Bayesian and Hierarchical Bayesian Based Regularization for Deconvolving the Distribution of Relaxation Times from Electrochemical Impedance Spectroscopy Data

Abstract The distribution of relaxation times (DRT) is a fast-growing methodology that is used to interpret data obtained from electrochemical impedance spectroscopy (EIS) experiments. However, the DRT deconvolution is often difficult to interpret. We tackle this issue by framing the DRT problem within a Bayesian statistical framework. This is critically important because, within such framework, we can introduce our experience regarding the EIS in the form of prior models and define the DRT as probability distribution function (pdf). The first such prior model is implemented in the regularization parameter and enables the detection of discontinuities in the DRT. This is critical for the recovery of various discontinuous elements including the Gerischer impedance. The second model focuses on the weights, which we can assign to each individual data point, and allows for the detection of outliers in EIS data. We also show that we can sample the DRT pdf given the data and the hypotheses on the regularization parameter and weights. By applying this framework to synthetic experiments and real EIS data, we show that we can obtain far more insight than with ridge regression (or Tikhonov regularization).