Classical least-squares analysis

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. 

Infrared data in the 1575-400 cm region (1218 points/spec-trum) from LTAs from 50 coals (large data set) were used as input data to both PLS and PCR routines. This is the same spe- tral region used in the classical least-squares analysis of the small data set. Calibrations were developed for the eight ASTM ash fusion temperatures and the four major ash elements as oxides (determined by ICP-AES). The program uses PLSl models, in which only one variable at a time is modeled. Cross-validation was used to select the optimum number of factors in the model. In this technique, a subset of the data (in this case five spectra) is omitted from the calibration, but predictions are made for it. The sum-of-squares residuals are computed from those samples left out. A new subset is then omitted, the first set is included in the new calibration, and additional residual errors are tallied. This process is repeated until predictions have been made and the errors summed for all 50 samples (in this case, 10 calibrations are made). This entire set of... 

Experience in this laboratory has shown that even with careful attention to detail, determination of coal mineralogy by classical least-squares analysis of FTIR data may have several limitations. Factor analysis and related techniques have the potential to remove or lessen some of these limitations. Calibration models based on partial least-squares or principal component regression may allow prediction of useful properties or empirical behavior directly from FTIR spectra of low-temperature ashes. Wider application of these techniques to coal mineralogical studies is recommended. 

X-ray structural calculations have been the subject of much study and careful development, essentially in the framework of classic least squares analysis. From the point of view of a modem data analyst, two sorts of increased realism appear vital recognition of systematic error and introduction of methods that do not depend crucially on utopian assumptions. 

Classical least-squares analysis using spectra of pure components... 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

Haaland and coworkers (5) discussed other problems with classical least-squares (CLS) and its performance relative to partial least-squares (PLS) and factor analysis (in the form of principal component regression). One of the disadvantages of CLS is that interferences from overlapping spectra are not handled well, and all the components in a sample must be included for a good analysis. For a material such as coal LTA, this is a significant limitation. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Calibration and mixture analysis addresses the methods for performing standard experiments with known samples and then using that information optimally to measure unknowns later. Classical least squares, iterative least squares, principal components analysis, and partial least squares have been compared for these tasks, and the trade-offs have been discussed (Haaland,... 

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... 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

The p and k matrix methods are two classical least squares approaches to multicomponent calibration. There are techniques based on factor analysis, however, that are increasingly popular these include the... 

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. 

This method of quantitative analysis is known as K matrix, or classic least squares (CLS). It has the advantage of being able to use large regions of the spectrum, or even the entire spectrum, for calibration to gain an averaging effect for the predictive accuracy of the final model. One interesting side effect is that if the entire spectrum is used for calibration, the rows of the K matrix are actually spectra of the absorptivities for each of the constituents. These will actually look very similar to the pure constituent spectra. 

Classical least-squares is a full spectrum method because all the digitized absorbances of the measured spectra are taken into account. A disadvantage of the CLS method is that all the interfering chemical components in the region of the spectrum being observed must be known and included in the analysis. Thus, spectral overlaps cause severe problems in this type of analysis unless explicitly taken into account. 

The term classical least squares (CLS) is often used to describe an extension of OLS analysis (as described earlier), CLS uses the entire spectrum, with each spectral point (i.e., discrete interval on the wavenumber axis) being considered a separate piece of in-... 

There are both gas- and liquid-phase data for toluene and the n-alkyl substituents ethyl, propyl, butyl, pentyl (liquid only), and decyl. Since these constitute a classic homologous series, we derive eqs 15 from a weighted least-squares analysis where hr is the number of carbon atoms in the substituent group. For most homologous series, n-RZ, the methyl-substituted derivative CH3Z is an outlier from tiie otherwise straight line. In the case of the n-alkyl benzenes, the correlation (r > 0.9998) is not im-... 

The operations described in this section are known as classical least-squares (CLS) regression, or sometimes the K-matrix method of quantitative analysis. 

What is meant by Equation (7.9) is a modeling of the observed spectra (A) by using the concentrations (C) and the component spectra (K). The target of modeling in Equation (7.9) is the matrix of the observed spectra (A). As this equation is an expansion of the classical form officer s law, it is called the classical least-squares (CIS) regression equation. The term regression means a process of analysis for predicting optimum values. 

ATR-FTIR is a useful analytical tool for multicomponent analysis that employs a mathematical data-treatment process. Also, Carolei and Gutz (2005) have used this technique combined with chemometrics, to determine three surfactants and water simultaneously in shampoo and in liquid soap without either sample dilution or pretreatment. The surfactants analysed were an amphoteric one (cocoamidopropyl betaine), two nonionic ones (coco diethanolamide in shampoo and alkylpolyglucoside in liquid soap), (minor components) and an anionic one (sodium lauryl ether sulfate). Overlapping bands and water absorption were resolved by two multivariate quantification methods classical least squares (CLS) and inverse least squares (ILS) (Massart et al., 1997, 1998). The wave numbers chosen for the calculation process were preferably those of maximum absorption of the minor components. This method can be applied during the production process but not in final product analysis because of interference caused by the fragrance added in the last step (Figure 7.1.2). 

In the analysis of multicomponent systems, there are four different classes of spectral problems [103]. In the first situation, all of the components and their spectra are known, and calibration data are available. In this case, the method of classical least-squares (see below) is appropriate for finding the quantity of each component. When proper calibration is carried out, this approach yields quantitative data for mixtures. In the second situation, the spectra of the components are not known, but the concentrations of the components of interest are known. This situation requires the use of a cross-correlation procedure. In the third situation, none of the components are... 

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