Classical least-squares regression method

Ragno G, Vetuschi C, Risoli A et al (2003) Application of a classical least-squares regression method to the assay of 1, 4-dihydropyridine antihypertensives and their photoproducts. Talanta 59 375-382... 

Haaland et al. [91] developed a so-called multi-window classical least-squares method for ICP-OES measurements [charge-couple device (CCD) detector arrays]. Essentially, it consisted in performing a classical least-squares regression in each of the spectral windows which were measured and combining the concentration predictions (for a given analyte). The methodology was compared with PLS and it proved superior and capable of handling interferences from several concomitants. 

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 classical least-squares method for multiple linear regression (MLR) to estimate G minimizes the sum of the squared residuals. Formally, this can be written as... 

Multivariate techniques are inverse calibration methods. In normal least-squares methods, often called classical least-squares methods, the system response is modeled as a function of analyte concentration. In inverse methods, the concentrations are treated as functions of the responses. The latter has some advantages in that concentrations can be accurately predicted even in the presence of chemical and physical sources of interference. In classical methods, all components in the system need to be considered in the mathematical model produced (regression equation). 

The fourth-derivative spectra of molybdenum complexes of tetramethyldithiocarbamate (tiram) fungicide were used for its quantification in commercial samples and in wheat grains [41], Atrazine and cyanazine were assayed in food samples by first- derivative spectrophotometry [42]. In order to improve results of assay, the first-derivative spectra of the binary mixture were subjected to chemometiic treatment (classical least squares, CLS principal component regression, PCR and p>artial least squares, PLS). A combination of first-derivative with PCR and PLS models were applied for determination of both herbicides in biological samples [42]. A first-derivative spectrophotometry was used as a reference method for simultaneous determination BriUant Blue, Simset Yellow and Tartrazine in food [43]. 

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. 

Studies on binary mixture samples frequently deal with classical least-squares, inverse least-squares, principal component regression and partial least-squares methods. These methods have been used for resolving mixtures of hydrochlorothiazide and spironolactone in tablets cyproterone acetate and estradiol valerate amiloride and hydrochlorothiazide ... 

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. 

Typically, NIR spectra contain bands that arise from OH, CH, and NH groups in a sample. NIR bands of individual analytes are generally broader and heavily overlapped than bands are in MIR spectra. As a result, it is difficult to determine an analyte concentration from only one or a few NIR absorbances. Nonetheless, full-spectrum multivariate calibration methods, such as partial least squares (PLSs), classical least squares (CLS), and principal components regression (PCR) have been used to accurately determine analyte concentrations from NIR spectra. 

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

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... 

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. ... 

Principal Component Regression, PCR, and Partial Least Squares, PLS, are the most widely known and applied chemometrics methods. This is particularly the case for PLS, for which there is a tremendous number of applications and a never-ending stream of proposed improvements. The details of these latest modifications are not within the scope of this book and we concentrate on the essential, classical aspects. 

For only one x-variable and one y-variable—simple x/y-regression—the basic equations are summarized in Section 4.3.1. Ordinary least-squares (OLS) regression is the classical method, but also a number of robust... 

Decision diamond Are the classical assumptions for fitting regression lines met N0 Clearly the measurements at the different x-levels differ in their variability. This can be shown by using the F-test. Another method is outlined in another chapter of this text (10). In this case weighted least squares will resolve the problem of heteroscedasticity or unequal variance across the graph. I have chosen weights of 1, 1, 0.1, 0.01 and 0.01 for the resolution of this problem. 

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. 

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