Weighted nonlinear least squares

The graphically deduced constants are subsequently refined by a weighted nonlinear least squares procedure [472]. Although the potentiometric method can be used in discovery settings to calibrate high-throughput solubility methods and computational procedures, it is too slow for HTS applications. It is more at home in a preformulation lab. 

Method III. The weighted nonlinear least squares parameter optimization procedure (34, 35) was applied to the entire set of points shown in Figure 5 for 0.1 M KNO-j. The value of C was fixed and the optimal values of and aHz were obtained. The model... 

Intensity/wavelength/time cross-sectional diagrams (or time-resolved fluorescence "contour" diagrams) are generated using a weighted nonlinear least squares polynomial surface procedure (20). Area-normalized TRE spectra can be used for convenient pictorial representation, since the absolute emission intensity of individual time-resolved spectra vary substantially with time after excitation. 

The plasma /8-carotene-dg and retinol-d4 concentration-time data were described using an empirical multiexponential description of the data using weighted, nonlinear least squares regression and the SAS NLIN procedure. Each observation was weighted by the reciprocal of its predicted value. 

In the Fisher approach, parameter estimates can be obtained by nonlinear least squares or maximum likelihood together with their precision, such as, a measure of a posteriori or numerical identifiabihty. Details and references on parameter estimation of physiologic system models can be found in Carson et al. [1983] and Landaw and DiStefano [1984]. Weighted nonlinear least squares is mostly used, in which an estimate 9 of the model parameter vector 0 is determined as... 

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

Selected entries from Methods in Enzymology [vol, page(s)] Association constant determination, 259, 444-445 buoyant mass determination, 259, 432-433, 438, 441, 443, 444 cell handling, 259, 436-437 centerpiece selection, 259, 433-434, 436 centrifuge operation, 259, 437-438 concentration distribution, 259, 431 equilibration time, estimation, 259, 438-439 molecular weight calculation, 259, 431-432, 444 nonlinear least-squares analysis of primary data, 259, 449-451 oligomerization state of proteins [determination, 259, 439-441, 443 heterogeneous association, 259, 447-448 reversibility of association, 259, 445-447] optical systems, 259, 434-435 protein denaturants, 259, 439-440 retroviral protease, analysis, 241, 123-124 sample preparation, 259, 435-436 second virial coefficient [determination, 259, 443, 448-449 nonideality contribution, 259, 448-449] sensitivity, 259, 427 stoichiometry of reaction, determination, 259, 444-445 terms and symbols, 259, 429-431 thermodynamic parameter determination, 259, 427, 443-444, 449-451. 

The misnormalized data of Lee et al.16) was interpreted in terms of two discrete relaxation processes. It was proposed that the relaxation function should be represented as the sum of two Williams-Watts functions. The slope at short times was claimed to be equal to the / for the faster of the two processes. Numerical calculations and graphical representations of exact relaxation functions with parameters equal to those reported by Lee et al.16) were carried out. They did not look even qualitatively similar to their reported data. The slope at the shortest times must be related to a weighted sum of both of the /3 values for the sum of two WW functions. If it was desired to fit the data to a sum of two WW functions, then this could easily be carried out with a nonlinear least squares routine. In most cases it would not be possible to obtain statistically independent values of all six parameters, but at least no further errors would be introduced by faulty manipulations of the data. The graphical procedure of Lee et al.16) cannot be recommended as of any worth in this problem. 

Once the phase problem is solved, then the positions of the atoms may he relined by successive structure-factor calculations (Eq. 21 and Fourier summations (Eq. 3) or by a nonlinear least-squares procedure in which one minimizes, for example, )T u ( F , - F,il(, )- with weights w lakcn in a manner appropriate to the experiment. Such a least-squares refinement procedure presupposes that a suitable calculalional model is known. 

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