## 2 Proposed Methodology
### Previous Works
#### 2.2.3 Generative Adversarial Network  
The GAN model, originally proposed to produce real-like images from Gaussian noise input, has also been developed to generate synthetic time-series data [50; 51; 52; 53]. The GAN model is composed of two deep learning models: generator and discriminator. The generator generates synthetic data by using Gaussian noise input. The discriminator gets real data and synthetic data as input data and discriminates whether the input data are real or synthetic. By using the training results of these models, parameters are updated to produce realistic synthetic data, which means that the synthetic data distribution \(\mathrm{p}_{z}\) has a similar distribution to the real data \(\mathrm{p}_{\mathrm{data(x)}}\). The loss function of the GAN model is written as Equation (3), which is the deformed shape of the Jensen-Shannon Divergence (JSD), where \(\mathrm{D(x)}\) is the discriminator output, and \(\mathrm{G(z)}\) is the generator output.  
\[\mathrm{maxminV(D,G)}=\mathrm{E}_{\mathrm{x}\sim\mathrm{P}_{\mathrm{data(x)}}} [\log\mathrm{D(x)}]+\mathrm{E}_{\mathrm{z}\sim\mathrm{P}_{\mathrm{z(z)}}}[ \log(1-\mathrm{D(G(z))})] \tag{3}\]  
The generator and discriminator keep training to deceive each other until the models reach the Nash Equilibrium. Figure 1 illustrates the training process of the general GAN model.  
Although the GAN model shows remarkable data augmentation performance, its mathematical nature raises several critical points. There are two main limitations to the application of the GAN model: gradient vanishing and mode collapse [54; 55]. Both problems arise from the loss function of the GAN model. The fundamental concept of the GAN model is a minimax game in which the discriminator tries to maximize the loss function while the generator attempts to minimize the loss function. In this process, if the discriminator is trained better than the generator, the gradient of the generator becomes 0 according to Equation (3), and the generator cannot be trained properly. This is called gradient vanishing. Also, when the generator generates synthetic data in multiple classes, the generator only tries to deceive the discriminator and does not consider the information about each class. As a result of this characteristic, the training process of the generator could be biased to specific weights, which means the generator only generates data belonging to a specific class, not evenly. Simply, the generator falls into a local minimum problem and cannot imitate the distribution in any other classes except one class. This is called mode collapse.  
Due to the drawbacks mentioned above, instability in training the GAN model often takes place. This makes it hard to converge the generator and discriminator, reducing the performance of the GAN model. Numerous unrevealed factors may exist in this instability, but one of the fundamental reasons is the loss function. The loss function that uses JSD could output 0 or too large a number, as mentioned above, leading to the submission of meaningless gradients to the generator and discriminator. To overcome these vulnerabilities, various GAN models that use different loss functions have been introduced.  
Figure 1: Training process of the GAN model.