Least-squared method

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

In steady-state problems 6/S.l = 1 and the time-dependent term in the residual is eliminated. The steady-state scheme will hence be equivalent to the combination of Galerkin and least-squares methods. 

For a specified mean and standard deviation the number of degrees of freedom for a one-dimensional distribution (see sections on the least squares method and least squares minimization) of n data is (n — 1). This is because, given p and a, for n > 1 (say a half-dozen or more points), the first datum can have any value, the second datum can have any value, and so on, up to n — 1. When we come to find the... 

Goodman, T.P., A Least-Squares Method for Computing Balance Corrections, ASME Paper No. 63-WA-295. 

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

Example 10. Calculate by the least squares method the equation of the best straight line for the calibration curve given in the previous example. 

Brown, C.W., "Classical and Inverse Least-Squares Methods in Quantitative Spectral Analysis", Spectrosc. 1986 (1) 23-37. 

Haaland, D.M. "Classical versus Inverse Least-Squares Methods in Quantitative Spectral Analyses", Spectrosc. 1987 (2) 56-57. 

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. 

Lindberg, W., et al. "Partial Least Squares Method for Spectrofluorimetric Analysis of Mixtures of Humic Acid and Ligninsulfonate , Anal. Chem. 1983, (55)643-648. 

Haaland, D.M., Thomas, E.V., "Partial Least-Squares Methods for Spectral Analysis 1. Relation to Other Quantitative Calibration Methods and the Extraction of Qualitative Information" Anal. Chem. 1988 (60) 1193-1202. 

Data can be fit to this equation by the nonlinear least-squares method. As it turns out, the Guggenheim approach for first-order kinetics is valid, even though the reaction... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Kinetic curves were analyzed and the further correlations were determined with a nonlinear least-square-method PC program, working with the Gauss-Newton method. 

This system of i + 2 equations is nonlinear, and for this reason probably has not received attention in the least-squares method (207). We are able to give an explicit solution (163) for the particular case when Xy = xj and m,- = m for all values of i that is, when all reactions of the series are studied at a set of temperatures, not necessarily equidistant, but the same for all reactions. Let us introduce... 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

Pigure 8.8a and b, respectively, show fluorescence autocorrelation curves of R6G in ethylene glycol and R123 in water at 294.4 K. The solid lines in these traces are curves analyzed by the nonlinear least square method with Eq. (8.1). Residuals plotted on top of the traces clearly indicate that the experimental results were well reproduced by the... 

So far no hypotheses are required concerning the true shape of the peak profile. Flowever, in order to avoid or reduce the difficulties related to the overlapping of the peaks, the experimental noise, the resolution of the data and the separation peak-background, the approach most frequently used fits by means of a least squared method the diffraction peaks using some suitable functions that allow the analytical Fourier transform, as, for example, Voigt or pseudo-Voigt functions (4) which are the more often used. 

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