## 3 Results
### GP for Belt Prognostics  
After system-level feature construction, a component-level HI (for the belt wear) has been constructed through the GP with the fitness function expressed by Equation (4). As the acquired signals have acceptable values of prognosability and trendability, greater weight is assigned to the monotonicity in order to obtain more monotonic HIs. Therefore, the weight of the monotonicity, \(w_{1}\), is set to 0.5 while both other weights are set to 0.25. As in the first case, the training set is made of run-to-failure trajectories F1, F2, F3, and F4. However, these trajectories have been divided into two input sets to extract an HI  
Figure 6: Constructed features through (**a**) clustering-based GP and (**b**) classification-based GP with an additional operating condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Classification-Based GP** & **Clustering-Based GP** & **PCA** & **ReliefF** \\ \hline DT & 92.8\% & 81.1\% & 92.2\% & 76.4\% \\ k-NN & 90.8\% & 70.8\% & 89.3\% & 69.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Classification accuracy (%) obtained with GP-based FCs, and feature extracted through PCA and ReliefF after adding a third machinery setting into the training set. for each condition. The fitness values of constructed HIs for both conditions during some GP runs are summarized in Table 10, including the RUL prediction mean error computed through Equation (2) on the signal acquired during test 5. As test 5 includes both C1 and C2 conditions, the error has been computed using \(HI_{\text{C1}_{i}},\ i=1,\ldots,4\), until the first setting is implemented and \(HI_{\text{C2}_{i}},i=1,\ldots,4\) for the remaining observations. Each constructed HI represents an optimal solution according to the GP fitness function. However, when a degradation model is built to compute the RUL, not all HIs provide good RUL prediction mean error results. For instance, \(HI_{\text{C1}_{4}}\) and \(HI_{\text{C2}_{3}}\) have the best fitness values, equal to 0.94 and 0.99, but they provide a high RUL mean prediction error. On the contrary, \(HI_{\text{C1}_{1}}\) and \(HI_{\text{C2}_{2}}\) provide a slightly lower fitness value, but a lower RUL mean prediction error. Hence, \(HI_{\text{C1}_{1}}\) and \(HI_{\text{C2}_{2}}\) are chosen as the best HIs for setting 1 and setting 2, respectively, since they provide the best fitness value among HIs with an acceptable prediction error. In this case, an error between 10% and 20% is acceptable because computed from the beginning of the test, when the belt was still in good health conditions. The constructed HIs for each trajectory of the training dataset are depicted in Figure 7. Table 11 summarizes the values of monotonicity, trendability, and prognosability of the two constructed HIs. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constructed HI** & **Fitting Function** & **Mean Error (\%)** \\ \hline \(HI_{\text{C1}_{1}}=f_{Skew}(x.a1)^{(f_{Kent}(x.a1)-f_{\text{RMS}}(y.a1))}\) & 0.89 & 14.2 \\ \(HI_{\text{C1}_{2}}=f_{Peak}(x.a3)/f_{Mean}(y.a1)\) & 0.87 & 33.4 \\ \(HI_{\text{C1}_{3}}=f_{CrestF}(y.a3)/f_{RMS}(y.a2)\) & 0.78 & 40.5 \\ \(HI_{\text{C1}_{4}}=f_{ImpulseF}(z.a2)/e^{f_{\text{RMS}}(z.a3)}\) & 0.94 & 130.1 \\ \(HI_{\text{C2}_{1}}=f_{Peak}(x.a2)/e^{f_{\text{RMS}}(x.a1)}\) & 0.985 & 36.6 \\ \(HI_{\text{C2}_{2}}=f_{Skew}(x.a3)\times f_{RMS}(z.a3)\times\log\left(\frac{ f_{ImpulseF}(y.a3)}{logf_{Mean}(y.a3)}\right)\) & 0.87 & 10.5 \\ \(HI_{\text{C2}_{3}}=f_{RMS}(z.a3)-\log\left(\frac{f_{\text{RMS}}(z.a3)}{f_{ SharpF}(z.a3)}\right)\) & 0.99 & 30.5 \\ \(HI_{\text{C2}_{4}}=f_{Skew}(x.a3)\) & 0.78 & 35 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: HI construction GP. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Health Indicator** & **Monotonicity** & **Trendability** & **Prognosability** \\ \hline \(HI_{\text{C1}_{1}}\) & 0.3999 & 0.1042 & 0.9095 \\ \(HI_{\text{C2}_{2}}\) & 0.2637 & 0.7599 & 0.7599 \\ \hline \hline \end{tabular}
\end{table}