## 4 Application to Rolling Bearings Fault Diagnosis
### Results and Analysis
Nevertheless, such improvements are limited for the cases of the three ball fault sizes because of their large dispersion. Secondly, the different fault sizes of the rolling bearing outer raceway are both well clustered with and without discriminative feature selection of the extracted nonlinear entropies. Additionally, the proposed discriminative feature selection method can better cluster and demarcate the normal as well as the three ball-fault-bearing conditions compared to that of the results of IMF1. To have a sense of the effect of the number of selected features on the classifying accuracy, a multi-class ESVM of ensemble number ten is constructed and trained with the extracted nonlinear entropy of the ten bearing conditions. Specifically, the kernel function is a radial basis function with unit bandwidth and the penalty factor C = 10,000. The results for class 3 (i.e., ball fault mode with fault size of 0.021 inches) are depicted in Figure 6.  
Although the accuracy under the four motor loads (i.e., 0 to 3 horsepower) varies differently with the increased number of features, a perfect recognition can be obtained based on the first three discriminative entropy features for this bearing fault class. Furthermore, a detailed analysis of the other nine bearing status shows that more than 85% training accuracy can be obtained if the number of discriminative features is set to be three. Therefore, the first three extracted entropy features are selected as the discriminative features for each class to diagnose rolling bearing faults. Figure 5: Scatter plots for nonlinear entropy features of the ten bearing fault classes: (**a**) selected discriminative features and (**b**) DisEn, PerEn, and SampEn of IMF1. Figure 6: Training classification accuracy for the bearing fault class 3. With the above analysis, a multi-fault diagnosis of rolling bearings can then be performed online. To validate the performance of the proposed approach, training and testing diagnostic results of the method as well as the non-discriminative features scenario (i.e., DisEn, PerEn, and SampEn of IMF1) under 0 horsepower motor load are summarized in Table 3. Interestingly, both the compromised training and testing accuracy for the two scenarios are in cases of ball fault mode (i.e., bearing fault classes 1, 2, and 3), which was highlighted for further research in the benchmark literature [41]. For the proposed method with the selected discriminative features, the training and testing accuracies are respectively compromised in fault classes 2 and 3 while the training accuracy is compromised in all three fault classes of ball fault mode and the testing accuracy is compromised in fault classes 2 and 3 for the non-discriminative features scenario. The corresponding misclassified testing samples for the proposed method and the non-discriminative features scenario under 0 horsepower motor load are plotted in Figure 3:0 It is observed that one testing sample in fault classes 2 and 3 is respectively classified into fault classes 6 and 8 for the non-discriminative feature scenario. However, only one testing sample from fault class 3 is misclassified into fault class 8 with the selected discriminative features. The above results can be partially explained by the entropy feature scatters as in Figure 5, where the related cluster boundaries are somehow overlapped. To draw comprehensively comparisons between the non-discriminative features (NDF) and the proposed discriminative features (DF) scenarios, testing results of all the bearing fault classes under four motor loads are averaged.