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 λ. This article studies the selection of λ using several cross-validation (CV) methods and the L-curve approach. A hierarchical Bayesian DRT (hyper-λ) deconvolution method is also analyzed, whereby λ_0, a parameter analogous to λ, is obtained through CV. The analysis of a synthetic dataset suggests that the values of λ 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-λ 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.

Theoretical and Experimental Study of the Distribution of Capacitive Times

“…Furthermore, the input and hidden layers of both DNNs were equipped with non-saturating exponential linear units [41], while the softplus activation function was used at the DNNs output layer to enforce the non-negativity constraint on the DCT [42]. Additionally, we used the following loss function to train the DNN: (29) 3 To deconvolve the DCT with the pyDRTtools, we discretized the integral in (2) with piecewise-linear functions, used the second derivative for the penalty in ridge regression, set the regularization parameter to 10 -3 , considered the full admittance spectrum, and excluded the capacitance, 𝐶 0 [24,38].…”

“…Unless otherwise specified, the maximum number of iterations was set to 1.00×10 5 . When needed, DCT peaks were separated using a modification of peak-separation algorithm available in pyDRTtools (Sections 5.2.1.1 and 5.2.1.3) [24,38], and the DCT was integrated with the trapezoidal rule (Section 5.2.1.3).…”

“…Although the DRT has been extensively studied using methods like Tikhonov regularization [24], Gaussian processes [25], and neural networks [26], the DCT method remains largely unexamined. This study aims to highlight the significance of the DCT, outline differences between the DRT and DCT, and demonstrate how the DCT can be used.…”

Organic Solvent-Based Li–Air Batteries with Cotton and Charcoal Cathode

“…Furthermore, as presented in Table II, lower C 4 capacitance was found for higher solvent polarity, and the C 4 value obtained for PC (2.05•10 −6 F) is in the order of typical capacitance values published in the literature for electrochemical double layers. 54 On the basis of impedance spectroscopy, DRT analysis was performed using Matlab software 55,56 to gain deeper insight into the contributions of different processes. DRT is an effective method for identifying the different electrochemical processes involved in the impedance spectra.…”

Demystifying the Distribution of Relaxation Times: A Simulation-Based Investigation into the Limits and Possibilities of Interpretation for Lithium-Ion Batteries

“…This regularization causes a smoothing of the distribution function and introduces an additional degree of freedom, namely the regularization parameter λ. Several methods have been developed in the literature to determine the optimal regularization parameter [14][15][16][17][18] and recent advancements include a regularizationfree approach using the Loewner method as an alternative to the DRT. 10 This aspect will not be the focus of this article, as our attention is directed toward the topic of interpretation, which has not yet received sufficient attention in the literature.…”