## 5 Conclusions  
In this study, we developed a similarity-based diagnostic approach for assessing aging in lithium-ion batteries during second life. Unlike conventional machine learning (ML) methods that rely on large, labeled datasets, intensive computational resources, or electrochemical characterization, our approach leverages interpretable similarity metrics to map observed aging behavior against known degradation trajectories, offering a pragmatic and generalizable diagnostic approach that can be extended to different chemistries and potentially other energy storage technologies. Five similarity methods were evaluated, including two time series-based and three tree-based approaches, demonstrating that Random Forest combined with Dynamic Time Warping offers the best trade-off between accuracy and robustness.  
We also employed a hybrid stacking approach, combining multiple similarity-based methods with linear regression to enhance our similarity analysis. Thus, the best-performing method pair is selected based on the root mean squared error (RMSE) and then used to train the combined model. For cycle 1, the Temporal Aggregation of Measures and Random Forest combined with Dynamic Time Warping yielded the best results, producing similarity percentages that closely matched the ground truth values of contributions.  
Based on our results, the proposed approach can effectively estimate the contribution of degradation trajectories. Although the current validation is based on synthetic data, our approach is generalizable to real-world settings, as it relies on interpretable similarity rather than black-box prediction (as in ML methods), making it well-suited for applications in battery second life, which are often challenged by limited historical data as well as uncertain and varying degradation paths.  
Future work will focus on expanding the trajectory library to include combinations of additional stress factors (e.g., depth-of-discharge, state-of-charge, and calendar aging), and consequently validating the performance of our approach using real-world datasets, as well as exploring its applicability to other energy storage technologies, such as supercapacitors.  
Conceptualization, D.G.; methodology, D.G.; software, D.G.; validation D.G.; formal analysis, D.G.; investigation, D.G.; resources, C.H.A. data curation, D.G.; writing--original draft preparation, D.G.; writing--review and editing, C.H.A.; visualization, D.G.; supervision, C.H.A.;project administration, C.H.A.; funding acquisition, C.H.A. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), eCAMION/JULE Inc. and Ford Motor Company of Canada Ltd.  
Simplified data is included in Appendix A (cycle 1).  
During the preparation of this study, the author [PERSON] used the tool ChatGPT (GPT-4) for the purpose of troubleshooting/debugging the R codes.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} BCP & Bayesian Change Point \\ C-rate & Charge/discharge rates \\ DOD & Depth-of-Discharge \\ DTW & Dynamic Time Warping \\ EIS & Electrochemical Impedance Spectroscopy \\ EoL & End of Life \\ EV & Electric Vehicle \\ LiB & Lithium-ion Batteries \\ LiP & Lithium Plating \\ nls & Nonlinear Least Squares \\ NMC622 & Nickel manganese cobalt oxide \\ PRFTN & Proximity in Random Forest Terminal Nodes \\ RF & Random Forest \\ RF-DTW & Random Forest with Dynamic Time Warping \\ RUL & Remaining Useful Life \\ SEI & Solid Electrolyte Interface \\ SOC & State of Charge \\ SOH & State of Health \\ TAM & Temporal Aggregation of Measures \\ \end{tabular}