## 2 Methodologies
### GAN Approach
In fact, most of the time, there are many generated low-quality samples (marked as red), as shown in Figure 5.  
The target of a GAN is to generate green samples instead of red samples. On the other hand, mode collapse restricts the diversity of the generated samples; that is, the generated samples are repetitive, similar, and lack modes, as shown in Figure 6.  
Figure 4: The one-dimensional Gaussian mixture distribution. Figure 5: The low-quality samples in actual situations. \[L(D,G)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}} [\log(1-D(G(z)))] \tag{1}\]0 The one-dimensional Gaussian mixture distribution. Because there are many regional Nash equilibrium states in a GAN, the parameter optimization is not a convex majorization problem. Even if a GAN enters a certain Nash equilibrium state and the loss function appears to converge, mode collapse may still occur. Mode collapse is usually accompanied by such a phenomenon: when the discriminator updates the parameters near the training samples, the gradient value is so large. Therefore, the solution of mode collapse is that the discriminator is near the training sample to impose a gradient penalty term \(\delta\), which is given by  
\[L(D,G)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}} [\log(1-D(G(z)))] \tag{1}\]1  
The proposed method attempts to construct a linear function near the training samples, because the linear function is a convex function and has a global optimal solution. The gradient penalty term is designed to improve GAN model and avoid mode collapse. By way of the reciprocal adversarial learning between \(D\) and \(G\), the capability of \(D\) and \(G\) is progressive. It has been proven that when \(P_{\text{g}}=P_{\text{data}}\), the distribution of the sample generated by \(G\) ideally matches the original fault-sample distribution. Moreover, the cross-entropy loss value converges to the Nash equilibrium when the output of \(D\) equals 0.5 [36]. Algorithm 1 shows the training algorithm of the GAN approach for generating new samples to augment the number of faulty samples. ```
1:For number of GAN training iterations do
2:For\(k\) steps do
3: Sample minibatch of \(m\) noise samples from noise prior \(P_{\text{z}}(z)\). 4: Sample minibatch of \(m\) examples from vibration data distribution \(P_{\text{data}}(x)\). 5: Update the \(D\) by ascending the stochastic gradient:
6:\(\
abla_{\theta_{d}}\frac{1}{m}\sum\limits_{i=1}^{m}\left[\log D\big{(}x^{i} \big{)}+\log\big{(}1-D\big{(}G(z^{i})\big{)}\big{)}\right]\)
7:endfor
8: Sample minibatch of \(m\) noise samples from noise prior \(P_{\text{z}}(z)\). 9: Update the \(G\) by descending the stochastic gradient:
10:\(\
abla_{\theta_{d}}\frac{1}{m}\sum\limits_{i=1}^{m}\log(1-D(G(z)))\)

endfor
```  
Any normal gradient-based learning regulation was used by the gradient-based updates, and the momentum was also used in the data augmentation experiments. \[L(D,G)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}} [\log(1-D(G(z)))] \tag{1}\]2 The duplicate samples and missing modes. As mentioned earlier, it is necessary to check the uncertainty of the generated samples. Therefore, a Gaussian-distribution cloud model was integrated into the GAN approach to evaluate this uncertainty. The mean \(\overline{X}\) and variance \(S^{2}\) of the generated samples are easily obtained. The entropy \(E_{\text{n}}\) of the generated samples is calculated as follows:  
\[L(D,G)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}} [\log(1-D(G(z)))] \tag{1}\]3  
where \(E_{\text{x}}\) refers to the expectation of the generated samples and is given by \(E_{x}=\overline{X}\). In the next step, the hyper entropy term (\(H_{e}\)) is the uncertainty index of \(E_{\text{n}}\). It incarnates the dispersion degree of the cloud droplets that make up the cloud model.