## 2 Materials and Methods
### Design of the Voice Disorder Detection Model using CGAN-IFCM
#### 2.2.2 Voice Disorder Detection Model Using IFCM
A new degree of membership is proposed as follows:  
\[u_{neighbor,pc}=\frac{C_{pc}\cdot\sum_{p=1}^{N_{c}}\
ormalsize{\left\|X_{p}-v_{ p}\right\|^{\frac{2}{m-1}}}}{\
ormalsize{\left\|X_{p}-v_{c}\right\|^{\frac{2}{m-1}}}} \tag{6}\]  
\[C_{pc}=\frac{\sum_{j\in SN(X_{p})}u_{jc}}{N_{n}} \tag{7}\]  
where \(C_{pc}\) is a conditional variable related to the level of participation of \(X_{p}\) in the \(c\)th cluster, \(SN(X_{p})\) is the square neighborhood centered at \(X_{p}\), and \(N_{c}\) is the number of feature vectors in the neighborhood. The joined degree of membership between \(u_{pc}\) and \(u_{neighbor,pc}\) is defined as follows:  
\[u_{VDC,pc}=\frac{(u_{neighbor,pc})^{\omega_{n}}(u_{pc})^{\omega_{0}}}{\sum_{q= 1}^{N_{n}}(u_{neighbor,pq})\Big{(}u_{pq}\Big{)}} \tag{8}\]  
\[v_{VDC,c}=\frac{\sum_{p=1}^{(0.9)(N_{1}+N_{2}+N_{3}+N_{4})}u_{VDC,pc}^{m}\cdot X _{p}}{\sum_{p=1}^{(0.9)(N_{1}+N_{2}+N_{3}+N_{4})}u_{VDC,pc}^{m}} \tag{9}\]  
where \(\omega_{n}\) and \(\omega_{0}\) are the control weighting factors of the importance between \(u_{neighbor,pc}\) and \(u_{pc}\). These are optimally designed via MOGA, which benefits the convergence of the model training. There are numerous combinations of \(\omega_{n}\) and \(\omega_{0}\) which may yield different \(u_{VDC,pc}\) and \(v_{VDC,c}\) values. Searching for the optimal \(\omega_{n}\) and \(\omega_{0}\) requires considerable computing power. Alternatively, a tradeoff is sought between the convergence of model training and computing power while maintaining favorable performance. The multiobjective optimization problem consists of four objectives, given by  
\[ \begin{cases}MAX&F_{1}=CH_{index}\\ MIN&F_{2}=\
ormalsize{\left\|U_{VDC}(^{b+1})-U_{VDC}(^{b})\right\|}\\ MAX&F_{3}=Specificity=TNR\\ MAX&F_{4}=Sensitivity=TPR\end{cases}  \tag{10}\]  
where \(CH_{index}\) is the Calinski-Harabasz (CH) index, which is important to determine the number of clusters [34]. \(CH_{index}\) evaluates the cluster validity based on the sum of the square between the cluster and the sum of the square within the cluster. Varying the number of clusters \(N_{c}\) yields distinct values of \(CH_{index}\): The higher the \(CH_{index}\) is, the better the solution. In addition, \(U_{VDC}=(u_{VDC,pc})(N_{1}+N_{2}+N_{3}+N_{4})\)-(\(N_{c}\)) is the fuzzy membership matrix, \(TNR\) is the true negative rate (equivalent to specificity), and \(TPR\) is the true positive rate (equivalent to sensitivity) of the voice disorder detection model. \(CH_{index}\), \(TNR\), and \(TPR\) are given by  
\[CH_{index}=\frac{SS_{BC}/(N_{c}-1)}{SS_{WC}((0.9)(N_{1}+N_{2}+N_{3}+N_{4})-N_{c})} \tag{11}\]  
\[Specificity=TNR=\frac{TN}{TN+FP} \tag{12}\]  
\[Sensitivity=TPR=\frac{TP}{TP+FN} \tag{13}\]where \(SS_{BC}\) is the sum of the square between clusters, \(N_{c}\) is the number of clusters, \(SS_{WC}\) is the sum of the square within the cluster. The larger the \(SS_{BC}\) is, the higher the degree of dispersion between the clusters becomes. The smaller the \(SS_{WC}\) is, the closer the relationship in the cluster. In addition, \(TN\) is the true negative, \(FP\) is the false positive, \(TP\) is the true positive, and \(FN\) is the false negative of the testing samples. The proposed MOGA-IFCM algorithm is then applied to solve the multiobjective optimization problem in Equation (10) [35; 36]. Generally, solving a multiobjective optimization problem has an overall run time complexity of O(GMN\({}^{2}\)), where G, M, and N are the number of generations, number of objectives, and population size, respectively. We adopted a hyper-grid scheme [37; 38] to reduce the time complexity. The calculation of individual sharing functions is restricted only to individuals in neighboring cells. Each cell contains a few individuals. The overall time complexity is thus reduced to O(GMN). Evolutionary algorithms (for instance, the genetic algorithm) tend to converge to a single solution as the diversity of the population diminishes [39]. This phenomenon is called genetic drift.