Least-Squares Fitting Methods

In Section 9.1 we saw that Beer s law states that absorbance at any wavenumber is a linear function of concentration for a single component. Errors in the measurement 

Despite the fact that we can readily calculate concentration, it is instructive to understand the operations that take place to generate the final answer. These operations form the basis of the multivariate methods of analysis that are used to analyze multicomponent mixtures. 

The basic mathematics for linear least-squares fitting can be found in most elementary texts on data handling, statistics, or quantitative analysis. The regression coefficient, m, or simply the slope of the line, can be calculated rather directly from the individual absorbance and concentration data. Once the regression coefficient is known, the intercept, d, can be calculated from the average absorbance and the average concentration values as well as the regression coefficient. 

After the curve has been calculated, it is beneficial to know the goodness of the fit, which is usually measured, but not always correctly, as the correlation 

A correlation coefficient of unity indicates that all the data fell exactly on the fine. values lower than unity indicate deviation from the line, and the lower the correlation coefficient, the greater the deviations of the point. Note that a value of 1 can arise either from noisy linear data or from nonlinear data. A correlation coefficient of zero indicates that the data were truly random and there is no correlation between the data and the curve of best fit. Of course, R values greater than 0.99 are desirable and indicate a good fit for the data. As R decreases the fit becomes worse and calculations based on the curve are likely to have larger errors. 

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The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

The system of equations (21.4) may be solved by the least squares fitting method, i.e. by looking for such values of unknowns, for which sum S of the differences squared between the left and right parts of each equation (21.4) are minimal ... 

The idea of expansion of physical quantities in powers of Z may also be applied to the semi-empirical parameters of the least squares fitting method as well as to the approximation of their theoretical, e.g. Hartree-... 

If the analysis of a dynamic NMR spectrum is carried out by an iterative least-squares fitting method, the results are accompanied by estimates of the errors. These are proportional to the square root of the sum of the squares of the deviations of the theoretical spectrum from the experimental one, as well as to the sensitivity of the sum to changes in the value of the parameter considered within the region where the sum attains a minimum. These estimates constitute a measure of the effects, on the resulting parameter values, of random errors. They do not include any effects due to systematic errors such as those involved in the assumed values of certain parameters. Moreover, because of the nonlinearity of the least-squares fitting procedure employed, estimates of the errors have only an approximate statistical significance (Section IV.B.2 and reference 67). 

Molecular geometry, namely, each atomic position ry in the unit cell, is conventionally determined by the least-squares fitting method based on the residue between IF qI and FC. A simple least-squares technique used in the conventional crystal structure analysis gives a slightly biased atomic position, because of the aspherical electron-density distribution caused by the chemical environment (15, 18). To obtain the unbiased deformation density, the atomic positions are commonly de-... 

The other values of k thus obtained are presented in the last column of Table 1.1. The mean value of k found from the time dependence of the total thickness of the AlSb layer by the least-squares fit method is equal to... 

The rapid development of computer technology has yielded powerful tools that make it possible for modem EIS analysis software not only to optimize an equivalent circuit, but also to produce much more reliable system parameters. For most EIS data analysis software, a non-linear least squares fitting method, developed by Marquardt and Levenberg, is commonly used. The NLLS Levenberg-Marquardt algorithm has become the basic engine of several data analysis programs. 

From these data it is suspected that the molecules of the solvate structure of lithium ion might be largely effected by the solvent molecules. Since the solubility of some lithium salts is relatively high in MN-dimethyl formamide (DMF), concentrated solutions can also be examined. In a previous study the solvate structure of lithium has been described in an 1.5 mol dm LiNCS solution in DMF [38]. A new XD measurement has been carried out for a LiCl solution of the same concentration. Table 1 hows the structural parameters for the lithium solvates in both solutions. The structural parameters were determined by a least-squares fitting method (LSQ). After the subtraction of the contributions ascribed to the intramolecular stmcture of the DMF molecules and to the assumed structure around the anions from the total structure function of the solution, the resulted difference curve was approximated by calculated model curves. The result is shown in Figure 1. 

FIGURE 3.10. (a) Time courses of the interfacial tension when hydrophilic anions (Cl", Br and I ) were in the oil phase. The concentration of these anions was 1.0 x 10 M. (b) Dependence of the desorption rates (ln(l/Ar) on the standard firee energy of ion transfer from the W/NB interface to the water phase (AGi w = AGo-w/2). The solid straight line with slope -l/RT (R gas constant 8.314 J K mor , T temperature 298 K) was obtained by the least-square fitting method. 

The least squares fitting method overcomes limitations of the peak finding method, but introduces some errors of its own. It is not as sensitive to the scan rate, as long as S >> 2/P (the Nyquist frequency), and its accuracy increases with increasing number of data points. It was found empirically (8) that the error was reduced if an integral number of cycles was used in the analysis. Some error is introduced due to the fact that first and second derivatives of the raw data have to be taken. 

Determination of a and 1 Gaussian Nonlinear Least Squares Fitting Method Given n couples of points (ti, C,) and (tif qi), let s minimize... 

Third, comparisons of the shape were carried out among the substructures with a least square fitting method (19). All the analyses described above were done by using the ACACS (Advanced Computer Aided Chemistry System), which has been developed through the joint cooperation of this company, Sumitomo Pharmaceuticals, and NEC (20). 

Determine the diffusion coefficients by fitting the MSD plots with a linear least square fitting method. 

Where do, fli and k (exponent of the attenuation factor) are obtained from measurements (Zimmermann Dostert, 2002). Their values are determined using a least-squares fitting method. The obtained attenuation parameters values are k = 0.75, do = 3.11.10 4 ttr, di = 1.76.10 and the phase is easily deducted based on L and C values. 

The least square fitting method was applied to determine the constants of the above functions from the experimental data. 

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