Complex nonlinear least squares fitting

The numerical values of the impedance parameters, as a function of the applied potential obtained with the complex nonlinear least-square fit program, using for the pore s wall surface the equivalent circuit from Figure 4.5.50, are given in Table 4.5.3 with the corresponding accuracy of the fit. 

Experimental data fitting using the ECM was performed with ZPlot s complex nonlinear least square fitting program. It was confirmed that the experimental data were fitted very well at all four temperatures and potentials, and this demonstrates that the proposed model in Fig. 12.13 is feasible for description of the methanol oxidation process. The charge transfer resistance (R ) of the methanol anode oxidation was obtained from the ECM fitting at the four different temperatures and potentials, as plotted in Fig. 12.14. 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

J. R. Macdonald, CNLS (Complex Nonlinear Least Squares) Immittance Fitting Program LEVM Manual Version 7.11, Houston, Texas (1999). 

A graphic illustration of these equations is presented in Fig. 11(b). Although, in simple cases, the process parameters may be obtained graphically, the best way to analyze the impedances is by the complex nonlinear least-squares approximation technique. The following parameters may be obtained from such fits 7 , R, and the Warburg coeffi-... 

An example of porous behavior was presented by Los et al for the hydrogen evolution reaction on LaP04-bonded Ni powder electrodes in 30% NaOH. Examples of the complex plane plots are shown in Fig. 36. Using the complex nonlinear least-squares (CNLS) fit, the parameters Ret, T, and C[Pg.215]

Usually an equivalent circuit is chosen and the fit to the experimental data is performed using the complex nonlinear least-squares technique. However, the model deduced from the reaction mechanism may have too many adjustable parameters, while the experimental impedance spectrum is simple. For example, a system with one adsorbed species (Section IV.2) may produce two semicircles in the complex plane plots, but experimentally, often only one semicircle is identified. In such a case, approximation to a full model introduces too many free parameters and a simpler model containing one time-constant should be used. Therefore, first the number and nature of parameters should be determined and then the process model should be constructed in consistency with the parameters found and the physicochemical properties of the process. 

In the following, the VALBOND-TRANS force field is parametrized for model octahedral complexes of Ir, with the aim to best capture relative energies calculated from DPT for different diastereomers. The (re-)parametrization of VALBOND-TRANS is important for successful use in further atomistic simulations. The fitting procedure can be conveniently carried out by using a recent combination of CHARMM and the I-NoLLS fitting environment [79, 80]. This parameter optimization environment can carry out interactive nonlinear least-square fittings to determine model parameters for a wide variety of applications. 

J.R. Macdonald, Complex Nonlinear Least Squares Immittance Fitting Program LEVM. Version 1989, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, U.S.A., 27599-3255. 

Presently, the most often used analysis is based on the complex nonlinear least-squares approximation of the impedance data acquired at a constant potential. The total impedance may be separated into the real and imaginary parts and fitted to the Randles model ... 

The problem of fitting impedances is nonlinear, and the method usually used is the complex nonlinear least-squares (CNLS) method [3, 24, 25, 616-618], In this method, a weighted sum of squares, S, of the differences between the experimental, Z i and Z"i, and the model, Z( caic and Z" caic impedances is minimized by choosing the best values of the adjustable parameters and minimizing the weighted differences between the experimental and model (calculated) impedances ... 

Complex nonlinear least squares avoids most of the weaknesses of earlier methods since it fits all the data simultaneously and thus yields parameter estimates associated with all, rather than half, the data. In addition, it provides uncertainty estimates for all estimated parameters, showing which ones are important and which unimportant in the model or equivalent circuit used for fitting and finally, it allows one to fit a very complex model, one having 5, 10, or even more unknown (free) parameters. None of the other methods can do this adequately, especially when several of the time constants of the model are close together. 

Figure 16.6 shows a typical complex nonlinear least-squares CNLS fit procedure that has been carried out on a measured impedance curve [14], The applied... 

The lowest energy phosphorescent states of this class of complexes have been found to undergo facile electron transfer reactions with a series of structurally related pyiidinium acceptors. Excited state reduction potentials [Cu4 / Cu4 +h.] of -1,71 (10), -1.55(10), and -1.56(10) V vs. SSCE for 12a [65], 12b [66], and 12c [72], respectively, have been estimated by a three-parameter, nonlinear, least-squares fit to the equation ... 

Besides, the photoexcited complex has also been found to react with a series of pyridinium acceptors such as MV [129]. The electron transfer nature of the photoreaction mechanism has been established by the appearance of the characteristic MV cation radical absorption in the transient absorption difference spectrum. The reaction has been shown to be reversible with a back-electron transfer rate constant of 1.5 x 10 dm mol" s" . From the oxidative quenching experiments with a series of structurally related pyridinium acceptors, an excited state reduction potential of [Au2 /AU ] of -1.6(1) V vs. SSCE [/fT In KV = 0.58(10) V vs. SSCE, = 0.9(K10) eV] has been estimated by three-parameter, nonlinear, least-squares fits to the equation ... 

Complex nonlinear least squares (CNLLS) regression fitting... 

Note Impedance spectra were modeled using complex nonlinear least-squares (CNLS) fitting program EQUIVCRT developed by B. A. Boukamp. The Chi-square function gives a good indication of the quality of the fit a value of... 

---