Least squares applications

Experimental Methods and Analysis of Kinetic Data Method of Least Squares Application to Specific Reactors Reactions of Complex Mechanism... 

Application. Merriman ( The Method of Least Squares Applied to a Hydraulic Problem, y, Franklin Inst., 23.3-241, October 1877) reported on a study of stream velocity as a function of relative depth of the stream. 

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

When evaluating gas concentrations in practical applications, a reterence spectrum is least squares fitted to the received absorption spectrum. This im proves the system accuracy, since the spectral fingerprint over the whole scanning range contributes to the result.- ... 

Figure 13-9. Timc-intcgralcd light output versus pulse length at various applied voltages. The lines arc least-square fils and their extrapolation yields the lime delay t,. Inset shows the liming between the application of a voltage pulse and the observed eleclroluniineseencc. Reproduced with permission from I22. Copyright 1998 by the American Physical Society.

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 ... 

Haaland, D.M., et.al. "Application of New Least-squares Methods for the Quantitative Infrared Analysis of Multicomponent Samples", Appl. Spec. 1982 (36) 665-673. 

Haaland, D.M. et.al. "Improved Sensitivity of Infrared Spectroscopy by the Application of Least Squares Methods", Appl. Spec. 1980 (34) 539-548. 

Rao, G.R., et. al. "Factor Analysis and Least-Squares Curve- Fitting of Infrared Spectra An Application to the Study of Phase Transitions in Organic Molecules", Appl. Spec. 1984, (38) 795-803. 

Haaland, D.M., Thomas, E.V., "Partial Least-Squares Methods for Spectral Analysis 2. Application to Simulated and Glass Spectral Data" Anal. Chem. 

This approach was applied to data obtained by Hausberger, Atwood, and Knight (17). Figure 9 shows the basic temperature profile and feed gas data and the derived composition profiles. Application of the Hougen and Watson approach (16) and the method of least squares to the calculated profiles in Figure 9 gave the following methane rate equation ... 

Statistical methods used in kinetic analyses have generally been based on a least-squares treatment. Reed and Theriault [494] have considered the application of this approach to data which obeys the first-order... 

Nowadays one very often fits data by means of a least-squares computer program. (Indeed, in Chapter 2 some data-fitting applications will be considered.) It is important to examine the residuals from the fits, to confirm that, with time, they lie randomly about the zero line. Examining them makes it is much easier to spot discrepancies. If the kinetic data are acquired with a computer-controlled instrument, then the data are already contained in a file that can be read by the fitting program. 

The method of least squares provides the most powerful and useful procedure for fitting data. Among other applications in kinetics, least squares is used to calculate rate constants from concentration-time data and to calculate other rate constants from the set of -concentration values, such as those depicted in Fig. 2-8. If the function is linear in the parameters, the application is called linear least-squares regression. The more general but more complicated method is nonlinear least-squares regression. These are examples of linear and nonlinear equations ... 

However, an error was made in the application of the temperature factor, which resulted in incorrect weights therefore the structure factors and their derivatives were recalculated on the basis of these parameters and a second least-squares treatment was carried out as described below. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Bennett KP, Embrechts MJ. An optimization perspective on kernel partial least squares regression. In Suykens JAK, Horvath G, Basu S, Micchelli J, Vandewalle J, editors. Advances in learning theory methods, models and applications. Amsterdam lOS Press, 2003. p. 227-50. 

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. ... 

The total residual sum of squares, taken over all elements of E, achieves its minimum when each column Cj separately has minimum sum of squares. The latter occurs if each (univariate) column of Y is fitted by X in the least-squares way. Consequently, the least-squares minimization of E is obtained if each separate dependent variable is fitted by multiple regression on X. In other words the multivariate regression analysis is essentially identical to a set of univariate regressions. Thus, from a methodological point of view nothing new is added and we may refer to Chapter 10 for a more thorough discussion of theory and application of multiple regression. 

However, our preoccupation is with the opposite application given a newly measured spectrum y , what is the most likely mixture composition and, how precise is the estimate Thus, eq. (36.2) is necessary for a proper estimation of the parameters B, but we have to invert the relation y =fix) = xB into, say, x = g y) for the purpose of making future predictions about x (concentration) given y (spectrum). We will treat this case of controlled calibration using classical least squares (CLS) estimation in Section 36.2.1. 

The model of eq. (36.3) has the considerable advantage that X, the quantity of interest, now is treated as depending on Y. Given the model, it can be estimated directly from Y, which is precisely what is required in future application. For this reason one has also employed model (36.3) to the controlled calibration situation. This case of inverse calibration via Inverse Least Squares (ILS) estimation will be treated in Section 36.2.3 and has been treated in Section 8.2.6 for the case of simple straight line regression. 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

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