Non-linear Least-squares Fitting

Modern EIS analysis uses a computer to find the model parameters that give the best agreement between a model’s impedance spectrum and a measured spectrum. A non-linear least squares fitting (NLLS) algorithm is used.

NLLS starts with initial estimates for all the model’s parameters. Starting from this initial point, the algorithm makes changes in several or all of the parameter values and evaluates the resulting fit. If the change improves the fit, the new parameter value is accepted. If the change worsens the fit, the old parameter value is retained. Next a different parameter value is changed and the test is repeated. Each trial with new values is called an iteration. Iterations continue until the goodness of fit exceeds an acceptance criterion, or until the number of iterations reaches a limit.

NLLS algorithms are not perfect. In some cases they do not converge on a useful fit. This can be the result of several factors including:

• An incorrect model for the data set being fitted.
• Poor estimates for the initial values.
• Noise

In addition, the fit from an NLLS algorithm can look poor when the fit’s spectrum is superimposed upon the data spectrum. It appears as though the fit ignores a region in the data. To a certain extent this is what happens. The NLLS algorithm optimizes the fit over the entire spectrum. It does not care if the fit looks poor over a small section of the spectrum.