## 4 Implementation of RUL Prediction Based on Proposed Method
### Feature Extraction and Selection  
During acquisition, vibration signals are often affected by noise. To remove meaningless components, wavelet decomposition and reconstruction were applied. Taking Experiment Group 1 as an example, the signal of bearing B1-1 was denoised using a discrete wavelet transform. The Daubechies 5 wavelet was chosen as the mother wavelet, and the signal was decomposed into three levels. The reconstructed signal after denoising is shown in Figure 10. It is clearl that the denoised signal was smoother with fewer outliers.  
Thirty-six features, as described in Section 3.1, were extracted from the denoised signal. To improve prediction accuracy and eliminate the influence of different feature magnitudes, the extracted features were smoothed using the Savitzky-Golay filter. Subsequently, the features were normalized to the range [0, 1]. The F-value for each feature was calculated according to Equation (16), and the results are shown in Figure 11. A threshold of 0.6 was set based on the highest-scoring features to filter out insensitive features. The selected features were [P2, P4, P5, P9, P10, P22, P25, P26, P27, P28], which are presented in Figure 12.  
Upon examining the results, it was observed that all features were sensitive to changes occurring throughout the entire life cycle of the bearing. For instance, the time-domain energy feature P4 remained relatively stable and low in magnitude during the early phase of bearing operation, corresponding to the normal running stage. This observation is consistent with the literature [32], reporting that vibration signals are steady and exhibit low amplitudes during this period. After approximately 15,000 s, the magnitude of P4  
Figure 11: Comprehensive index screening results.  
Figure 10: Comparison diagram before and after signal noise reduction.  
began to increase, albeit with a gentle slope, suggesting a gradual rise in signal energy. This aligns with the literature [32], reporting that amplitude fluctuations emerge as degradation begins. Around 27,000 s, a sharp increase in energy was observed, which corresponded to the abrupt changes in vibration amplitude typically seen during the terminal failure phase. These results confirm that the selected features effectively captured the bearing's degradation behavior over its full service life.  
However, the rate at which trends change differed across the features, likely due to the varying types of degradation information encoded in each feature. Additionally, some features, such as P2 and P4, exhibited very similar trends. Figure 13 displays the Pearson correlation matrix heatmap, which illustrates the relationships between features. From Figure 13, it is evident that some features were highly similar, with P2 and P4 being linearly correlated.  
Although the comprehensive index effectively filtered out features that were insensitive to bearing degradation, it did not account for the relationships among features. Excessive correlation between features leads to redundant information within the feature set, which not only increases its dimensionality but also adds to the model's complexity and computational burden [40]. To address this issue, further feature selection was performed. Based on hierarchical clustering combined with the elbow method proposed in Section 3.3, clustering analysis was conducted on the sensitive feature set. The resulting elbow plot is shown in Figure 14a. As observed, the inflection point occurred at \(K\) = 6. Although the error  
Figure 12: Sensitive feature set.  
Figure 13: Sensitive feature set correlation coefficient heat map.  
reduction rate was faster at \(K=3\), the decline in error did not level off after this point, which does not conform to the typical \"elbow\" pattern. Therefore, the dendrogram was pruned into six clusters, as shown in Figure 14b. The final clustering results are presented in Table 4. The feature with the highest F-score in each cluster was selected as the representative, forming the following optimal feature set: {P2, P5, P9, P10, P25, P28}.