## 2 Methodologies
### Data Augmentation  
Like other deep learning methods, there are many parameters in the SDAE model. A vast number of training samples are necessary for training the SDAE so as to avoid overfitting and improve the generalization. In addition, training samples are balanced or imbalanced, which can cause differences in the diagnostic results [44,45]. The accuracy of fault diagnosis is usually affected by the imbalance of datasets. However, datasets A, B, C, and D were all balanced.  
According to the previous introduction (Section 2.2), we needed to check the Nash balance point to verify whether \(G\) and \(D\) were in the trained convergence state. The result for all data reached the Nash equilibrium point. The calculation result, based on the Label 2 type of Dataset A, is shown in Figure 12 as an example. This method was applicable to other datasets.  
Figure 13 shows the original vibration sample of the Label 2 type of Dataset A and its corresponding generated samples. Other fault-type generated samples were similar. The generated samples differed from the original sample and the original training samples were extended by the generated ones.  
The uncertainty analysis for the generated samples of the Label 2 type of Dataset A is shown in Figure 14. The analysis of other generated fault samples was similar. We concluded that the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**G structure** & **D Structure** & **SDAE Structure** & **Reconstruction Error** \\ \hline
[3,2,1] & [3,2,1] & [5,4,3,1] & 0.063 \\
[3,3,2] & [3,3,2,1] & [4,4,3,1] & 0.073 \\
[3,3,2,1] & [3,3,2] & [4,2,1] & 0.091 \\
[2,4,2,1] & [4,2,1] & [3,3,1] & 0.058 \\
[4,3,1] & [4,3,2,1] & [5,3,2,1] & 0.082 \\
[4,4,1] & [4,4,1] & [3,3,2,1] & 0.046 \\
[4,3,2,1] & [1,4,2,1] & [3,2,1] & 0.094 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Reconstruction error of the different network structures.  
Figure 12: Probability value of the D output.  
Figure 13: Original and generated samples.  
generated samples conformed to the 36 principle and that the membership function obeyed a Gaussian distribution using the cloud model. Therefore, the generated samples were feasible for SDAE training.  
Enough training samples were obtained to investigate how many training samples were sufficient and how well the SDAE performed. Different numbers of training samples as generated by the GAN approach were fed into the SDAE to test its performance. These amounts were selected as 700, 750, 800, 850, 900, 950, 1000, 1050, and 1100 samples in the experiment. The results with varying amounts of training samples are shown in Figure 15.  
It is evident that the performance of the SDAE for fault diagnosis improved as the volume of the training samples increased, with an increase of 30% in accuracy from 700 samples to 1100 samples. With 1000 training samples, the accuracy of the fault diagnosis was above 80%. When the samples were increased to 1100, the accuracy of the SDAE peaked at 92.01%. Considering the cost of the calculation, we made a compromise by expanding the sample size to 1100 without increasing the amount of data. The random-noise data were inputted into the GAN for data augmentation and were converted into vibration fault data, based on adversarial learning. This approach makes up for the shortcomings of other methods, such as oversampling. Because GAN data augmentation expands the count of the training samples, the training performance of the SDAE is intensified and the generalization of the SDAE is further increased.