Selecting the Regularization Parameter in the Distribution of Relaxation Times

Abstract

Electrochemical impedance spectroscopy (EIS) is used widely in electrochemistry. Obtaining EIS data is simple with modern electrochemical workstations. Yet, analyzing EIS spectra is still a considerable quandary. The distribution of relaxation times (DRT) has emerged as a solution to this challenge. However, DRT deconvolution underlies an ill-posed optimization problem, often solved by ridge regression, whose accuracy strongly depends on the regularization level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> <mml:mo>.</mml:mo> </mml:math> This article studies the selection of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> using several cross-validation (CV) methods and the L-curve approach. A hierarchical Bayesian DRT (hyper- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> ) deconvolution method is also analyzed, whereby <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:math> a parameter analogous to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> </mml:math> is obtained through CV. The analysis of a synthetic dataset suggests that the values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> selected by generalized and modified generalized CV are the most accurate among those studied. Furthermore, the analysis of synthetic EIS spectra indicates that the hyper- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> approach outperforms optimal ridge regression. Due to its broad scope, this research will foster additional research on the vital topics of hyperparameter selection for DRT deconvolution. This article also provides, through pyDRTtools, an implementation, which will serve as a starting point for future research.