Regression analysis classical least squares

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. ... 

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. 

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. 

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... 

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. 

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. 

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. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

What is the equivalent four-parameter linear model expressing y, as a function of jci and xfl Use matrix least squares (regression analysis) to fit this linear model to the data. How are the classical factor effects and the regression factor effects related. Draw the sums of squares and degrees of freedom tree. How many degrees of freedom are there for SS, 55, and SS 7... 

Besides the classical Discriminant Analysis (DA) and the k-Nearest Neighbor (k-NN), other classification methods widely used in QSAR/QSPR studies are SIMCA, Linear Vector Quantization (LVQ), Partial Least Squares-Discriminant Analysis (PLS-DA), Classification and Regression Trees (CART), and Cluster Significance Analysis (CSA), specifically proposed for asymmetric classification in QSAR. 

Under circiim.stances in which the molecular descriptors are highly intercorrelated (e.g., molecular connectivity indices), there are statistical limitations with respect to the use of a classical multiple regression analysis. Such data sets can be satisfactorily treated by the application of principal components regression (PCR) and partial least squares (PLS) methods.Numerous environmental QSAR model.s use... 

Most researchers who have worked with discrete event simulation are familiar with classical statistical analysis. By classical, we mean those tests that deal with assessing differences in means or that perform correlation analysis. Included in these tests are statistic procedmes such as t-tests (paired and unpaired), analysis of variance (univariate and multivariate), factor analysis, linear regression (in its various forms ordinary least squares, LOGIT, PROBIT, and robust regression) and non-parametric tests. 

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