Fourier transform factor analysis

Antoon, M.K., et. al. "Factor Analysis Applied to Fourier Transform Infrared Spectra", Appl. Spec. 1979, (33) 351-357. 

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). 

J.-H. Jiang, Y. Ozaki, M. Kleimann and H.W. Siesler, Resolution of two-way data from on-line Fourier-transform Raman spectroscopic monitoring of the anionic dispersion polymerization of styrene and 1,3-butadiene by parallel vector analysis (PVA) and window factor analysis (WFA), Chemom. Intell. Lab. Syst., 70, 83-92... 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

Infrared analyses are conducted on dispersive (scanning) and Fourier transform spectrometers. Non-dispersive industrial infrared analysers are also available. These are used to conduct specialised analyses on predetermined compounds (e.g. gases) and also for process control allowing continuous analysis on production lines. The use of Fourier transform has significantly enhanced the possibilities of conventional infrared by allowing spectral treatment and analysis of microsamples (infrared microanalysis). Although the near infrared does not contain any specific absorption that yields structural information on the compound studied, it is an important method for quantitative applications. One of the key factors in its present use is the sensitivity of the detectors. Use of the far infrared is still confined to the research laboratory. 

In conclusion, the analysis of spectra properly recorded to 185 nm, or lower where possible, can give useful estimates of secondary structure content, but the content of turns and of P-structure should be interpreted with caution. Fourier transform infrared spectroscopy (FTIR) provides better estimates of the latter. When using the results of far-UV CD determination to characterize reproducibility of folding for different samples, it is important first to compare the spectra visually and to look for possible trends or factors that may explain small differences, rather than to rely solely on comparison of derived secondary structure contents. 

The discussion of the previous section suggests that the linear combination of the shifted and scaled Fourier transforms of the analysis window in Equation (9.72) must be explicitly accounted for in achieving separation. The (complex) scale factor applied to each such transform corresponds to the desired sine-wave amplitude and phase, and the location of each transform is the desired sine-wave frequency. Parameter estimation is difficult, however, due to the nonlinear dependence of the sine-wave representation on phase and frequency. 

Core and valence electrons can be divided based on a criterion of whether their distributions and populations are changed or not owing to bond formation. The electron population analysis conventionally uses spherical form factors, f, which are given by the spherically averaged Fourier transform of electron-density distributions. 

The Fourier coefficients in crystallographic analysis are the measured structure factor amplitudes of diffraction maxima and correspond to the Fourier transform of the periodic density. Numerical solution of the phase problem enables the Fourier transformation that synthesizes the unit-cell electron-density function and hence the three-dimensional molecular structure. Quantum-chemical computations assume the molecular structure and calculate Fourier coefficients for a limited basis set to redefine the electron density. 

An operation or transformation links these two spaces, such as Fourier transformation or factor analysis. 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

Antoon MK, D Esposito L, Koenig JL (1979) Factor analysis applied to Fourier transform infrared spectra Appl Spectrosc 33 351-357... 

In the most simplistic means of defining particle shape, measurements may be classified as either macroscopic or microscopic methods. Macroscopic methods typically determine particle shape using shape coefficients or shape factors, which are often calculated from characteristic properties of the particle such as volume, surface area, and mean particle diameter. Microscopic methods define particle texture using fractals or Fourier transforms. Additionally electron microscopy and X-ray diffraction analysis have proved useful for shape analysis of fine particles. 

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