Incorporation of Error Structure

Three approaches have been documented in the literature for incorporating the error structure of impedance data into interpretation strategies. One approach has been to assume a standard form for the stochastic errors. Two models are commonly used. Zoltowski and Boukamp advocated use of modulus weighting. Use of a modulus weighting strategy invokes an assumption that the standard deviation is proportional to the frequency-dependent modulus Z(o ) of the impedance, i.e., 

A second approach has been to use the regression procedure to obtain an estimate for the error structure of the data. A sequential regression is employed in which the parameters for an assumed error structure model, e.g., equations (21.19) and (21.20), are obtained directly from regression to the data. In more recent work, the error variance model was replaced by 

The third approach is to use experimental methods to assess the error structure. Independent identification of error structure is the preferred approach, but even minor nonstationarity between repeated measurements introduces a significant bias error in the estimation of the stocheistic variance. Dygas emd Breiter report on the use of intermediate results from a frequency-response analyzer to estimate the variance of real and imaginary components of the impedance. Their approach allows assessment of the variance of the stochastic component without the need for replicate experiments. The drawback is that their approach cannot be used to assess bias errors and is specific to a particular commercial impedance instrumentation. Van Gheem et have proposed a structured multi-sine 

Measurement models, developed for impedance spectroscopy by Agarwal et 56,86,262 generally applicable and can be used to estimate both stochastic and bias errors of a measxuement from imperfectly replicated impedance measurements. Orazem et al. used a measurement model approach to show that a general model for the error structure could take the form 

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