## 2 Materials and Methods
### Design of the Voice Disorder Detection Model using CGAN-IFCM
#### 2.2.1 Generation of Additional Training Data Using CGAN  
Imbalanced classes were observed in the databases SVD and VOICED. This imbalance causes a significant deviation between the sensitivity and specificity of the voice disorder detection model, particularly in VOICED [16; 17; 18]. The goal is to increase the number of training samples to balance the classes. A recent state-of-the-art article presented the development and progress of the evolution of GAN [22]. The basic GAN proposal has a key limitation: its noise vector does not have restrictions and may lead to fatal theoretical issues. As a result, different kinds of solutions have been proposed. The following relevant articles have received a large number of citations: auxiliary classifier GAN (ACGAN) [23], CGAN [24], and information maximizing GAN (InfoGAN) [25]. In this paper, CGAN is selected to increase the number of training samples for classes with fewer samples. The first reason is that various types of information (the information included in Table 1) are included in SVD and VOICED. This information is conditional information that fits well with the theory of CGAN. Further, conditional information is introduced in both the generator and discriminator to reduce the aforesaid fatal theoretical issue. Conditional generation is then introduced, in which CGAN is restricted to generate new data for the class of the largest sample size. Therefore, in the binary detection model, the class of candidates with voice disorders is restricted in both SVD and VOICED. For the multi-class detection model, the class of healthy candidates and the class of candidates with hyperkinetic dysphonia are restricted in SVD and VOICED, respectively. To avoid the excessive generation of training data, there is another restriction that involves limiting the number of generated samples of a class to, at most, the original sample size quantity of the corresponding class.  
Figure 1 shows the system workflow of CGAN. Given noise vector \(n\) and conditional variable \(c\), generator \(G\) captures data distribution \(x\), and discriminator \(D\) determines whether a sample was from the original dataset or generated dataset. Both \(G\) and \(D\) are conditioned. Basically, \(n\) and \(c\) serve as the inputs of \(G\), \(G\) serves as the input of \(G(n|c)\), and \(c\) serves as the input of \(x\). To train a discriminator, \(D(x;\theta_{d})\) gives a single scalar representing the probability that \(x\) came from the training data instead of \(p_{data}(x)\). To train the generator distribution \(p_{data}(x)\) using data distribution \(x\), the generator constructs a mapping function from Gaussian noise distribution \(p_{n}(n)\) to the data space as \(G(n;\theta_{g})\). The training of \(D\) and \(G\) occurs at the same time, where parameters \(\theta_{d}\) for \(D\) are adjusted to minimize \(logD(x|c)\), and parameters \(\theta_{g}\) for \(G\) are adjusted to minimize \(log(1-D(G(n|c)))\) for the conditional variable \(c\). Mathematically, the objective function is expressed as follows:  
\[\min_{G}\max_{D}V(D,G)=E_{x\sim p_{data}(x)}[\log(D(x|c))]+E_{x\sim p_{n}(n)}[ \log(1-D(G(n|c)))] \tag{1}\]  
where \(V(D,G)\) is the value function.  
Figure 1: System workflow of conditional generative adversarial network (CGAN).