## 4 Engineering Application
### Application to Fault Diagnosis of Rolling Bearing  
In order to demonstrate the effectiveness of the proposed SSFCM-ASCA-ID-SVDD method, experiments were implemented by contrast with constrained-K-means (CK-means) [46] in the clustering phase and with SCA in the parameter optimization phase. Similarly, support vector machine (SVM) and SVDD with relative distance decision strategy (RD-SVDD) were applied for comparison during decision making phase. In the proposed method, the first step was to decompose the signal of each type of fault into a set of components with VMD, whose mode number \(K\) needed to be determined in advance with central frequency observation method. If the value of \(K\) is too large, center frequencies of adjacent IMFs may be too close, and mode mixing will emerge. The center frequencies of all IMFs when taking different \(K\) are listed in Table 2, where \(K\) was ascertained by the sample of inner race fault with diameter of 0.007 inches (L1). As illustrated in Table 2, when \(K\) was set to 5, the first two center frequencies were relatively close, which meant that excessive decomposition occurred. The same conclusion can also be seen in Figure 6, during the iterative calculation of VMD, if \(K\) was 5 or greater, some center frequencies of IMFs would be relatively close to each other. For example, when \(K\) was 5, IMF\({}_{1}\) and IMF\({}_{2}\) were relatively close, while \(K\) was 6, IMF\({}_{2}\) and IMF\({}_{3}\) were relatively close. Therefore, parameter \(K\) was set to 4 in this research. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fault Location** & **Diameter (inches)** & **Fault Label** & **Number of Samples** \\ \hline Inner race & 0.007 & L1 & 59 \\ Inner race & 0.014 & L2 & 59 \\ Inner race & 0.021 & L3 & 59 \\ Ball & 0.007 & L4 & 59 \\ Ball & 0.014 & L5 & 59 \\ Ball & 0.021 & L6 & 59 \\ Outer race & 0.007 & L7 & 59 \\ Outer race & 0.014 & L8 & 59 \\ Outer race & 0.021 & L9 & 59 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the vibration samples used in the experiment. Figure 5: Experiment stand and its measurement sensors in bearing data center. The decomposition results of samples with different fault types are illustrated in Figure 7, from which it can be seen that waveforms of different fault samples are different to some extent. After VMD decomposition, FuzzyEn values were calculated to construct the fault feature vectors, and the fractal dimension \(m\) was set 2 while the positive real number \(r\) was set as 0.2. The first three FuzzyEn vectors of different fault samples (L1-L9) are reported in Table 3.  
Figure 6: Center frequency distribution with different \(K\) values. Figure 7: The variational mode decomposition (VMD) decomposition results of signals with different faults: (**a**) inner race (0.007 inches); (**b**) inner race (0.014 inches); (**c**) inner race (0.021 inches); (**d**) ball (0.007 inches); (**e**) ball (0.014 inches); (**f**) ball (0.021 inches); (**g**) outer race (0.007 inches); (**h**) outer race (0.014 inches); (**i**) outer race (0.021 inches). The feature vectors of samples from different fault types were divided randomly into two parts with 40 and 19 vectors, respectively, the first part was used for training and the second one for testing. In order to better illustrate the characteristics and effects of SSFCM, training samples with ratio of 0.3 and 0.5 were randomly selected as labeled samples, while the rest were samples with unknown fault types. The weighting parameter \(w\) of SSFCM was set to 2 while the balance coefficient \(\alpha\) was set to 1/0.3 and 1/0.5, respectively, according to the ratio. Figure 8 depicts the 2D projection of training samples after clustering. The figure shows that training samples are gathered into nine clusters under the supervision of labeled samples and distributed near the center of each cluster. The contour-map displays the samples with the same partitioning values.