## 4 Engineering Application
### Application to Fault Diagnosis of Rolling Bearing
In the optimization phase, the proposed ASCA approach was applied to optimize the penalty factor \(C\) and the kernel parameter \(\sigma\), where the individual number was set to 30 and iteration times were set to 100. Considering the constraints of the optimization problem of SVDD, \(C\) should subject to \(C\geq 1/n\), where \(n\) is the number of target objects. Likewise, \(C\) should be subject to \(C<1\) or else it loses constraint on Lagrange multipliers. Taken together, searching ranges of \(C\) and \(\sigma\) were [1/\(n\), 1] and [2-10, 210], respectively. The coefficient \(k\) was set to 3 for KNN decision. During optimization process, the fitness value was calculated by five-fold cross validation on the training samples. The optimization effect with the given (\(C\), \(\sigma\)) was measured by average accuracy of cross validation, then the optimal (\(C\), \(\sigma\)) was obtained. The convergence procedure of the adaptive parameter optimization is depicted in Figure 9a. It can be seen that the absolute average fitness value of all individuals increases rapidly at early stage of iterations and then tends to be stable, which means the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fault Label**} & \multirow{2}{*}{**Sample Number**} & \multicolumn{4}{c}{**Fuzzy Entropy for Different IMFs**} \\ \cline{3-6}  & & **IMF1** & **IMF2** & **IMF3** & **IMF4** \\ \hline \multirow{3}{*}{L1} & 1 & 1.7996 & 1.8052 & 1.1744 & 0.6738 \\  & 2 & 1.7691 & 1.8104 & 1.1649 & 0.6638 \\  & 3 & 1.8286 & 1.7767 & 1.1695 & 0.6738 \\ \hline \multirow  
individuals were moving closer to the global optimal solution. The shaded part shows the distribution of the convergence curve in 10 optimization experiments. The comparison of ASCA and SCA is shown in Figure 9b, where all convergence experiments use the same FuzzyEn values and each convergence curve is the average of 10 experiments. The figure shows that both ASCA and SCA have a good convergence effect, but ASCA keeps a better convergence effect than SCA in the whole iterations. Besides, in the middle and later stages of iterations, the convergence curve of SCA decreases slightly, while ASCA remains stable. With the optimal parameters (\(C\), \(\sigma\)) and labeled training samples, the ID-SVDD model was trained and applied to recognize testing samples. For an in-depth verification of the proposed method, diagnosis results were averaged over 10 runs and the training samples were chosen at random in every run. In this application, normalized mutual information (NMI) as well as accuracy (ACC) as two widely used evaluation metrics are employed to evaluate the diagnosis results [47], which reflects the matching degree between classification results and real sample distribution information. The scope of two metrics is both [0, 1]. The value closer to 1 indicates a better classification performance of the model. In addition, corresponding deviation scope for each result was also calculated as a reference for evaluation. The optimal (\(C\), \(\sigma\)) is presented corresponding to the best accuracy among 10 runs. Figure 8: 2D projection of the training samples (0.5 labeled). Figure 9: Convergence procedure of adaptive parameter optimization (0.5 labeled): (**a**) Distribution of convergence curves in 10 runs, (**b**) comparison of different optimization methods. In contrastive experiments, parameters of CK-means, SCA, and RD-SVDD were set similar to the proposed method. That is, training samples with a ratio of 0.3 and 0.5 were randomly selected as labeled samples for CK-means clustering; SCA had 30 individuals and iterated 100 times; the ranges of \(C\) and \(\sigma\) of RD-SVDD were [1/\(n\), 1] and [2-10, 210] respectively. Considering the difference between SVDD and SVM, the searching scopes of parameter \(C\) and \(g\) of SVM were both set as [2-10, 210] respectively. Moreover, the optimal parameters \(C\) and \(\sigma(g)\) of all contrastive methods were selected in the same way the proposed method done.