## 5 Experimental Results and Analysis
### Experiment 1: Analysis of Ball Bearing Faults  
First, the presented method was studied through the analysis of ball bearing faults. The type of the bearing in the study is LYC 6205E. Table 1 describes its main parameters. Four conditions  
Figure 5: Flowchart of the presented method. Figure 6: (**a**) The experiment rig; (**b**) its structure diagram. (i.e., the normal, inner-race fault, outer-race fault and rolling element fault) of the bearing were simulated, respectively. Rectangular grooves with a depth of 1.5 mm and a width of 0.2 mm were cut on the surface of inner-race, outer-race, and rolling element by wire cutting machine to simulate the faults of inner-race, outer-race, and rolling element. Figure 7 shows the faulty ball bearings with inner-race fault, outer-race fault, and rolling element fault, respectively. The motor speed was kept as 1470 rpm during the whole experiment. The defect frequencies of the test bearing can be calculated based on the expressions for the primary bearing-induced frequencies in Appendix A. It should be noticed that the feature frequency of rolling element is twice the roller rotational spin speed. Table 2 lists the calculated defect frequencies of the test bearing. Figure 8 displays the time waveform of the vibration signals that represent the four states of the test bearing. frequency and rolling element fault frequency of the enhanced envelope spectrum were selected to form the fault feature vector as shown in Figure 9d. Firstly, the vibration signal of each state was divided into 40 subset signals. The signal length of each subset signal is 6400. Subsequently, the improved fast spectral correlation was employed to extract the fault features of all subset signals. The obtained fault feature vectors are shown in Figure 10. It is evident that the curve of the same fault mode has the same trend and the curves of different fault modes are easily distinguished. Each state can get 40 sets of fault feature vectors, among which 10 sets were used for training and 30 sets were used for testing. The specific setting of the training set and test set is shown in Table 3. The label of the normal state, inner-race fault, outer-race fault and rolling  
Figure 8: The waveform of (**a**) the normal; (**b**) inner-race fault; (**c**) outer-race fault; (**d**) rolling element fault. Figure 9: (**a**) The time waveform of an inner-race fault signal; (**b**) the improved fast spectral correlation spectrum; (**c**) the corresponding enhanced envelope spectrum; (**d**) the fault feature vector. element fault are set as 0, 1, 2 and 3 for classification, respectively. Finally, the obtained fault feature vectors were used for training and testing by using RF and PSO-RF. The classification results are shown in Figure 11. As shown in Figure 11, the classification accuracy of RF and PSO-RF is 99.1667% and 100%, respectively. It demonstrates that the PSO method improves the classification accuracy of RF. To further validate the effectiveness of the improved fast spectral correlation, it was compared with the EMD-SVD [18] and MPE [19] methods. For an equitable comparision, the classifier of PSO-RF was used to identify the fault feature vectors obtained using EMD-SVD and MPE. Figure 12 shows the state classification results of EMD-SVD and MPE. In Figure 12a, there are two samples of normal state that are mistakenly classifed as the inner-race fault. A sample of inner-race fault is mistakenly classified as normal state in Figure 12b. It can be seen that the classification accuracy of EMD-SVD and MPE are 98.3333% and 99.1667%, respectively. It indicates that the fault feature vectors extracted using the improved fast spectral correlation can express the fault chatacteristics better than EMD-SVD and MPE. In order to illustrate the superiority of the selected classifier used in this paper, it was compared with two classifiers, ELM and SVM. Figure 13a,b show the classification results obtained by respectively conducting ELM and SVM on the fault feature vectors.