## 2 Methodology
### FC-SMOTE Algorithm Handing  
As one of the standard oversampling methods used to deal with class-imbalanced datasets, SMOTE artificially generates new minority class samples. It combines the generated samples with the initial training set to balance the training dataset. SMOTE generates synthetic samples in three steps. Firstly, it selects a random minority sample \(\vec{a}\) for observation. Then, it selects the sample \(\vec{b}\) among its k nearest minority sample neighbours. Finally, it creates a new sample \(\vec{x}\) by randomly interpolating two samples. The \(\vec{x}\) function is shown in Formula (1). \[\vec{x}=\vec{a}+\vec{w}\times(\vec{b}-\vec{a}), \tag{1}\]  
where \(w\) is the random weight in [0; 1]. However, SMOTE randomly selects a minority sample to oversample, which may generate minority class samples in majority regions in the presence of noise, as shown in Figure 2a. The cluster-based SMOTE can avoid the generation of noise by oversampling only in safe areas, as shown in Figure 2b. Besides, when considering the clinically imbalanced dataset dealing with the pathological voice, FC-SMOTE focuses on within-class imbalance by expanding the sparse minority areas to generate more differentiated minority samples to avoid underfitting the classifier. Thus, it can generate as many differentiated and representative minority samples as possible to improve the VPD system performance. FC-SMOTE consists of three steps--clustering, filtering, and oversampling. In the clustering step, after setting the number of clusters to be divided according to the number of categories, the fuzzy c-means (FCM) clustering algorithm is used to cluster the input samples to specify the number of clusters. The FCM algorithm is one of the most widely used fuzzy clustering algorithms and a popular iterative method for finding groups. It first selects the number of clusters. Then, it assigns coefficients between \([0,1]\) randomly to  
Figure 1: The architecture of the voice pathology detection (VPD) system using machine learning. initialize the membership matrix. The centroid of the cluster is calculated as the average of all points and weighted according to the degree of the cluster to which they belong. The position of the centroids for each cluster and coefficient in the clusters for each data point is updated. Repeat until the algorithm has converged. In the filtering step, the safe area for oversampling was selected based on the fuzzy set's spatial distribution. In the safe area, the Euclidean matrix of the membership degree of the minority samples and the centroid of a cluster is calculated to obtain the density distribution matrix to perform the next step of oversampling. Therefore, the filtering step allocates more generated samples to the sparse minority samples space in the safe area rather than in the dense minority sample space in order to generate as many minority samples with significant differences as possible to overcome the underfitting problem of the classifier caused by oversampling. Finally, in the oversampling step SMOTE is used in each selected cluster based on sampling weight to balance the number of samples in the minority and majority classes. The safe areas used for oversampling are selected based on a fuzzy set. A fuzzy set is a set that has degrees of membership between 0 and 1. In the following formula, the fuzzy set is represented with a tilde character(\(\sim\)). We define the safe area \(\widehat{\mathbf{C}}_{(k,safe)}\) as shown in Formula (2). \[ \begin{cases}\widehat{\mathbf{C}}_{(k,safe)}=\widehat{\mathbf{C}}_{k}-\sum_{i =1,j\
eq k}^{j}(\widehat{\mathbf{C}}_{k}\cap\widehat{\mathbf{C}}_{i})\quad(k =1,2,...,j)\\ \widehat{\mathbf{U}}=\sum_{i=1}^{j}\widehat{\mathbf{C}}_{i}.\end{cases}  \tag{2}\]  
In Formula (2), \(j\) is the class number of minority samples and \(\widehat{\mathbf{U}}\) is the membership matrix universal set.