Classical least-squares regression

MLR is based on classical least squares regression. Since known samples of things like wheat cannot be prepared, some changes, demanded by statistics, must be made. In a Beer s law plot, common in calibration of UV and other solution-based tests, the equation for a straight line... 

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. 

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... 

To obtain adequate statistics, the calibration range should be established by at least 10 independently prepared solutions. If a linear calibration model is assumed and the parameters of the model (slope and intercept) are determined by classical least-squares regression, the assumptions of the regression must hold. Namely ... 

With classical least-squares regression we were able to increase the number of absorbances (dependent variable) essentially without limit, but here the absorbances are the independent variable and the number of absorbances cannot exceed the number of calibration spectra. Consequently, the wavenumbers at which the absorbance is measured should be picked with care to ensure that the absorbances at those wave-number positions are reflective of the overall contribution of those components to the spectrum. An examination of Eqs. 9.15 shows that there must be at least one analytical wavenumber for each component. Equations 9.15 represent two components consequently, there are two analytical wavenumbers, vi and V2. 

Beer s Law for a Multicomponent System CLS (Classical Least Squares) Regression... 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

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... 

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. 

If a calibration function is used with coefficients obtained by fitting the response of an instrument to the model in known concentrations of calibration standards, then the uncertainty of this procedure must be taken into account. A classical least squares linear regression, the default regression... 

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... 

Let e, be the rth residual in the ordinary least squares regression of y on X in the classical regression model and let s, be the corresponding true disturbance. Prove that plim(e, - e,) = 0. 

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... 

PARTIAL LEAST-SQUARES REGRESSION 6.7.1 Classical PLSR... 

Partial Least Squares regression (PLS) is usually performed on a - data matrix to search for a correlation between the thousands of CoMFA descriptors and biological response. However, usually after - variable selection, the PLS model is transformed into and presented as a multiple regression equation to allow comparison with classical QSAR models. 

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]. 

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