Comparison of Parametric and Nonparametric Methods for the Analysis and Inversion of Immittance Data: Critique of Earlier Work

Abstract

Recently, two methods for the estimation of discrete and/or continuous distributions of relaxation times from small-signal electrical frequency-response data have been compared. For discrete-line distributions, the parametric method used was found to be inferior in some ways to the nonparametric one, which involved Tikhonov regularization, and it was concluded that the parametric one could not be employed to estimate continuous distributions at all. Here it is shown by Monte Carlo simulation that both conclusions are incorrect. The same data situations analyzed in the earlier work were reanalyzed using a complex nonlinear least-squares parametric method that has been employed to estimate discrete-line distributions since 1982 and continuous ones since 1993. Quite different results from those presented earlier were obtained, and the original parametric method was shown to be far superior to the nonparametric one for the estimation of discrete-line distributions, since inversion is unnecessary and resolution is far greater. For continuous or mixed distribution inversions, the parametric method was again superior, and it allows unambiguous distinction between discrete-line points and those associated with a continuous distribution, while the nonparametric inversion method does not allow such distinction and approximates all distributional points as continuous-distribution ones. The parametric method used and described here is also valuable for other data analysis tasks other than those involving inversion. Some of its error characteristics are investigated herein, and the importance of matching the weighting error-model to the form of the errors in the data is illustrated. It was found that with normally distributed random errors added to exact data, the distributions of estimated parameters were not normal but were closer to normal for proportional errors than for additive ones.