## 2 Materials and Methods
### The Methodology  
The methodology consists of two main steps: the training phase, whose framework is depicted in Figure 2, and the testing phase, which is depicted in Figure 3.  
After signal acquisition under conditions described in Table 4, nine typical time-domain features, summarized in Table 5, are extracted for each signal of tests 1, 2, 3, and 4 used for the GP training step. As the system is provided with three triaxial accelerometers, the original feature set contains 81 features. The length of the signal segments for feature \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test** & **Setting** & **Run-to-Failure Trajectories** & **Duration (min)** \\ \hline
1 & C1 & F1 & 38.4 \\
2 & C1 & F2 & 170.5 \\
3 & C2 & F3 & 157.9 \\
4 & C2 & F4 & 74.2 \\
5 & C1-C2 & F5 & 328.1 + 667.1 \\
6 & C1-C2-C3 & F6 & 208.9 + 232.2 + 995.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Available datasets. Figure 3: Framework for GP-based feature evaluation (testing). Figure 2: Framework for GP training. extraction is equal to 1 s (12,800 observations). Two different datasets are created from the so-obtained feature set and used for the setting recognition and the belt prognostics. Both datasets contain all the extracted features plus the label for each observation. The label may assume only two values corresponding to setting C1 and C2 for the setting recognition goal. The label C1 is assigned to tests 1 and 2, while the label C2 is assigned to tests 3 and 4. The label assumes continuous values for prognostics and expresses the second in which the observation has been collected, starting from 0. As feature observations are extracted every 1 s, the label increases by one along with the dataset, and the last value corresponds to the belt's duration. Note that the two training sets are used separately for the GP training to find two distinct HIs for the two machinery settings. The dataset created for system setting recognition is provided to train GP with a classification-based fitness function and a clustering-based fitness function. In the first case, a supervised approach is adopted during the training, while an unsupervised learning approach is adopted in the clustering-based GP. The datasets created for prognostics are provided to a GP aiming to construct a component-level feature revealing the degradation processes of the belt during the run-to-failure trajectories F1, F2, F3, and F4. The output of the training phase is represented by three features, CF1, CF2, and CF3, that correspond to the best feature for the classification-based GP, clustering-based GP, and GP for HI construction, respectively. The performances of the features extracted for setting recognition, i.e., CF1 and CF2, are evaluated in terms of training accuracy and training time provided through two typical classification models, i.e., the DT and the k-NN [36]. The performance of the constructed feature for prognostics, i.e., CF3, is evaluated by considering its ability to predict the belt Remaining Useful Life. First, the obtained HI for each system setting is smoothed through a moving average method. Then, an exponential model is applied to one of the available trajectories for each condition, and the accuracy of RUL prediction is computed through Equation (2), which represents the mean prediction error. \[\textit{meanerror}\%_{0}=\frac{1}{N}\sum_{n=1}^{N}\frac{|\textit{realRUL}_{n} -predictedRUL}{\textit{realRUL}_{n}}\times 100 \tag{2}\]  
where \(N\) is the number of observations. The testing phase has two main goals, i.e., to assess the generalization ability of the models built on the constructed features and to assess the ability of constructed features to recognize novel settings [37], where novel means that they are not included in the GP training process. Therefore, all the CFs are extracted from two other datasets during the testing phase. The first dataset includes the trajectory F5, collected under known system settings (C1 and C2), while the second dataset includes trajectory F6, collected under