Comparison and application of two methods for the least squares analysis of immittance data

Abstract

Two different, recent methods for complex nonlinear least squares fitting of the small-signal ac response of dielectric and partially conducting systems are described and compared. These methods, which are also appropriate for fitting mechanical relaxation, nuclear magnetic relaxation, and light-scattering complex data, simultaneously fit the real and imaginary parts of the data to an appropriate model. For small-range response data, the weighted, extended least squares, vector-minimization approach (Program LEVM) and the much more complicated unweighted matrix-determinant method are shown to yield indistinguishable results. For small-range data measured on a dielectric polymer, weighting is found to be unnecessary. Verification of some of the results is accomplished by Monte Carlo simulation. For large-range data sets, such as those usually arising from electrical response measurements on ionic or electronic conductors, data variances generally vary widely (high heteroscedasticity). Some kind of weighting is then essential in order to obtain low-bias fitting-model parameter estimates with small estimated standard deviations. This is demonstrated by fitting wide-range data for the admittance of hydrogen-doped lithium nitride over a wide frequency range. The results of fitting with the several different weighting schemes available as choices in the vector-minimization approach are compared and discussed. Such fitting was carried out using the general-purpose program LEVM, which runs rapidly on a PC/AT or equivalent machine. It is the only available complex nonlinear least squares fitting program which can use the data to produce least-squares estimates of the parameters of an error-variance (weighting) model as well as those of the fitting model. Thus, the data themselves can be used to determined the most appropriate weighting. An excellent fit of the Li 3N data was obtained, demonstrating that iterative reweighting, using variable weights proportional to the magnitude of the real and imaginary parts of the fitting-model function raised to a power near unity (a form of extended least squares), was the most appropriate way to treat the high heteroscedasticity of the data and to obtain minimum-bias, high-precision parameter estimates.