## 2 Methodologies
### GAN Approach  
The GAN approach was inspired by a zero-sum game, and it is able be used to generate many fault signals to achieve data augmentation. The GAN approach consists of two models, namely the generative model (\(G\) model) and the discriminant model (\(D\) model). Usually, both the \(G\) and \(D\) models apply a neural network structure, such as the typical multilayer perception network. The input of the \(G\) model is a stochastic noise signal \(Z=(z_{1},z_{2},\ldots,z_{\rm m})\) with a data distribution \(P_{\rm z}\). The output of the \(G\) model is the generated signals or samples \(G(z)\), with a data distribution \(P_{\rm g}\) that resemble the true distribution \(P_{\rm data}\) of the original experiment signal. The relationships among the \(P_{\rm z}\), \(P_{\rm g}\) and \(P_{\rm data}\) are shown in Figure 2. For example, the generative model can convert a random-noise signal into multiple new signals and transform the data in generated data space to the true data space through the loss. The input of the \(D\) model is the generated samples \(G(z)\) and the true and fault samples \(X=(x_{1},x_{2},\ldots,x_{\rm n})\). The output of \(D\) model is the probability that the generated fault samples are true samples. The likelihood function \(\rm L(\textit{D},G)\) of \(G\) and \(D\) is given by  
\[L(D,G)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}} [\log(1-D(G(z)))] \tag{1}\]  
During the training procedure in GAN, \(G\) is first initialized and fixed; then, \(D\) is trained by maximizing the loss function in Equation (2). \(D\) is upgraded using the stochastic gradient ascending. \[\mathrm{max}_{D}L(D)=\mathrm{E}_{x\sim P_{\rm data}}[\log D(x)] \tag{2}\]  
The trained \(D\) is fixed, and then \(G\) is promoted by minimizing the loss function in Equation (3) with the descending stochastic gradient. \[\mathrm{min}_{G}L(G)=\mathrm{E}_{z\sim P_{\rm z}}[\log(1-D(G(z)))] \tag{3}\]  
The basis of generating vibration fault samples is to ensure their accuracy. Under this premise, the target of \(D\) is to recognize the true or false fault samples by exporting the logical value. The \(G\) and \(D\) models are alternately trained until the GAN model reaches the Nash equilibrium [34]. By combining Equations (2) and (3), the cross-entropy loss of \(\rm L(\textit{D},G)\) is calculated as follows:  
\[\mathrm{min}_{G}\mathrm{max}_{D}\big{\{}L(D,G)=\mathrm{E}_{x\sim P_{\rm data}} [\log D(x)]+\mathrm{E}_{z\sim P_{\rm z}}[\log(1-D(G(z)))]\big{\}} \tag{4}\]  
With many peaks, the data distribution of the vibration signals is highly complex and multi-modal. Each peak is called a mode, and each mode represents the concentration of similar failure samples. Mode collapse means that \(G\) only exports part of multiple failure modes, and misses other failure modes. In mode collapse, the generated failure samples belongs to a limited set of modes, when \(G\) thinks it can fool \(D\) by locking a single mode. \(D\) eventually finds that the samples in the single mode are fake. However, \(G\) is just locked into the single mode, which fundamentally limits the generated samples' diversity to affect the GAN model performance. According to the manifold distribution law [35], high-dimensional data of the same category is often concentrated near a low-dimensional manifold. The ideal situation for \(G\) is to map the input noise to the manifold where the training data is located, and correspond to the probability distribution of the training data. For example, the probability distribution of a training data set is a simple one-dimensional Gaussian mixture distribution, including two peaks, as shown in Figure 3.  
Ideally, the generated samples are as shown in Figure 4 (marked in green). The positions of the generated samples are almost under the two peaks, and the samples conform to the probability distribution of the training set. However, the ideal situation is almost impossible.