Fitting Levenberg-Marquardt method

Solver for non-linear data fitting tasks. Several examples are based on the fitting tasks already solved by the Newton-Gauss-Levenberg/Marquardt method in the earlier parts of this chapter. 

We perform nonlinear fitting using the Levenberg-Marquardt method implemented in the MRQMIN routine [75], From the experimental end, eight families of data are involved, namely, x (T) and x jT) at four frequencies, taken from Ref. 64. From the theory end, we employ formulas (4.121)-(4.124) with the numerical dispersion factors. The results of fitting are presented in Figures 4.6 and 4.7 and Table I. 

Nonlinear least-squares fitting in the time domain (Levenberg-Marquardt method)... 

The procedure was tested on simulated time domain MRS data where the model data consisted of metabolite peaks at 3.2, 3.0 and 2.0 ppm representing choline, creatine and IV-acetylaspartate (NAA) respectively, with corresponding values of Ak of 1.0, 1.0 and 3.0 units.89 White noise of specified standard deviation, crt, was then added. The Levenberg-Marquardt method requires suitable initial values for each of the nine parameters being fitted. The initial values of the three frequencies were taken as their known values. An exponentially decaying curve with a constant offset parameter was fitted, using a nonlinear least-squares fit, to the envelope of the free induction decay, Mv(t), in order to obtain an initial value for T and for the amplitudes, each of which was taken to be one-third of the amplitude of the envelope. The constant offset was added to account for the presence of the noise. 

In the case of complex stoichiometries, and when several complexes can coexist in solution, data must be processed using several wavelengths simultaneously. This requires specific software. For instance, the commercially available SPEC FIT Global Analysis System (V3.0 for 32-bit Window Systems) deserves attention. This software uses singular value decomposition and nonlinear regression modeling by the Levenberg-Marquardt method [8]. 

Figure 3.8 Example of parameter redundancy in nonlinear models. Symbols were generated using the model given by Eq. (3.97). Solid line is the predicted fit using Eq. (3.98). The biexponential model predicted values cannot be distinguished from data generated using a triexponential equation. Starting values were 10,1,10, and 0.25. Model was fit using the Levenberg— Marquardt method within the NLIN procedure in SAS.

The Local Least Squares algorithm performs an independent fit for each individual peak. The calculation is thereby restricted to the range around the band maximum. This drastically reduces the amount of data required for the calculation, enhancing the speed compared to the Levenberg-Marquardt method. Some loss of precision versus the Levenberg-Marquardt method occurs. The Local Least Squares algorithm has some conditions ... 

The data from spin-lattiee relaxation time measurements were fitted using the Levenberg-Marquardt method according to the monoexponential function presented in Equation (1). 

Relative fluorescence output versus scan number was fit to one or more exponentials using the Levenberg-Marquardt method. The results are presented in Figure 12-14. 

Figure 3 shows the dissolution kinetics and Table 4 summarises the dissolution parameters derived from the rapid and slow dissolution phases by fitting equation 2 to the experimental data using a non linear least squares fitting routine based on the Levenberg Marquardt method. Table 5 summarises the radiochemistry results with imcertainties. 

The fitting procedure In this block the differences between experimental and theoretical data are minimized. A weighted least squares cost function is formulated. The Gauss-Newton and Levenberg-Marquardt method are implemented to minimize this cost fxmction and eventually provide the parameter values which best describe the data. Moreover, the standard deviations of the estimated parameters are also calculated. 

Step 2 Non-linear regression (Levenberg-Marquardt method) of the correct solution (eq. 10) holding tp and Pp constant at 0. In order to fit the data by a non-linear regression, estimates of the parameters b and have to be supplied. The estimate for the parameter b is computed by means of eq. 9 assuming an... 

We have fitted the adsorption data to the Dubinin - Radushkevich - equation /ll/, using the nonlinear Levenberg - Marquardt method... 

Fitting in the frequency domain is readily visualized graphically and the well-established Levenberg-Marquardt method85 is straightforward to implement. This method is applicable to any lineshape function. For example, Marshall et al 2 have used V ( f), an approximation to the Voigt function, when fitting... 

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. 

In order to fit experimental data, a standard Levenberg-Marquardt (LM) method for minimizing is usually performed. This method has two main drawbacks ... 

The differential equations are solved by the BDF method. For the fitness criterion we use a (weighted) least squares sum, which is minimised by a Levenberg-Marquardt algorithm. 

Instead of using standard non-linear fitting routines (such as the steepest descendent or the Levenberg-Marquardt routine [31 ]), in the speeial ease of the Laplacian curve it appears more convenient to adopt substantially the same approach as earlier used by Maze and Burnet [21, 32], which can be defined as an iterative linear method. 

The Arrhenius form of the parameters are used, i.e., ki=Aie, Ki=Aje, and the exponent % were estimated using the nonlinear regression software package SAS (Statistical Analysis Software). The Proc Model (with Marquardt-Levenberg method) and Fit Procedures in SAS were used for this purpose. The results are shown in Table 1 and Table 2. 

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