Least-squares procedures, weighted

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

Ej is determined by the weights (through Oj which is a function of NET, see eq. (44.8)). Note that this error is in fact the same as the error term used in a usual least squares 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. 

Use of Multiple-Curve and Weighted Least-Squares Procedures with Confidence Band Statistics... 

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

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. 

As part of a study of biscyclic dibenzacridines, the crystal structure of 1,2-8,9-dibenzacridine (72) has been investigated by Mason (1957, 1960). The space group was shown to be Pna2v thus, with four molecules in the unit cell, no molecular symmetry is required. The structure was determined from an examination of the weighted reciprocal lattice and trial and error methods, refinement being by two-dimensional least-squares procedures. At the conclusion of the analysis,... 

Assuming that we have measured a series of concentrations over time/ we can define a model structure and obtain initial estimates of the model parameters. The objective is to determine an estimate of the parameters (CLe, Vd) such that the differences between the observed and predicted concentrations are comparatively small. Three of the most commonly used criteria for obtaining a best fit of the model to the data are ordinary least squares (OLS)/ weighted least squares (WLS)/ and extended least squares (ELS) ELS is a maximum likelihood procedure. These criteria are achieved by minimizing the following quantities/... 

Avdeef has recently reported the refinement of partition coefficients and ionization constants of multi-protic substances based on a generalized, weighted, non-linear least-squares procedure and pH titration curve. This method allows for the determination of pKa and logP values of multiprotic substances with fairly close ionization constants. 

A different modification was proposed by Cammarata and Yau (213). They arbitrarily assigned the observed biological activity of the unsubstituted compound to be the constant term. By this they assumed that there is no experimental error in determining the biological activity of the unsubstituted compound. However, since all the measured values contain experimental error, the weights given to the different activity contributions,, obtained from a least-squares procedure will not be equal. JThis is inconsistent with the definition of the least-squares method (214). 

The physical properties determined using the ECD are important to different areas of chemistry. Analytical chemistry deals with how much and what are involved in a chemical reaction. Expressed differently, it establishes what we refer to as the QQQ quantitation, qualitative identification, and the quality of the results. The determination of the electron affinities of the chlorinated biphenyls, dioxins, and phenols and the prediction of the response of the ECD and NIMS are important to qualitative and quantitative analyses of environmental pollutants [21]. Polarographic reduction in solutions likewise gives accurate and precise qualitative and quantitative results. The quality of the analyses is expressed by the random and systematic uncertainties in the reported values. These are obtained from the same principle of weighted least squares used to obtain information from ECD data. Wentworth has described the application of the general least-squares procedure to chemical problems [22, 23]. 

Vi being the variance of observation i. More details about weighted least-squares procedures can be found in the monograph by Bevington and Robinson [1]. 

Several of the procedures for deriving structural parameters from moments of inertia make use of the method of least squares. Since the relation between moments of inertia and Cartesian coordinates or internal coordinates is nonlinear, an iterative least squares procedure must be used.18 In this procedure an initial estimate of the structural parameters is made and derivatives of the n moments of inertia with respect to each of the k coordinates are calculated based on this estimate. These derivatives make up a matrix D with n rows and k columns. We then define a vector X to be the changes in the k coordinates and a vector B to be the differences between the experimental moments and the calculated moments. We also define a weight matrix W to be the inverse of the ma-... 

In most real experiments, some measurements are made with greater precision than others (strong unblended lines versus weak or blended lines a combined fit to microwave and optical data). A weighted least-squares procedure is appropriate. Typically, each measurement is weighted by the square of the reciprocal of its estimated uncertainty. 

If the values of the original dependent variable have equal expected errors, an unweighted least-squares fit is appropriate if we use that variable in our procedure. However, if we take a function of the original variable in order to use a linear fit, then the original expected errors, which are all equal, will not generally produce equal errors in the new variable, and the weighted least-squares procedure is preferred. 

Ten ml of 1 mM or 5 mM aqueous solutions of the samples were pre-acidified to pH 1.8-2.0 with 0.5 M HCl, and were then titrated alkalimetrically to some appropriate high pH (maximum 12.5). The titrations were carried out at 25.0 0.1 °C, at I = 0.1 M ionic strength using NaCl, and under N2 atmosphere. The initial estimates of pJC values were obtained by difference plots (nH vs. pH, where nn is the average number of bound protons) and were then refined by a weighted non linear 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 pK values along with the standard deviations were calculated."... 

Figure 17 is based on series of conductance measurements on LiClO in PC/DME mixtures as a function of solvent composition (weight % of PC = ), electrolyte concentration m, and temperature 0. For every solvent composition 4, a x-m-function, see Fig. 7, was established by a least-squares procedure. The maximum specific conductances from these fimctions, x , are then plotted in Fig. 17 as a function of solvent composition and temperature. Figures 18 a and b show the viscosity and the relative permittivity of the mixed solvent as a function of the same parameters. The conductance behaviour of the LiClO /PC/DME-system can be understood from the competition between the solvent viscosity, ion solvation and ion aggregation to ion pairs, triple ions and higher aggregates A comprehensive study of this... 

Correlation relationships. A standard least-squares procedure (Kuo, 1965) was used to establish the correlation relationships among the geometric and vibrational parameters. The geometric data were used with weights inversely proportional with the e qperimental errors reported in the original studies. The vibrational data were given equal weights. 

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