Nonlinear least squares procedures

Processing all peaks for the model by nonlinear least square procedure ... 

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. 

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. 

A convenient way to illustrate the behavior of the model for the example of ZnS deposition is to plot the measured deposition rate, r(d, ZnS), as a function of the incident-flux rate of one element when the incident-flux rate of the second element is fixed. An example is shown in Figure 13 for the deposition of ZnS as a function of the incident-flux rate of sulfur at a substrate temperature of 200 °C. Experimental data points and curves representing the best-fit model predictions are shown for each of four zinc incident-flux rates. A nonlinear least-square procedure was used to obtain the following values for the model parameters that best fit equation 40 to the experimental data 8(Zn) = 0.6-0.7, 8(S) = 0.5-0.7, and K(ZnS) > 1015 cm2-s/ZnS. 

The biexponential rate equation associated with this model was fitted to the experimental data using a nonlinear least squares procedure. Pharmacokinetic constants for the two-compartment model were calculated by standard methods. The fraction amount absorbed as a function of time was estimated by the Loo-Riegelman method using the macroscopic rate constants calculated from the intravenous data. The slope of the linear part of the Loo-Riegelman plot combined with the total amount absorbed (quantitated by depletion analysis of the saturated donor solution) was used to calculate the zero-order rate constant for buccal permeability. 

One of the major accomplishments of the Wentworth group has been the application of the nonlinear least-squares procedure to chemical problems. This has been generally recognized by the fact that two articles on the subject published in the Journal of Chemical Education in 1965 have become classics. The following is taken from the March 31, 1986, Current Contents issue ... 

As graduate students in the Wentworth laboratory, we were all required to use nonlinear least squares initially, sometimes with a mechanical calculator whose best feature was that it could take precise square roots. Later, as we tested the ECD method, we dropped the least-squares procedure for the simpler determination of a slope through straight lines. This was generally correct, but as we learned when the dissociative mechanism of the ECD was included, there was a need to use the complete ECD equation. Hirsch studied the temperature dependence of the ECD and sought new correlations of the electron affinities and hydrogen bond strengths. In order to obtain the thermodynamic parameters for the complexes from the data, a nonlinear least-squares procedure to include data determined by other experiments was developed [51]. This procedure was applied to the ECD data for the multistate model. 

In this chapter the experimental ECD and NIMS procedures for studying the reactions of thermal electrons with molecules and negative ions are described. Gas phase electron affinities and rate constants for thermal electron attachment, electron detachment, anion dissociation, and bond dissociation energies are obtained from ECD and NIMS data. Techniques to test the validity of specific equipment and to identify problems are included. Examples of the data reduction procedure and a method to include other estimates of quantities and their uncertainties in a nonlinear least-squares analysis will be given. The nonlinear least-squares procedure for a simple two-parameter two-variable case is presented in the appendix. 

In order to interpret these values further, it is necessary to have values for the overlap integrals (10). We obtained these as follows An approximate real wavefunction which fits the outer regions of the 5/function well (within 0.4% for af,Slater-type orbitals was fit to the probability amplitude calculated from a relativistic SCF wavefunction. The result is... 

The following instructions are written for Excel 2003 for Windows. If you have a later version or an earlier version of this spreadsheet, there might be small differences in the procedure. There are also small differences in Excel for the Macintosh computer. The Excel spreadsheet will carry out least squares fits in two different ways. You can carry out linear least squares in a worksheet, or you can carry out linear and various nonlinear least squares procedures on a graph. The advantage of... 

Typically, 10 ml of 0.5 to 10 mM solutions of the samples were preacidified to pH 1.8-2.0 with 0.5 M HCl, and were ttien titrated alkalimetrically to some appropriate high pH (maximum 12.0). The titrations were carried out at 25.0 0.1 °C, at constant ionic strength using NaCl, and under an inert gas atmosphere. The initial estimates of pKg values were obtained from Bjerrum difference plots (nH vs. pH) and then were refined by a wei ted nonlinear least-squares procedure (Avdeef, 1992,1993). For each molecule a minimum of three and occasionally five or more separate titrations were performed and the average pXa values along with the standard deviations were calculated."... 

Although a more complicated nonlinear least squares procedure has been described by Tsai and Whitmore [1982] which allows analysis of two arcs with some overlap, approximate analysis of two or more arcs without much overlap does not require this approach and CNLS fitting is more appropriate for one or more arcs with or without appreciable overlap when accurate results are needed. In this section we have discussed some simple methods of obtaining approximate estimates of some equivalent circuit parameters, particularly those related to the common symmetrical depressed arc, the ZARC. An important aspect of material-electrode characterization is the identification of derived parameters with specific physicochemical processes in the system. This matter is discussed in detail in Sections 2.2 and 3.3 and will not be repeated here. Until such identification has been made, however, one cannot relate the parameter estimates, such as Rr, Cr, and y/zc, to specific microscopic quantities of interest such as mobilities, reaction rates, and activation energies. It is this final step, however, yielding estimates of parameters immediately involved in the elemental processes occurring in the electrode-material system, which is the heart of characterization and an important part of IS. 

The three parameters h, and q were derived from mean activity coefficients by nonlinear least-squares procedures. The hydration indexes were found to decrease from 12 for Mg2+ and 11 for Ca2+ to 5.2, 3.8, and 2.5 for Li, Na", and id", respectively. Pan (64) has preferred to fix the ion-size parameter in the hydration equation at the sum of the crystallographic radii of the ions. If this is done, he has found that much larger values of h (about 21, 13, and 7 for the lithium, sodium, and potassium halides, respectively) are obtained from mean activity coefficients. 

An alternate way to determine the parameters A m, and 6 is to fit the data using a nonlinear least-squares procedure. However, all such methods require reasonable initial choices for the parameters so the method outlined in the previous paragraph is a necessary preliminary. 

Langhals (1982a, b) fitted values of (30) for a number of binary mixtures by Equation 2.27, using a nonlinear least-squares procedure. 

Detailed quantitative analysis of amide I and amide II bands is not always an easy task because there is considerable overlapping of a number of bands due to various different secondary structures. The most popular data processing techniques that avoid overlapping in estimation of protein secondary structures involve either conversion of the spectrum to its second derivative or reduction of the width of the bands by Fourier self-deconvolution of the amide I region to a sum of Lorentzian band components (Figure 19.2) with a nonlinear least-squares procedure [18, 33,35]. 

Sometimes it is necessary to try two or more families of functions to determine which gives the best fit. If it is possible to transform different possible formulas into linear form, we can use linear least-squares to test the possibilities. Nonlinear least-squares procedures can also be carried out. If there is no back reaction, the concentration of a single reactant undergoing a first-order chemical reaction is given by. 

For subsequent usage in molecular valence-only calculations compact valence basis sets were generated, i.e., the pseudovalence orbitals were fitted by using a nonlinear least-squares procedure, similar to the one for fitting the potentials, to a linear combination of Gaussian functions... 

The terms G, and 0, are found by simply fitting Eqs. (14.7.9) and (14.7.10) to measured G (co) and G"(co) data using a nonlinear least-squares procedure [18]. In doing so, the choice of N is crucial a small value of N can lead to errors, whereas a large value of N cannot be justified given the normal errors associated with measuring G and G". The set of relaxation times 0, and moduli G, is called... 

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