## 4 Application to Rolling Bearings Fault Diagnosis
### Comparisons and Discussions  
In this subsection, our proposed method is further compared with some other related methods. For vibration signal preprocessing, the ICEEMDAN algorithm is encapsulated in our method. The two algorithms of CEEMDAN and EEMD have also been frequently utilized in literature for bearing signal decomposition. So, to study the preprocessing algorithm for multi-fault diagnosis performance, fault feature extraction and selection are also performed with the IMFs from CEEMDAN and EEMD to classify the ten bearing fault conditions. The results under 0 horsepower motor load are listed in Table 4. The three algorithms all successfully classify the nine classes except class 3 from the ball fault mode. For this bearing state class under 0 horsepower motor load, two testing samples are misclassified with EEMD and only one is misclassified with CEEMDAN and ICEEMDAN. From the misclassified testing samples in Figure 9 for EEMD and CEEMDAN and in Figure 7b for ICEEMDAN, all the off-target samples of class 3 are labeled class 8 (i.e., outer raceway fault mode with fault size of 0.014 inches). Actually, the time waveforms of class 3 and 8 in Figure 3 are very similar to each other, which complicates the correct classification of class 3. In addition, the average recognition rates by the three preprocessing algorithms for the ten bearing fault classes under four motor loads are summarized in Figure 10. It is also observed that only classes 2 and 3 do not have perfect recognition and that the best performance is obtained with the case of the ICEEMDAN algorithm, which further verifies the effectiveness of the proposed hybrid method. Figure 8: The average recognition rates with non-discriminative and selected discriminative features for the ten bearing fault classes. In the proposed method, the classifier is a multi-class ESVM. To evaluate the performance of the ESVM, three other pattern recognition models of neural network (NN), k-Nearest Neighbor (kNN), and plain multi-class SVM are also tested. For related settings, the SVM uses the same parameters as ESVM, k = 1 for the kNN classifier, and NN is of multilayer perceptron type with the hidden node of 25. The classification results under 0 horsepower motor load are listed in Table 5. It can be concluded that NN has the worst performance with misclassifications in four bearing conditions while the other \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Fault Class**} & \multicolumn{3}{c}{**Classification Accuracy (\%)**} \\ \cline{2-4}  & **EEMD** & **CEEMDAN** & **ICEEMDAN** \\ \hline
1 & 100 & 100 & 100 \\
2 & 100 & 100 & 100 \\
3 & 83.33 & 91.67 & 91.67 \\
4 & 100 & 100 & 100 \\
5 & 100 & 100 & 100 \\
6 & 100 & 100 & 100 \\
7 & 100 & 100 & 100 \\
8 & 100 & 100 & 100 \\
9 & 100 & 100 & 100 \\
10 & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The bearing fault diagnosis results by the three preprocessing algorithms. Figure 10: The average recognition rates by the three preprocessing algorithms for the ten bearing fault classes. Figure 9: Misclassified testing samples with preprocessing by **(a)** EEMD and **(b)** CEEMDAN. three classifiers miss the target in the hardest ball fault mode. To be specific, the kNN misclassifies two testing samples both in classes 2 and 3, SVM misses two targets in class 3, and ESVM only misclassified one testing sample in class 3. Detailed analysis can be seen from Figure 11, where the misclassifications are marked. For the multi-class SVM, the testing samples are erroneously classified into not only class 8 but also class 2, which is of poor performance compared to ESVM in Figure 7b. As regards the NN model, eight testing samples are misclassified, resulting in the poorest recognition rates. Finally, kNN classifies two testing samples from class 2 to class 6 and from class 3 respectively to class 8 and class 1.