Classical least-squares analysis results

The next section of this paper describes the use of classical least-squares analysis of FTIR data to determine coal mineralogy. This is followed by promising preliminary results obtained using factor analysis techniques. 

Results of classical least-squares analysis of FTIR spectra of ten coals using forty-two reference minerals were evaluated with regard to reproducibility and accuracy as described below. 

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. ... 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

The existence of a multitude of new physico-chemical methods of analysis, side by side with the classical methods of chemical analysis, urgently raises the question of finding out rational criteria for comparing the results obtained by various analytical methods. The development and introduction of new analytical methods take place considerably faster than their standardization...It is already evident that an analyst must be as thoroughly familiar with the methods of modern mathematical statistics as the geodesist is with the method of least squares. 

The resulting spectro-chromatograms (SCG) are 3D-representations of the tar matrices with the UV-absorbances as function of the retention time in the gel column and the wavelength of absorption, respectively fig. 3). Sections of the SCG parallel to the retention time axis at 215 nm UV-absorption ("tar profiles" in the following) enable quick qualitative tar characterization. For the quantitative evaluation of the SCG, chemometric methods such as factor analysis and the classical least squares method are applied. This requires the set-up of a spectral library which contains the SCG of the quantitative important tar compounds. 

The next example of an OTC map was treated first using a direct classic least square (DCLS) method, and then with more sophisticated multivariate analysis methods. The tablet was mapped over 800 X 800 gm with 10 gm steps. The data were baseline-corrected and normalized before being subjected to an unsupervised multivariate analysis. The first set of results was produced using univariate analysis (Figure 11.8a), when a manual exploration revealed three distinguishable and... 

However, multicomponent quantitative analysis is the area we are concerned with here. Regression on principle components, by PCR or PLS, normally gives better results than the classical least squares method in equation (10.8), where collinearity in the data can cause problems in the matrix arithmetic. Furthermore, PLS or PCR enable a significant part of the noise to be filtered out of the data, by relegating it to minor components which play no further role in the analysis. Additionally, interactions between components can be modelled if the composition of the calibration samples has been well thought out these interactions will be included in the significant components. 

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