Factor analysis fitness components

The terms factor analysis, principal components analysis, and singular value decomposition (SVD) are used by spectroscopists to describe the fitting of a two-way array of data with a general bilinear model. We will use the term factor analysis in this sense, although this term has a somewhat different meaning in statistics. SVD is a specific algebraic procedure, discussed by Henry and Hofrichter and briefly later in this chapter, whose use alone is often not the best way to fit a general bilinear model. 

The integrated intensities of the fitted component peaks should then be related to the electron population of different valence states, subject to correction factors, according to the same equation used earlier for quantitative analysis of survey XPS spectra (Eq. 3) [10]. Because photoelectron KEs are similar throughout the valence band region, spectrometer-dependent factors and IMFP values can be assumed to be the same for all states, so that the equation simplifies to ... 

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. 

The analysis of any multicomponent resin or composite is greatly facilitated when the spectrum of that material is expressed by a linear combination of a finite set of pure component spectra. The entire process may be separated into three steps calculation of the number of species present, identification of each of those species, and curve fitting of the spectra of these species to the spectra of the composites7). The technique for determining the number of components in the mixture is called factor analysis or major component analysis and has been described in detail in a number of publications 28,29). Factor analysis is concerned with a matrix of data points. So, in matrix notation we can write the absorbance spectra of a number of mixtures as ... 

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. ... 

Higher order differences derivatives Difference ratios Fourier coefficients Principal components scores Factor analysis scores Curve-fitting coefficients Partial least-squares scores... 

Although extensively studied in nonaqueous solvents (see below), Raman studies of aqueous solutions of thiocyanates are scarce. Kato [99] and Rothschild and Perrot [100] published two spectroscopic studies on the dynamics of SCN ion in concentrated solutions and molten alkali (Li, Na, K) thiocyanates. Antic-Jovanovic and co-workers have published two papers on the ionic association in Mg VSCN aqueous solutions [101,102]. Using factor analysis to calculate the number of components in the C=S and C=N band envelopes, they found spectroscopic evidence of two different ionic pairs, namely [MgNCS] and [Mg(NCS)2]. In the i/(C=N) Raman spectral region, three components were fitted at 2068, 2087, and 2108.5 cm and assigned to free thiocyanate, 1 1 complex, and 1 2 complex, respectively. The successive association equilibria from the hydrated magnesium ion, [Mg(OH2)6], imply the loss of one or two water molecules from the first solvation sphere, which means that the magnesium ion retains its octahedral coordination in both ionic pairs. 

Ng and Shurvell [108] have applied factor analysis to deconvolve the complex Raman spectra of acetic acid in aqueous solution. As can be appreciated in Fig. 6, up to five different components can be fitted. These were assigned by the authors in the following manner carbonyl stretching mode of the monomer, symmetric C=0 stretch of the cyclic dimer, terminal and hydrogen-bonded carbonyl group from linear dimer and/or polymer, and water bending band. The monomer-cyclic dimer equilibrium constant was calculated as 0.06 mol Vdm at concentrations below 7 M,... 

There are now several curve-fitting methods available on most AES/XPS systems. Among these may be mentioned principal component analysis (factor analysis), linear least-squares fitting, the maximum entropy method, the pattern recognition method (derived from factor analysis) and difference spectra. 

In principle, valence band XPS spectra reveal all the electronic states involved in bonding, and are one of the few ways of extracting an experimental band structure. In practice, however, their analysis has been limited to a qualitative comparison with the calculated density of states. When appropriate correction factors are applied, it is possible to fit these valence band spectra to component peaks that represent the atomic orbital contributions, in analogy to the projected density of states. This type of fitting procedure requires an appreciation of the restraints that must be applied to limit the number of component peaks, their breadth and splitting, and their line-shapes. 

A or As). It is necessary to first establish a reliable experimental database for the property of interest, and then to fit it, by means of a statistical analysis code, to (usually) three or four of the quantities, appropriately selected, as computed for the molecules in the database. If the interaction involves multicomponent systems, as does solvation, then only one component may vary. For example, a relationship could be developed for a series of solutes in a particular solvent, or a given solute in different solvents. In doing so, we have always sought to use as few of the computed quantities as is consistent with a good correlation, since they can provide insight into the physical factors that are involved in the interaction this becomes obscured if many terms are involved. 

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