## 7 Experimental Verification
### Bearing Reliability Prediction  
In response to the issue of reliability prediction for rolling bearings, this study proposes a reliability prediction model that integrates AdaBoost ensemble learning and Mixed Kernel Relevance Vector Machine (MKVM) algorithms. In order to verify the effectiveness of this method, bearings 3-1, 3-2, and 3-4 in the XJTU-SY rolling bearing dataset and bearing 5 in the bearing vibration data from the University of Cincinnati were selected for reliability prediction experiments. Taking bearing 3-1 as an example, the feature sample set of bearing 3-1 after noise reduction is first extracted, and 12 initial feature parameter sets consisting of features in the time domain, frequency domain, and time-frequency domain are obtained. Dimensionality reduction is then performed using the PaCMAP algorithm, and the resulting three-dimensional feature vector set after PaCMAP dimensionality reduction is selected as the degradation feature information. Bearing 3-1 recorded a total of 2496 time points of data, with the first 2138 samples used for model training and the remaining 400 samples serving as the test set. Through multiple grid searches to determine the mixed kernel function coefficient (as shown in the first subplot of Figure 14), it was found that when the coefficient is set to 0.73, the model achieves the lowest test error (MAE) of 0.029, along with optimal error stability. Further experiments revealed that when the coefficient fluctuates within the range of 0.7 to 0.75, the variation in MAE remains below 0.005, indicating low sensitivity of the model performance to this parameter and strong robustness.Simultaneously, when the number of base learners reaches 10, the test error stabilizes (as shown in the second subplot of Figure 14). Additional experiments demonstrated that increasing the number of base learners (e.g., to 15 or 20) improves the MAE by less than 0.002 but significantly increases computational costs. Therefore, selecting 10 base learners achieves an optimal balance between performance and efficiency. Based on these findings, the mixed kernel function coefficient is ultimately set to 0.73, and the AdaBoost algorithm is configured with 10 base learners. After determining these parameters, the training data are used to predict the bearing's reliability. The prediction results from the AdaBoost-MKVM model are then input into a logistic regression model to calculate the regression coefficients, thereby determining the operational reliability of bearing 3-1. Furthermore, the effectiveness of this prediction method is validated using data from bearings 3-2 and 3-4, with the relevant results presented in Figure 15.  
Figure 14: Test results. The reliability curve of bearing 3-1 is shown in Figure 15a. It can be observed that bearing 3-1 operated relatively stably in the early stages, but a serious failure occurred in the last 150 points of operation, resulting in a rapid decline in reliability. Figure 15b shows the reliability curve of bearing 3-2. During operation, the reliability continues to fluctuate slightly, indicating that faults gradually accumulate and eventually lead to faults. Figure 15c shows the reliability curve of bearing 3-4, where the reliability remains relatively stable until the last 300 points, and then drops sharply to zero during the last 100 points. Figure 15d shows the reliability curve of bearing 5 of the University of Cincinnati. Due to the small number of sample points of this bearing, the last 300 samples were selected as the test set. It can be seen in the figure that the bearing began to completely fail at the last 20-30 sample points. Comprehensive analysis shows that the predicted reliability values of the four kinds of bearings are in good agreement with the actual reliability trend, which indicates that the proposed method can accurately predict the bearing reliability change.