## 2 Materials and Methods
### Design of the Voice Disorder Detection Model using CGAN-IFCM
#### 2.2.2 Voice Disorder Detection Model Using IFCM  
As shown in Table 1, the voice disorder detection model can be formulated as a binary detection problem and a multi-class (4-class) detection problem. We selected a multi-class detection model for the illustration in this subsection because it represents the expected application through which the voice disorder detection model could diagnose actual types of voice disorders. To design the feature vector, there are two typical approaches: (i) follow clinical and expert guidelines; and (ii) if the first approach is not available, a thorough investigation of the proper features should be carried out. For the voice disorder detection problem, we follow the first approach because there is a clinical acoustic analysis technique available for the formal assessment of voice disorders [26; 27]. Four voice quality-based parameters are chosen. These qualities are harmonic to the noise ratio (HNR), shimmer, jitter, and fundamental frequency (f\({}_{0}\)). Since the estimations of these parameters are a standard approach, and feature extraction is not the focus of this paper, we follow [PERSON]'s algorithm to measure HRN [28], jitter (as the cycle-to-cycle variation of fundamental frequency) [29], shimmer (as the peak-to-peak amplitude variation in decibels) [29], and f\({}_{0}\) (as the maximal of autocorrelation function) [30]. In the implementation and performance evaluation, we adopt k-fold cross-validation, with k = 10 as the typical order [31; 32]. We define N\({}_{i}\) as the total number of samples in class 1, where \(N_{1}\), \(N_{2}\), \(N_{3}\), and \(N_{4}\) correspond to healthy candidates, those with hyperkinetic dysphonia, those with hypokinetic dysphonia, and those with reflux laryngitis, respectively. The feature vector is \(X_{p}=\{HRN_{p}\), \(jitter_{p}\), \(shimmer_{p},f_{0,p}\}\) for \(p=[1\), \(0.9(N_{1}+N_{2}+N_{3}+N_{4})]\), where \(0.9(N_{1}+N_{2}+N_{3}+N_{4})\) is equivalent to 90% of the training dataset. In general, the FCM problem is defined to minimize the objective function [33]. Intuitively, in each cluster, the data points should be close to the cluster center. Therefore, the objective is defined as a minimization of the intra-cluster variance, which is equivalent to a maximization of intra-cluster similarities. The less the variance, the greater the similarities among the data points. For example, a variance of zero results in maximal similarity:  
\[F_{\text{FCM}}=\sum_{p=1}^{(0.9)(N_{1}+N_{2}+N_{3}+N_{4})}\sum_{c=1}^{N_{c}} \mu_{pc}^{m}\big{\|}X_{p}-v_{c}\big{\|}^{2},1\leq m\leq\infty \tag{2}\]  
\[\text{s.t.}\hskip 56.905512 pt\sum_{c=1}^{N_{c}}u_{pc}=1 \tag{3}\]  
where \(N_{c}\) is the optimally designed total number of clusters using the multiobjective genetic algorithm (MOGA), \(u_{pc}\) is the degree of membership of \(X_{p}\) in the \(c\)th cluster, \(v_{c}\) is the cluster center of the \(c\)th cluster, and m controls the fuzziness of the resulting partition. The value \(m=2\) is common and is selected according to [33]. Taking the partial derivative, we have two iterative solutions, \(v_{c}\) and \(u_{pc}\), given by  
\[v_{c}=\frac{\sum_{p=1}^{(0.9)(N_{1}+N_{2}+N_{3}+N_{4})}u_{pc}^{m}X_{p}}{\sum_{ p=1}^{(0.9)(N_{1}+N_{2}+N_{3}+N_{4})}u_{pc}^{m}} \tag{4}\]  
\[u_{pc}=\frac{\sum_{p=1}^{N_{c}}\big{\|}X_{p}-v_{p}\big{\|}^{\frac{2}{m-1}}}{ \big{\|}X_{p}-v_{c}\big{\|}^{\frac{2}{m-1}}}. \tag{5}\]  
Nonetheless, the existing FCM algorithm [33] does not include the relationship between \(X_{p}\) and its neighbor \(X_{p}\), which limits its performance in clustering problems. Feature vector \(X_{p}\) is indeed strongly correlated (sharing similar characteristic) to its neighbor \(X_{p}\) due to the nature of characteristics in thefeature space. These \(X_{p}\) values have a high probability to be grouped into the same cluster. As a result, the rationale of the proposed work is to introduce the interaction between \(X_{p}\) and its neighbor \(X_{p}\) into the fuzzy membership function.