## 4 Application to Rolling Bearings Fault Diagnosis
### Results and Analysis  
After the above experimental configuration, training and testing vibrational samples for different bearing conditions under four motor loads are prepared. One vibration signal sample for each of the ten fault classes under 0 horsepower motor load is illustrated in Figure 3, which demonstrates that it is problematic to discern each bearing fault class directly from the time waveforms, especially for the ball and inner raceway fault modes. Then, the ICEEMDAN algorithm is utilized to preprocess time waveform signals with the parameters chosen as detailed in Section 3.1, and the results for one sample of bearing fault class 7 (i.e., outer raceway fault mode with fault size of 0.007 inches) under 0 horsepower motor load are displayed \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fault Mode**} & \multirow{2}{*}{**Fault Size in** **Inches**} & \multirow{2}{*}{**Class Labels**} & **Number of** **Training** & **Number of** **Testing Samples** \\  & & & **Samples** & **Testing Samples** \\ \hline \multirow{3}{*}{Ball} & 0.007 & 1 & 47 & 12 \\  & 0.014 & 2 & 47 & 12 \\  & 0.021 & 3 & 47 & 12 \\ \hline \multirow{3}{*}{Inner Raceway} & 0.007 & 4 & 47 & 12 \\  & 0.014 & 5 & 47 & 12 \\  & 0.021 & 6 & 47 & 12 \\ \hline \multirow{3}{*}{Outer Raceway} & 0.007 & 7 & 47 & 12 \\  & 0.014 & 8 & 47 & 12 \\  & 0.021 & 9 & 47 & 12 \\ \hline \multicolumn{2}{c}{Normal state} & 0 & 10 & 95 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the bearing multi-fault conditions and their class labels. Figure 3: One vibration signal sample for each of the ten bearing fault classes. in Figure 4. For comparison purposes, the decompositions by CEEMDAN and EEMD are performed with the same parameters. It can be observed from Figure 4 that the periodic impulses generated by an outer raceway point fault are successful detected by all three algorithms. However, there are ten IMFs using CEEMDAN and EEMD algorithms while nine IMFs using ICEEMDAN. Similar results are also found for other vibration samples. Thus, more compact decomposition results can be obtained with the ICEEMDAN algorithm. In the following, the obtained IMFs are further processed to extract fault features of the rolling bearing. Specifically, nonlinear entropy features of DisEn, PerEn, and SampEn are extracted for fault diagnosis of rolling bearings. Before calculation of the three entropy features, the related parameters are to be set. Based on the analysis in Section 3.2 and with some initial trials, the parameters are finally chosen for all cases of bearing conditions under four motor loads as \(c=8\), \(m=2\), and \(\tau=1\) for DisEn; \(m=6\) and \(\tau=1\) for PerEn; and \(m=2\) and \(0.2\)std(\(\mathbf{d}\)) for SampEn. Actually, the parameters of DisEn and SampEn show limited effects on the extracted features; the embedding dimension has a relatively greater influence on PerEn. To compromise between calculating complexity and effectiveness, these parameters are determined as mentioned above. The scatter plots of entropy features under 0 horsepower motor load are shown in Figure 5. In previous studies, the IMF1 from such decomposition is usually selected as the components for further analysis; thus, the three kinds of entropy features of IMF1 are also plotted in Figure 5. Some observations can be made. Firstly, different bearing conditions are easier to be discerned in the extracted entropy  
Figure 4: Decomposition results for one vibration signal sample of bearing fault class 7 by (**a**) ensemble empirical mode decomposition (EEMD), (**b**) complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), and (**c**) improved CEEMDAN (ICEEMDAN). feature space than the original time waveforms (refer to Figure 3). This is especially evident for the fault mode of the inner raceway, since the three classes with increasing fault sizes are well clustered and demarcated from the remaining seven classes.