## 4 Experimental Results and Analysis
### Effectiveness of Online Fault Diagnosis  
The effectiveness of the proposed online fault diagnosis system is evaluated by considering its classification performance using the \(k\)-NN classifier. For each test case, the DSK is used as a training set for the \(k\)-NN classifier, where \(k_{kmn}\) is set to 5 (the number of nearest neighbors \(k\) is equal to the nearest integer to \(\sqrt{N_{dsk}}\)[25]). The evaluation process includes five progressive test cases \(t\), each containing an evaluation set \(C_{eval,\ t}\times N_{eval,\ t}\). To calculate the classification accuracy before updating the system, the DSK of the previous test case \(C_{dsk,\ t-1}\times N_{dsk,\ t-1}\times F_{dsk,\ t-1}\) is used as a training set and \(C_{eval,\ t}\times N_{eval,\ t}\)\(\times F_{dsk,\ t-1}\) is used as a test set. To calculate the classification accuracy after updating the system, the DSK of the current test case \(C_{dsk,\ t}\times N_{dsk,\ t}\times F_{dsk,\ t}\) is used as a training set and \(C_{eval,\ t}\times N_{eval,\ t}\times F_{dsk,\ t}\) is used as a test set. In the classification process, we use \(k\)-cv (cross-validation), where \(k_{cv}\) is set to 3. Thus, \(N_{eval,\ t}\) is randomly divided into thirds, and each third is used for testing during each \(k\)-cv iteration. The cross-validation process is executed \(N_{iteration}\) times (\(N_{iteration}\) = 10 in this study), and the generalized classification performance is calculated by averaging the classification accuracy. Table 4 presents the diagnosis performance for the five test cases in terms of the average sensitivity per class and the average classification accuracy (\(ACA\)), defined in Equations (10) and (11), respectively. \[\text{Sensitivity}=\frac{N_{\text{TP}}}{N_{\text{TP}}+N_{\text{FN}}}\times 100( \%), \tag{10}\]  
\[ACA=\frac{\sum\limits_{N_{\text{class}}}N_{TP}}{N_{samples}}\times 100(\%) \tag{11}\]  
where \(N_{TP}\) is the number of correctly classified samples in a fault class, \(N_{FN}\) is the number of incorrectly classified samples in a fault class, and \(N_{samples}\) is the total number of test samples. Figure 14 shows the average classification performance for different datasets. The proposed method detects the unknown faults well and updates the DSK (i.e., fault modes and optimum fault signatures). The updated DSK then provides improved classification performance. For each test case, when new faults are introduced, the classification performance initially degrades because the system does not have enough information about the new faults. However, the performance improves once the DSK is updated with information on the new fault modes. The classification accuracy using the proposed method is high for test case 1 because it involves only two well-distributed classes. For test case 3, the classification accuracy of the proposed method is almost 100% because the unknown data contains no new faults. Otherwise, the classification performance decreases with an increase in new faults (small or big cracks). However, after updating the DSK, the classification performance improves by about 21%. Furthermore, classification accuracies for more severe faults (e.g., bigger cracks) are better than those for less severe faults (e.g., smaller cracks) because more severe faults contain more separable features. According to the experiment, the proposed model can detect the newly accounted fault in data. However, in a real monitoring environment like industry, the quality of bearing degrades very slowly. According to the suggestion of the American Bearing Manufacturing Association (ABMA) and the American Rolling Bearing Company, the minimum L10 life of a continuous 24 h operated bearing is 60,000 h, which is around seven years. The proposed model might be deployed in the real industry environment, where a chunk of data will be captured a certain interval, i.e., once a day, once a week, once a month, or even more.