## 3 Results and Discussion
### Distribution Function of Relaxation Times (DFRT) Analysis  
The distribution function of relaxation times (DFRT) is given by a function, \(R_{\rm p}\cdot G(\tau)\), which can be given by [44; 45]:  
\[Z(\omega_{\rm i})=R_{\infty}+R_{\rm p}\int_{0}^{\infty}\frac{\gamma(\tau)}{1+{ \rm j}\omega_{\rm i}\tau}{\rm d}\tau=R_{\infty}+R_{\rm p}\int_{-\infty}^{\infty }\frac{G(\tau)}{1+{\rm j}\omega_{\rm i}\tau}{\rm d}{\rm ln}\tau \tag{7}\]  
with \(G(\tau)=\tau\cdot\gamma(\tau)\) and \(\int_{-\infty}^{\infty}G(\tau){\rm d}{\rm ln}\tau=1\). The term \(\tau\) is the time constant, \(\omega\) is the angular frequency, \(R_{\infty}\) is the ohmic resistance, and \(R_{\rm p}\) is the polarisation resistance given by \(R_{\rm p}=R_{0}-R_{\infty}\), where \(R_{0}\) represents the impedance at \(\omega\to 0\). The distribution function \(\gamma(\tau)\) satisfies the normalisation condition \(\int_{0}^{+\infty}\gamma(\tau)=1\). Nonetheless, the impedance data must reflect a linear, time-invariant, and causal system [32; 33; 34]. Hence, the quality of the impedance data was initially evaluated in terms of _Kramers-Kronig_ (KK) relations. The comparison is presented in Figure 4 for the real and imaginary parts of the impedance dispersions from Figure 3a,b. This is often represented in terms of a residual plot, which is obtained by the following two well-known relations [34]:  
\[\Delta_{\rm real}(\omega_{\rm i})=\frac{Z_{\rm data,real}(\omega_{\rm i})-Z_{ \rm fit,real}(\omega_{\rm i})}{|Z_{\rm data}(\omega_{\rm i})|} \tag{8}\]  
\[\Delta_{\rm imag}(\omega_{\rm i})=\frac{Z_{\rm data,imag}(\omega_{\rm i})-Z_{ \rm fit,imag}(\omega_{\rm i})}{|Z_{\rm data}(\omega_{\rm i})|} \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline
**Circuit Element** & **Wet** & **Low Humidity** \\ \hline L/H & 5.13 \(\times\) 10\({}^{-7}\) (2.33\%) & 3.69 \(\times\) 10\({}^{-7}\) (2.41\%) \\ \(R_{\infty}\) /\(\Omega\) & 12.28 (1.16\%) & 13.25 (1.86\%) \\ \(R_{\rm high}\)/\(\Omega\) & 4.79 (6.45\%) & 2.40 (23.23\%) \\ \(Q_{\rm high}\)/S s\({}^{\rm\varphi}\) & 2.08 \(\times\) 10\({}^{-2}\) (11.90\%) & 8.30 \(\times\) 10\({}^{-2}\) (21.76\%) \\ \(\phi_{\rm high}\) & 0.25 (7.06\%) & 0.16 (21.88\%) \\ \(Y_{0}\)/S s\({}^{-1/2}\) & 8.15 \(\times\) 10\({}^{-2}\) (1.36\%) & 0.99 (93.56\%) \\ \(K_{\rm G}\)/s\({}^{-1}\) & 1.78 (1.38\%) & 0.74 (73.81\%) \\ \(R_{\rm low}\)/\(\Omega\) & - & 2.99 (73.81\%) \\ \(Q_{\rm low}\)/S s\({}^{\rm\varphi}\) & - & 1.49 (125.73\%) \\ \(\phi_{\rm low}\) & - & 0.76 (47.52\%) \\ \hline \end{tabular}
\end{table} 
Table 1: Fitting results using the equivalent circuit model (EQM) model (error % inside parenthesis) to the data obtained at 450 \({}^{\circ}\)C in O\({}_{2}\) atmosphere. where \(\omega=2\pi f\). From Figure 4, it becomes clear that the imaginary part can well reconstruct the real part (and vice-versa) of the experimental data with values lower than \(\pm\)0.2% (Figure 4). A further analysis is related to the _Bayesian_ Hilbert transform (BHT) [32] test that is depicted in Figure 5, obtained from both EIS dispersions. In this figure, the blue dots represent the mean impedance, while the green shading area was calculated based on the number of experimental points that lie within three standard deviations (3\(\sigma\)) [32]:  
\[S_{3\sigma,\text{re}}=\frac{1}{M}\sum_{m=1}^{M}1\big{(}\big{|}R_{\infty}+Z_{ \text{BHT,re}}(\omega_{\text{m}})-Z_{\text{exp,re}}(\omega_{\text{m}})\big{|} \leq 3\sigma(\omega_{\text{m}})\big{)} \tag{10}\]  
\[S_{3\sigma,\text{im}}=\frac{1}{M}\sum_{m=1}^{M}1\big{(}\big{|}\omega_{\text{m}} L_{0}+Z_{\text{HT,im}}(\omega_{\text{m}})-Z_{\text{exp,im}}(\omega_{\text{m}}) \big{|}\leq 3\sigma(\omega_{\text{m}})\big{)} \tag{11}\]  
where \"1\" is the indicator function, which can take the value of 1 if its argument is true and 0 when the argument is false, 3\(\sigma\) (\(\omega_{\text{m}}\)) is the standard deviation multiplied by 3 (i.e., three credible intervals), \(\omega_{\text{m}}\), and \(R_{\infty}\) and \(L_{0}\) are obtained by the _Bayesian_ regression.