## 5 Case Study
### Characterization of the Measurement Error and Its Effect  
The degradation increments in Table 1 were measured using a vision system with special software applications to measure crack propagations. As \(\sigma\) is unknown, we performed an R&R study to assess the performance of the measurement system and to determine how much of the observed variation is due to the measurement system variation, i.e., \(\sigma\). The study was performed under the next characteristics: a total of three people were selected to perform \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Mean** & **Sd** & **MC Error** & \(\mathbf{p_{0.025}}\) & \(\mathbf{p_{0.5}}\) & \(\mathbf{p_{0.975}}\) \\ \hline \(v\) & 22.55 & 3.094 & 0.01476 & 16.93 & 22.39 & 29.02 \\ \(u\) & 0.01664 & 0.002687 & \(1.76\times 10^{-5}\) & 0.0126 & 0.01658 & 0.02309 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Obtained estimations for the observed degradation. Figure 4: BGR graphs for parameters of the observed degradation gamma process, (**a**) \(v\), (**b**) \(u\). the study, 10 devices were selected, and three replicates were performed, making a total of 60 readings. The results of the gage R&R study are presented in Table 3 and Figure 5. It can be noted from Table 3 that the total variation contribution of the repeatability and reproducibility are 3.83% and 0.00%, respectively, which makes the total gage R&R contribution at 3.83%. The general rule says that if the total gage R&R contribution is less than 10%, the measurement system is acceptable, which is the case of this study. From Figure 5, it can be noted that indeed the measurement system performs well, and that most of the variation comes from the part-to-part variation. From Table 3, the standard deviation of the gage R&R study is \(\sigma_{R\&R}=0.0006058\), which is the total variation due to the measurement system. Thus, we consider \(\sigma_{R\&R}\) as an estimation of \(\sigma\), such that \(\vartheta=\sigma_{R\&R}=0.0006058\), and use this value to perform the deconvolution approach presented in Section 3. Considering the estimated parameters in Table 2 for the gamma process, we estimated \(q_{g}\) for every \(t_{j}\). Then, we implemented the deconvolution approach considering \(\vartheta=0.0006058\) and \(p=1000\). In Figure 6, a comparison of the observed degradation paths and the obtained true deconvoluted degradation paths is provided by presenting the box plots and mean for every \(t_{j}\). It can be noted from Figure 6 that the variation in every \(t_{j}\) was reduced in the true deconvoluted paths. In addition, the mean degradation in every \(t_{j}\) is smaller in the true deconvoluted paths than the observed paths as in the work of [PERSON] et al. (2018). It is obvious, that the reduction in the variation at every \(t_{j}\) will cause variations in the mean degradation, i.e., degradation rate, as can be noted in both degradation paths. Indeed, these conditions will have an impact on the first-passage time distributions. As the true degradation function does not have an analytical closed form, we consider to fit the obtained true degradation to the gamma, IG, GBM and Wiener stochastic processes. Stochastic models are considered to describe temporal uncertainty, such as the observed  
Figure 5: Graphs obtained from the gage R&R study applied to the measurement system. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Source** & **StdDev (SD)** & **Study Variation (6*SD)** & **\% Study Variation** \\ \hline Total gage R\&R & 0.0006058 & 0.0036347 & 3.83 \\ Repeatability & 0.0006058 & 0.0036347 & 3.83 \\ Reproducibility & 0.000000 & 0.000000 & 0.00 \\ Operators & 0.0000000 & 0.0000000 & 0.00 \\ Part to Part & 0.0157952 & 0.0947713 & 99.93 \\ Total Variation & 0.0158068 & 0.0948409 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Gage R&R variation contribution. degradation is modeled with the gamma process. Although, a simple approximation can be defined when obtaining cumulative sums of the true deconvoluted variables.