Complex nonlinear least-squares

In such cases, KK analysis turns out to be particularly helpful, as demonstrated in the discussion of Section 4.4.5. 

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. 

The procedure described above is not really a CNLS approach unless/ and/ are the real and imaginary parts of a complex variable. But as we have seen, Z and 6 are not, although ln Z and 9 and Z and Z are. Since we sometimes are inter- 

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Complex nonlinear least-squares regression (CNLS)... 

J. R. Macdonald, J. Schoonman, and A. P. Lehnen, "The Applicability and Power of Complex Nonlinear Least Squares for the Analysis of Impedance and Immittance Data," Journal of Electroanalytical Chemistry, 131 (1982) 77-95. 

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

VanderNoot69has studied poorly separated faradaic and diffusional processes. He has found that a complex, nonlinear, least-squares regression is capable of extracting kinetic information from impedance measurements when the ratio of the charge-transfer process time constant tf=... 

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. 

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. 

However, complex nonlinear least-squares (CNLS) approximation can recover both kinetic and mass transfer parameters in this case. For faster processes this might be difficult [146]. 

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 ... 

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. 

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... 

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