## 2 Preliminaries
### Semi-Supervised Standard Fuzzy C-Means Clustering (sSFCM) Algorithms  
The semi-supervised fuzzy clustering method in [14] added a supervisory matrix \(\overline{U}\) that represents the supervision of an element forced to belong or to not belong in a cluster. \(\overline{U}_{ik}>0\) when the supervised element \(i\) is placed into cluster \(k\), and \(\overline{U}_{ik}=0\) when there is no supervision. This additional supervised membership grade \(\overline{U}_{ik}\) was added to the objective function to be minimized, and is shown as follows:  
\[J(U,V)=\sum_{i=1}^{N}\sum_{k=1}^{C}\left|U_{ik}-\overline{U}_{ik}\right|^{m}D _{ik}^{2}, \tag{4}\]  
where \(\sum_{j=1}^{C}U_{ij}=1\), \(\sum_{j=1}^{C}\overline{U}_{ij}\leq 1\), \(0\leq U_{ik}\leq 1\), for all \(1\leq i\leq N\) and \(1\leq k\leq C\). The sSFCM algorithm, shown below, works to minimize \(J(U,V)\) through many iterations.  
**Summary of steps for the sSFCM algorithm:**  
**Input:** the dataset \(X=\{X_{1},X_{2},\ldots,X_{N}\}\), and the supervised membership grade \(\overline{U}\).  
**Output:** the partition of \(X\) into \(C\) clusters.  
* Step 1: Initialize value for \(V\), let \(l=0\), set \(\varepsilon>0\) and \(m>1\).
* Step 2: At the \(l-th\) loop, update \(U\) according to the formula: \[U_{ik}=\overline{U}_{ik}+\left(1-\sum_{j=1}^{C}\overline{U}_{ij}\right)\left( \sum_{j=1}^{C}\left(\frac{D_{ik}}{D_{ij}}\right)^{\frac{2}{m-1}}\right)^{-1}\] (5)
* Step 3: Update \(V\) for the next step \((l+1)\), according to the formula: \[V_{k}=\frac{\sum_{i=1}^{N}{\left|U_{ik}-\overline{U}_{ik}\right|}^{m}X_{i}}{ \sum_{i=1}^{N}{\left|U_{ik}-\overline{U}_{ik}\right|}^{m}}\] (6)
* Step 4: If \(\left|\left|V^{(l)}-V^{(l+1)}\right|\right|<\varepsilon\), then go to Step 5; otherwise, let \(l:=l+1\), and return to Step 2.
* Step 5: End.  
**A numerical example of the sSFCM algorithm**[14]:  
A data set with 20 elements \(X=\{X_{1},X_{2},\ldots,X_{20}\}\), \(X_{i}\in R^{2}\), as seen in Table 1, is to be divided into two clusters. Implement the sSFCM algorithm with \(m=2\) for 3 cases:The clustering results indicate that sample points 9 and 10 belong to cluster 2 in case 1 and case 2 and to cluster 1 in case 3, showing the impact of the added semi-supervised component in the sSFCM algorithm through the supervised membership grade \(\overline{U}\). Although in case 2, the attempt to place points 9 and 10 in cluster 1 was not successful, the points were much closer to cluster 1 compared to case 1.  
In this study, instead of using the matrix \(\overline{U}\), we propose the use of multiple fuzzification coefficients as the mean to make the original unsupervised FCM algorithm a semi-supervised one.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**i** & **(X\({}_{\mathbf{i1}},\mathbf{X_{i2}}\))** & **i** & **(X\({}_{\mathbf{i1}},\mathbf{X_{i2}}\))** & **i** & **(X\({}_{\mathbf{i1}},\mathbf{X_{i2}}\))** & **i** & **(X\({}_{\mathbf{i1}},\mathbf{X_{i2}}\))** \\ \hline \(1\) & \((0,4.5)\) & 6 & \((3.5,5.5)\) & \(11\) & \((9,0)\) & \(16\) & \((10,0)\) \\ \hline \(2\) & \((0,5.5)\) & 7 & \((5.25,4.5)\) & \(12\) & \((9,2.5)\) & \(17\) & \((10,2.5)\) \\ \hline \(3\) & \((1.75,4.5)\) & 8 & \((5.25,5.5)\) & \(13\) & \((9,5)\) & \(18\) & \((10,5)\) \\ \hline \(4\) & \((1.75,5.5)\) & 9 & \((7,4.5)\) & \(14\) & \((9,7.5)\) & \(19\) & \((10,7.5)\) \\ \hline \(5\) & \((3.5,4.5)\) & 10 & \((7,5.5)\) & \(15\) & \((9,10)\) & \(20\) & \((10,10)\) \\ \hline \end{tabular}  
* Case 1: unsupervised, \(\overline{U}_{ik}=0\), for all \(1\leq i\leq 20\), \(1\leq k\leq 2\) (standard FCM algorithm). Case 2: semi-supervised, attempting to place data points 9 and 10 into cluster 1, \(\overline{U}_{9,1}=\overline{U}_{10,1}=0.3\), otherwise, \(\overline{U}_{ik}=0\). Case 3: semi-supervised, attempting to place data points 9 and 10 into cluster 1, \(\overline{U}_{9,1}=\overline{U}_{10,1}=0.6\), otherwise, \(\overline{U}_{ik}=0\).  
\begin{tabular}{c c c c c} \hline  
Semi-Supervised Fuzzy C-Means Clustering Algorithm with Multiple Fuzzification Coefficients (sSMC-FCM)