximum, it would not alter the main conclusion that a novel absorption band is present in the hybrid photocatalysts. 

Most thermodynamic studies of the equilibria between hydrogen-bonded complexes of phenols and their free component molecules have been conducted in a diluting solvent. Binary solutions of phenols (phenol o-cresol ) in the pure base propionitrile have also been studied by means of Raman and IrW3,i44 spectrometry. Factor analysis of the v(C=N) band indicates the formation of a 1 1 complex over a large concentration range. However, this procedure is not recommended for the determination of equilibrium constants because these exhibit a strong concentration dependence. 

Wul-6 (In contrast to Marcus-Hush which refers to the theory of electron transfer activation, the Mulliken-Hush equation describes the preexponential factor of the rate constant. We spell out Mulliken-Hush each place it occurs in this chapter and use the acronym MH to refer to only Marcus-Hush.) In practice, however, FCWD(O) cannot be extracted from experimental spectra, and one needs a theoretical model to calculate FCWD(O) from experimental band shapes measured at the frequencies of the corresponding electronic transitions. This purpose is achieved by a band shape analysis of optical lines. 

The accuracy of quantification methods may be improved using multivariate methods since these have the advantage of analysing variation across the whole spectmm rather than the intensity of individual bands [11, 65]. The use of IR spectroscopy with factor analysis [66], partial least squares [67,68] and neural networks [69], and FT-Raman with factor analysis [70] and principal component analysis [71] has been demonstrated. 

Features due to g-anisotropy are in many cases difficult to differentiate from those due to hfs in X-band spectra. The advantage of multi-frequency measurements in this case is demonstrated in Fig. 4.7 for a semiquinone complex of monovalent copper in a frozen toluene solution. The g-anisotropy is clearly resolved at the 2 mm band because of the better g-factor resolution. The hfs due to Cu and the two P atoms are not resolved at this band. The analysis of the hfs observed in the X-band spectmm is, however, simplified when the principal values of the g-tensor are known. A simulation based on the analysis is shown at the bottom of the figure with parameters given in the legend. 

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. 

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