## 4 Validation of the Proposed Method
### Case 1: Experiment Results on the CWRU Bearing Dataset and Performance Analysis
#### 4.2.3 Bearing Fault Diagnosis Experiment  
Fault Diagnosis Results and Evaluation of the Proposed INPSO Optimization Method  
During the optimization stage, the proposed INPSO approach was used to optimize the penalty factor \(c\) and the kernel parameter \(g\), where the number of particles was set to 20, the number of iterations was 100, and the search ranges of \(c\) and \(g\) were set to [0.001,10]. Fitness values were calculated by threefold cross validation for the training samples. A group of samples was randomly selected as the test set, and the proposed 1 DCNN was applied for feature extraction. The extracted feature vectors were respectively fed into three classifiers (SVM, PSO-SVM and INPSO-SVM), where the parameter settings of the PSO and INPSO algorithms were the same. Each method was performed 10 times and the diagnosis accuracy of three classifiers are listed in Table 5. As can be seen from Table 5, in the standard SVM, the default parameters \(c\) and \(g\) of SVM respectively were 1 and \(1/n\), where \(n\) was the data dimension. The fault diagnosis accuracy of the test set was 93.6% without the parameters tuning process of SVM. As a contrast, the PSO algorithm was used to optimize SVM parameters, and the average fault diagnosis accuracy of PSO-SVM was 90.56% with a standard deviation of 10.52. This means that the diagnosis accuracy was not stable and the parameter optimization process of the PSO was easily trapping into local optimums. In order to further evaluate the effectiveness of the proposed INPSO method, with the same test set, SVM optimized by INPSO was applied to do fault diagnosis experiments and the experimental results revealed that the average accuracy of the test set could reach 96.97%, and the standard deviation was only 0.69. Therefore, we could conclude that the diagnosis accuracy and robustness of INPSO-SVM was better than PSO-SVM and SVM method, which meant that the proposed INPSO overcame the shortcomings of original PSO, e.g., trapping into local optimums, and had a stronger global optimization ability. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Run Number**} & \multicolumn{3}{c}{**Accuracy**} \\ \cline{2-4}  & **SVM** & **PSO-SVM** & **INPSO-SVM** \\ \hline
1 & 94.67 & 75.33 & 95.33 \\
2 & 93.67 & 97 & 97 \\
3 & 91 & 75.33 & 97.33 \\
4 & 95.33 & 97.33 & 97 \\
5 & 94.67 & 97.33 & 97 \\
6 & 94.67 & 97.33 & 96.33 \\
7 & 94 & 96.33 & 97.67 \\
8 & 91.67 & 97.33 & 97 \\
9 & 94.67 Figures 8 and 9 depict the convergence curves of INPSO algorithm compared with the original PSO algorithm in the first trial. The optimal fitness value of PSO and INPSO algorithm in the first iteration were both 70.72%. With the progress of iteration, the accuracies of the SVM optimized by two algorithms rose rapidly, while the particles fell into local optimums in the second iteration. At this point, the fitness achieved 80.87%. After a couple of iterations, the optimal fitness value of PSO algorithm did not change, while Levy flight helped the particles jump out of the local optimums and find a better global solution in the proposed INPSO algorithm. The accuracy of the proposed INPSO-SVM method was \(\sim\)100%. Figure 7: Scatter plots of the feature visualization by t-SNE: (**a**) feature distribution of raw data, (**b**) feature distribution of convolutional layer 1, (**c**) feature distribution of convolutional layer 2, (**d**) feature distribution of convolutional layer 3, (**e**) feature distribution of convolutional layer 4, and (**f**) feature distribution of the fully connected layer. was higher than that of the PSO-SVM, reaching up to 100.0% in the training process. The training results also showed that the proposed INPSO approach had a better performance in the parameters tuning process and INPSO-SVM achieved a higher diagnosis accuracy.