Stochastic Errors in Impedance Measurements

Regression problems in impedance spectroscopy may become ill-conditioned due to improper selection of measurement frequencies, excessive stochastic errors (noise) in the measured values, excessive bias errors in the measured values, and incomplete frequency ranges. The influences of stochastic errors and foequency range on regression are demonstrated by examples in this section. The issue of bias errors in impedance measurement is discussed in Chapter 22. The origin of stochastic errors in impedance measurements is presented in Chapter 21. 

Remember 21.1 The stochastic errors in impedance measurements arise from an integration of time-domain signals that contain noise originating from the electrochemical cell and the instrumentation. 

The following steps may be taken to reduce the role of stochastic errors (see Section 21.2) in impedance measurements. 

In the absence of instmment-induced correlations, stochastic errors in the frequency-domain are normally distributed. The appearance of a normal distribution of frequency-domain stochastic errors can be regarded to be a consequence of the Central Limit Theorem applied to the methodology used to measure the complex impedance. ° This result validates an essential assumption routinely used during regression analysis of impedance (and other frequency-domain) data. 

Qsp were applied for the validation of electrochemical impedance data. Agarwal et al. described an approach that eliminated problems associated with direct integration of the Kramers-Kronig integral equations and accoimted explicitly for stochastic errors in the impedance measurement. 

Roy and Orazem (2008) conducted impedance measurements to gain insight into flooding of a single PEMFC. The flooding of gas-diffusion layer pores in the fuel cell has been associated with increases in the internal cell resistance and in the impedance response of the fuel cell. The formation and removal of water droplets is an inherently stochastic process which increases the stochastic errors observed in impedance measurements. A measurement technique oriented toward... 

The contributions to the error structure of impedance measurements Eire described in Section 21.1. Impedance measurements entail a compromise between minimizing bias errors, minimizing stochastic errors, and maximizing the information content of the resulting spectrum. The parameter settings described in this section may not apply to all impedance instrumentation. 

The greatest sensitivity is observed for plots of residual errors. Residual errors normalized by the value of the impedance are presented in Figures 20.5(a) and (b), respectively, for the real and imaginary parts of the impedance. The experimentally measured standard deviation of the stochastic part of the measurement is presented as dashed lines in Figure 20.5. The interval between the dashed lines represents the 95.4 percent confidence interval for the data ( 2cr). Significant trending is observed as a function of frequency for residual errors of both real and imaginary parts of the impedance. 

The measurement model method for distinguishing between bias and stochastic errors is based on using a generalized model as a filter for nonreplicacy of impedance data. The measurement model is composed of a superposition of line-shapes that can be arbitrarily chosen subject to the constraint that the model satisfies the Kramers-Kronig relations. The model presented in Figure 21.8, composed of Voigt elements in series with a solution resistance, i.e.. 

Impedance measurements are, in general, heteroscedastic, which means that the variance of the stochastic errors is a strong function of frequency. It is important, therefore, to use a weighting strategy that accoimts for the frequency dependence of the stochastic errors. 

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... 

In the more practical case where the impedance is sampled at a finite number of frequencies, r, x) represents the error between an interpolated function and the "true" impedance value at frequency x. This error is seen in Figure 22.3, where a region of Figure 22.2 was expanded to demonstrate the discrepancy between a straight-line interpolation between data points and the model that conforms to the interpolation of the data. This error is composed of contributions from the quadrature and/or interpolation errors and from the stochastic noise at the measurement frequency (v. Effectively, equation (22.75) represents a constraint on the integration procedure. In the limit that quadrature and interpolation errors are negligible, the residual errors r( c) should be of the same magnitude as the stochastic noise r(o ). 

If impedance measurements are carried out at the same sensitivity scale for the real and imaginary components, the stochastic errors of the real and imaginary impedances will be similar, and one can use modulus weighting. Modulus weighting assumes the same statistical weights for real and imaginary parts, and they are proportional to the impedance modulus. This means that small and large impedances contribute in a similar way to the sum of squares and are equally important. 

While the nature of the error structure of the measurements is often ignored or understated in electrochemical impedance spectroscopy, recent developments have made possible experimental identification of error structure. Quantitative assessment of stochastic and experimental bias errors has been used to filter data, to design experiments, and to assess the validity of regression assumptions. 

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