## 3 Case Study
### Feature Extraction and Selection  
To compare the relevant data features, \(n=53\) segments of acceleration time series data were extracted for each state. Using Equations (1)-(9), the combination feature vectors of the 53 segments were obtained, where \(\varepsilon=200\) and \(\delta=0.102\). Table 2 displays the typical results of the feature value calculations corresponding to states A, B, C, and D in the time series data. Figure 9: Time–frequency spectrograms of acceleration signals for four states: (**a**) state A, (**b**) state B, (**c**) state C, and (**d**) state D.  
State C exhibited the highest \(\overline{R}\) = 3.07 and \(R_{rms}\) = 4.51 values, signifying a greater average acceleration and variability, respectively. In contrast, state D displayed the lowest \(\overline{R}\) = 0.15 and \(R_{rms}\) = 0.16 values, indicating lower average acceleration and lesser variability. This corresponds with the forced excitation in the pumping state and the small ambient vibration in the idling state. The \(R_{pp}\) values portray considerable variation across states, with state C demonstrating the highest \(R_{pp}\) = 29.3, and state D was the lowest, \(R_{pp}\) = 0.40. State D also showed the highest \(R_{H}\) = 0.93, suggesting a more homogeneously distributed power spectrum. The initial comparison between states A and B, considering the first four features, revealed only minor differences. However, the feature \(R_{cc}\) introduced a significant divergence. In state B, the value of \(R_{cc}\) was 142, while state A recorded an \(R_{cc}\) value of zero. This suggests that state B had a smaller offset magnitude, as evidenced by the higher frequency of intersections between the vibration curve and the offset axis. On the other hand, state A demonstrated a larger offset magnitude, with no intersections occurring between the vibration curve and the offset axis over 30 s. Consequently, \(R_{cc}\) becomes a key factor in differentiating between states A and B. Among all states, state D showed the highest \(R_{cc}\) value of 1255, indicating that its vibration curve was very close to the offset axis. Therefore, \(R_{cc}\) is also a significant feature for distinguishing state D.  
Figure 10 shows the scatter plots of feature values for 53 segments in each state. In state A, the segments were distributed as \(\overline{R}\) = 1.0-3.5, \(R_{rms}\) = 1.0-5.0, \(R_{pp}\) = 0.5-12.0, \(R_{H}\) = 0.5-0.9, and \(R_{cc}\) = 0. This wide range indicates the climbing state has a scattered distribution, and will likely be confused with other states. State B segments were distributed as \(\overline{R}\) = 0.2-1.0, \(R_{rms}\) = 0.2-2.1, \(R_{pp}\) = 0.7-38, \(R_{H}\) = 0.7-1.0, and \(R_{cc}\) = 36-408. Except for an outlier of \(R_{pp}\) = 37.4 in Figure 10a, \(R_{pp}\) was 0.7 to 14. This consistent distribution signifies the HBM operating state, distinguishable from others. In state C, the segments were distributed as \(\overline{R}\) = 1.4-4.0, \(R_{rms}\) = 1.9-6.0, \(R_{pp}\) = 11-40, \(R_{H}\) = 0.5-0.8, and \(R_{cc}\) = 0-16. This indicates that under the pumping state, the segments  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**State** & \(\overline{R}\) & \(R_{rms}\) & \(R_{pp}\) & \(R_{H}\) & \(R_{cc}\) \\ \hline A & 1.10 & 1.11 & 1.84 & 0.82 & 0 \\ B & 0.44 & 0.52 & 3.66 & 0.81 & 142 \\ C & 3.24 & 4.85 & 29.3 & 0.85 & 2 \\ D & 0.14 & 0.16 & 0.40 & 0.93 & 1255 showed relatively high root mean square values, mean values, and peak-to-peak values, exhibiting a large variation range. In Figure 10b, the overlapping ranges of \(\overline{R}\), \(R_{cc}\), and \(R_{H}\) make it difficult to distinguish between states A and C. However, Figure 10a shows that due to the differing \(R_{pp}\) ranges, states A and C can be relatively well differentiated. The segments of state D were distributed within the following ranges: \(\overline{R}\) = 0.14-0.15, \(R_{rms}\) = 0.15-0.17, \(R_{pp}\) = 0.3-0.6, \(R_{H}\) = 0.92-0.94, and \(R_{cc}\) = 1230-1416.