Least-squares analysis univariate

This method measures total unsaturation in a sample by the multivariate analysis of Fourier transform infrared spectra. It correlates the absorbance in the spectral regions corresponding to two major types of unsaturation with their concentrations. This is an extension of univariate least squares analysis that correlates a single band absorbance height or area with concentration. 

Lagtime, 75 Laplace transform, 82 Larmor precessional frequency, 155, 165 Laser pulse absorption, 144 Lattice energy, 403 Law of mass action, 60, 125 Least-squares analysis linear, 41 nonlinear, 49 univariate, 44 unweighted, 44, 51 weighted, 46, 51, 247 Leaving group, 9, 340, 349, 357 Lennard-Jones potential, 393 Lewis acid-base adduct, 425 Lewis acid catalysis, 265 Lewis acidity, 426... 

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 ... 

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. 

Traditionally, the determination of a difference in costs between groups has been made using the Student s r-test or analysis of variance (ANOVA) (univariate analysis) and ordinary least-squares regression (multivariable analysis). The recent proposal of the generalized linear model promises to improve the predictive power of multivariable analyses. 

Raman spectroscopy can also directly benefit TE analysis by non-invasively monitoring the growth and development of ECM by different cells on a multitude of scaffold materials exposed to various stimuli (e.g. growth factors, mechanical forces and/or oxygen pressures). Indeed the non-invasive nature of Raman spectroscopy enables the determination of the rate of ECM formation and the biochemical constituents of the ECM formed. Univariate (peak area, peak ratios, etc.) and multivariate analytical techniques (e.g. principal component analysis (PCA)) can be used to determine if there are any significant differences between the ECM formed on various scaffolds and/or cultured with different environmental parameters, and what these biochemical differences are. Least square (LS) modelling, for example, could allow the quantification of the relative components of the ECM formed (Fig. 18.3) [4, 38],... 

Univariate and multivariate spectroscopy was applied to the analysis of spironolactone in presence of chlorthalidone [17]. Satisfactory results were obtained by partial least squares regression, with the calibration curve being linear over the range 2.92 -14.6 pg/mL. A kinetic-spectrophotometric method was described for the determination of spironolactone and canrenone in urine that also used a partial least-squares regression method [18]. After the compounds were extracted from urine, the spectra were recorded at 400 - 520 nm for 10 minutes at 30 second intervals. The relative error was less than 5%. 

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... 

Examination of the univariate distribution of 5-FU clearance revealed it to be skewed and not normally distributed suggesting that any regression analysis based on least squares will be plagued by non-normally distributed residuals. Hence, Ln-transformed 5-FU clearance was used as the dependent variable in the analyses. Prior to analysis, age was standardized to 60 years old, BSA was standardized to 1.83 m2, and dose was standardized to 1000 mg. A p-value less than 0.05 was considered to be statistically significant. The results from the simple linear regressions of the data (Table 2.4) revealed that sex, 5-FU dose, and presence or absence of MTX were statistically significant. 

Two fundamentally different statistical approaches to biomarker selection are possible. With the first, experimental data can be used to construct multivariate statistical models of increasing complexity and predictive power - well-known examples are Partial Least Square Discriminant Analysis (PLS-DA) (Barker Rayens, 2003 Kemsley, 1996 Szymanska et al., 2011) or Principal Component Linear Discriminant Analysis (PC-LDA) (Smit et al., 2007 Werf et al., 2006). Inspection of the model coefficients then should point to those variables that are important for class discrimination. As an alternative, univariate statistical tests can be... 

J. M. Andrade, G. Cristoforetti, S. Legnaioli, G. Lorenzetti, V. Palleschi and A. A. Shaltout, Classical univariate calibration and partial least squares for quantitative analysis of brass samples by laser-induced breakdown spectroscopy, Spectrochim. Acta, Part B, 2010, 65, 658-663. 

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. 

As far as the quantitative evaluation of vibrational spectra is concerned, IR and NIR spectroscopy follow Beer s law, whereas the Raman intensity JRaman is directly proportional to the concentration of the compound to be determined (Figure i),i iS Si To compensate laser fluctuations, in many cases, quantitative Raman spectroscopy is performed with an internal reference signal in the vicinity of the analytical band. For Raman and IR spectroscopy, quantitative analysis can be performed by either univariate evaluation of band heights/ areas or multivariate evaluation (e.g., partial least-squares (PLS) regression) of large spectral regions. Due to the overlap of many absorption bands for the quantitative analysis of NIR spectra, predominantly multivariate chemometric procedures are applied. For an in-depth study of the precautions, pitfalls, and limitations, which have to be observed or may be encountered in the measurement of vibrational spectra, the reader is referred to the pertinent literature. " ... 

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