## 3 Results and Discussion
### Distribution Function of Relaxation Times (DFRT) Analysis
This metric considers that \(0\leq S_{3\sigma}\leq 1\), where scores closer to 1 have lower residuals. Figure 4: Residual plots obtained for the _Kramers-Kronig_ (KK) relations from the electrochemical impedance spectroscopy (EIS) dispersion at 450 \({}^{\circ}\)C in both wet (\(p_{\text{H2O}}\sim 10^{-2}\) atm) and low humidity (\(p_{\text{H2O}}\sim 10^{-5}\) atm) O\({}_{2}\) atmosphere. In wet conditions, both the real and imaginary experimental part falls within this interval, with the majority of the residuals represented inside the 3\(\sigma\) credible interval, suggesting that the EIS data are consistent. However, the dispersion indicates a minor time drift in the EIS data acquired in low humidity conditions. Particularly, in the case of the imaginary part, a significant deviation at low frequencies is observable. Such deviation is significantly larger in the case of the analysis in low humidity conditions, due to the difficulty in fitting the low-frequency tail below 0.1 Hz (Figure 3b). This notable result may well explain the relatively large errors obtained by the EQM fitting in Table 1, highlighting the importance of studying the quality of the EIS spectra when analysing the results. The scores presented in Table 2 also corroborate this conclusion. Another useful score, \(S_{\mu}\), consists in summing 1 to the negative of the relative distance between the mean \(\mu_{\rm{DFT}}\) and \(\mu_{\rm{BHT}}\) vectors. The score \(S_{\mu}\) varies between 0 and 1 and, when both \(\mu_{\rm{DFT}}\) and \(\mu_{\rm{BHT}}\) vectors are similar higher scores can be obtained, as follows [32]:  
\[S_{\mu,\rm{re}}=1-\frac{\mu_{\rm{DFT,re}}-\mu_{\rm{BHT,re}}}{\mu_{\rm{DFT,re}}+ \mu_{\rm{BHT,re}}} \tag{12}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Metric**} & \multicolumn{2}{c}{**Score/\%**} \\ \cline{2-3}  & **Wet** & **Low Humidity** \\ \hline \(S_{\rm{k\sigma,re}}\) & 76.19 & 76.19 \\ \(S_{\rm{k\sigma,im}}\) & 100.00 & 100.00 \\ \(S_{\mu,\rm{re}}\) & 99.58 & 98.39 \\ \(S_{\mu,\rm{im}}\) & 99.54 & 86.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scores (%) obtained for the _Bayesian_ Hilbert transform experiments. Figure 5: Residuals of the _Bayesian_ Hilbert transform (BHT) data: Wet conditions (**a**) real and (**b**) imaginary; low humidity conditions (**c**) real and (**d**) imaginary. \[S_{\mu,\text{im}}=1-\frac{\mu_{\text{DFTT,im}}-\mu_{\text{BHT,im}}}{\mu_{\text{DFTT,im}}+\mu_{\text{BHT,im}}} \tag{13}\]  
Table 2 depicts the scores (%) obtained for the _Bayesian_ Hilbert transform (BHT) experiments by both described metrics. The high \(S_{\mu}\) values are indicative that the real part can recover the imaginary part well and vice-versa, of the spectra obtained in both wet and low humidity conditions, with values in the range of 86-100%. This result can be observed by the fitting curves depicted in Figure 3a,b. Nonetheless, while an apparent good match to the experimental data can be observed, this result emphasises that different metrics should be used for a deep understanding of the results. This is especially important when the accuracy of the determination of the fitting parameters from the EQM may be questionable, as it is the case of the results presented in Table 1. Hence, the use of both metrics allows determining, with more depth, the quality of the impedance data, being able to detect minor deviations from linearity or stability, thus, emphasising the need for a much deeper analysis. Figure 6 depicts the DFRT analysis performed on the LNO+BCY10 composite cathode in both wet (\(p_{\text{H2O}}\sim 10^{-2}\) atm) and low humidity (\(p_{\text{H2O}}\sim 10^{-5}\) atm) conditions at 450 \({}^{\circ}\)C in O\({}_{2}\) atmosphere.