Linear least-squares fitting methods

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. 

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

Following the determination of the hydrolytic rate constants of an epimeric pair of tricyclic nitriles (78) by a non-linear, least-squares fitting method, a novel eigenvalue-eigenvector analysis of the sensitivity coefficients permitted maximization of the kinetic dataJ ... 

A Fortran code based on a gradient searching non-linear least square fitting method (55) was deveioped and used to fit SANS intensities in an absolute scale. Quality of the fits is uniformly excellent for Plutonic micellar systems in their entire range of the diswdered micellar phases. As an illustration, foe calculated particle structure factor P(k), the inter-particle structure factor S(k) and foe absolute intensity distribution I(k) from a typical fit is shown in Figure 3 together with foe measured one. 

We shall not treat the methods of fitting nonlinear equations, those that are not linear in the parameters, in detail, but we shall remind the reader that nonlinear least squares does not lead to a closed solution for the parameters, as in linear least squares. The method of nonlinear least squares requires a set of tentative values of the parameters, followed by an iterative process that is stopped when successive results are close... 

Four methods of performing this calculation have been proposed, the tracer property, linear programming, ordinary linear least squares fitting, and effective variance least squares fitting. 

One way of linearizing the problem is to use the method of least squares in an iterative linear differential correction technique (McCalla, 1967). This approach has been used by Taylor et al. (1980) to solve the problem of modeling two-dimensional electrophoresis gel separations of protein mixtures. One may also treat the components—in the present case spectral lines—one at a time, approximating each by a linear least-squares fit. Once fitted, a component may be subtracted from the data, the next component fitted, and so forth. To refine the overall fit, individual components may be added separately back to the data, refitted, and again removed. This approach is the basis of the CLEAN algorithm that is employed to remove antenna-pattern sidelobes in radio-astronomy imagery (Hogbom, 1974) and is also the basis of a method that may be used to deal with other two-dimensional problems (Lutin et al., 1978 Jansson et al, 1983). 

For consecutive or parallel electrode reactions it is logical to construct circuits based on the Randles circuit, but with more components. Figure 11.16 shows a simulation of a two-step electrode reaction, with strongly adsorbed intermediate, in the absence of mass transport control. When the combinations are more complex it is indispensable to resort to digital simulation so that the values of the components in the simulation can be optimized, generally using a non-linear least squares method (complex non-linear least squares fitting). 

If the function may be made linear with respect to its unknown parameters by a suitable transformation, then it may be fitted by the Linearized Least Squares method (10) so as to minimize the root mean square error in the original (untransformed) space. The essence of this technique is to use weighted (linear) least squares to effect a non-linear least squares fit. Assume that the equation has been transformed into an equal variance space and let... 

Figure 3.4 illustrates two lifetime spectra collected by methods similar to those outlined above, (a) exhibits the non-exponential shoulder region associated with the annihilation of non-thermalised positrons. After thermalisation (essentially at time zero for condensed matter) the spectra are sums of exponential components associated with each decay mode, and a background component B, A] = 2 A, exp(-Ajt,) + B. For long lifetime components (> Ins) each X can be extracted by non-linear least squares fitting. For short X values characteristic of condensed matter, however, a... 

In anal3rtical chemistry, developii a calibration curve or modelling a phenomenon often requires the use of a mathematical fitting procedure. Probably the most familiar of these procedures is linear least-squares fitting [1]. Criteria other than least-squares for defining the best fit have been developed for linear parameters when the data possibly contain outliers [2,3]. Sometimes, the model equation to be fit is nonlinear in the parameters. This requires appeal to other fitting methods [4]. 

Non-linear least-squares fitting by the Marquardt method [19,20] appears to be the most commonly used technique for hiexponential fluorescence decay analysis, at least for a time-domain measurement such as used here [21,22]. Fitting by this method requires evaluation of the derivatives of the model equation (Equation... 

The RATIO method table (Table I) includes provision for specifying upper and lower limits of integration for both primary and reference bands with the peak area evaluation procedure. The practical limits of the integration can be determined empirically by evaluating a set of spectra stored on microfloppy disks with varying limits set in the appropriate locations in the method table. Optimum limits can be determined from the calibration plots and related error parameters. The calibration plots shown in Figures 4 and 5 indicate that both evaluation procedures, peak height and peak area provide essentially the same level of precision for the linear least squares fit of the data. The error index and correlation coefficients listed on each table are both indicators of the relative scatter in the data from the least squares fit line. The correlation coefficient is calculated as traditionally defined in statistics. 

The continuum model has been applied to an experimental study of the solvent effect on the 6-chloro-2-hydroxypyridine/6-chloro-2-pyridone equilibrium in a variety of essentially non-hydrogen-bonding solvents (Beak et al., 1980). In this study, a plot of log A nh/oh) versus (e - 1)/ (2e + 1), the solvent dielectric term, yielded a linear least-squares fit with a slope of 2.5 0.2, an intercept of -1.71, and a correlation coefficient of 0.9944. This result was used to estimate the gas phase free-energy difference of 9.2 kJ mole-1, which compares favorably with the observed value of 8.8 kJ mole-1 for this system. The authors also reported that alcohol solvents are correlated fairly well in this study but that other solvents seem to be divided into two classes, those that are electron-pair donors and those that are electron-pair acceptors in a hydrogen bond. The hydrogen bonding effect is assumed to be independent from the reaction field effect and is included in the continuum model by means of the Kamlet and Taft (1976) empirical parameters. The interested reader is referred to the original paper for a detailed discussion of the method and its application. 

---