## 2 Preliminaries
### Derivation of the Proposed sSMC-FCM Algorithm
\]  
we can obtain \(V_{k}\) using  
\[\frac{\partial L}{\partial V_{k}}=\frac{\partial\left[\sum_{i=1}^{N}{U_{ik}} ^{m_{ik}}(X_{i}-V_{k})^{2}\right]}{\partial V_{k}}=\sum_{i=1}^{N}{U_{ik}}^{m_ {ik}}*2*(X_{i}-V_{k})*(-1)=0,\]  
hence, we have  
\[\sum_{i=1}^{N}{U_{ik}}^{m_{ik}}X_{i}=\sum_{i=1}^{N}{U_{ik}}^{m_{ik}}V_{k},\]  
or  
\[V_{k}=\frac{\sum_{i=1}^{N}{U_{ik}}^{m_{ik}}X_{i}}{\sum_{i=1}^{N}{U_{ik}}^{m_{ ik}}} \tag{10}\]  
And next, we calculate \(U_{\text{ik}}\) using  
\[\frac{\partial L}{\partial U_{ik}}=m_{ik}{U_{ik}}^{m_{ik}-1}{D_{ik}}^{2}- \lambda_{i}=0, \tag{11}\]  
which implies the following:  
\[U_{ik}=\left(\frac{\lambda_{i}}{m_{ik}{D_{ik}}^{2}}\right)^{\frac{1}{m_{ik}-1}} \tag{12}\]  
Combining with \(\sum_{j=1}^{C}U_{ij}=1\), we consider the two following cases:  
Case 1:  
For unsupervised elements \(i\), \(m_{ij}=M\) for all \(1\leq j\leq C\), Equation (12) becomes:\[U_{ik}=\left(\frac{\lambda_{i}}{M\text{D}_{ik}{}^{2}}\right)^{\frac{1}{M-1}}= \left(\frac{\lambda_{i}}{M}\right)^{\frac{1}{M-1}}\left(\frac{1}{D_{ik}{}^{2}} \right)^{\frac{1}{M-1}}=\left(\frac{\lambda_{i}}{M}\right)^{\frac{1}{M-1}}\left( \frac{1}{D_{ik}}\right)^{\frac{2}{M-1}} \tag{13}\]  
Since \(\sum_{j=1}^{C}U_{ij}=1\), we have \(\sum_{j=1}^{C}\left(\frac{\lambda_{i}}{M\text{D}_{ij}{}^{2}}\right)^{\frac{1}{ M-1}}=1\), or \(\left(\frac{\lambda_{i}}{M}\right)^{\frac{1}{M-1}}\sum_{j=1}^{C}\left(\frac{1}{ D_{ij}{}^{2}}\right)^{\frac{1}{M-1}}=1\), or  
\[\left(\frac{\lambda_{i}}{M}\right)^{\frac{1}{M-1}}=\frac{1}{\sum_{j=1}^{C} \left(\frac{1}{D_{ij}{}^{2}}\right)^{\frac{1}{M-1}}}=\frac{1}{\sum_{j=1}^{C} \left(\frac{1}{D_{ij}{}}\right)^{\frac{2}{M-1}}} \tag{14}\]  
Replacing (14) into (13), we obtain  
\[U_{ik}=\frac{1}{\sum_{j=1}^{C}\left(\frac{D_{ik}}{D_{ij}{}^{2}}\right)^{\frac{ 2}{M-1}}}=\left(\sum_{j=1}^{C}\left(\frac{D_{ik}}{D_{ij}}\right)^{\frac{2}{M-1 }}\right)^{-1} \tag{15}\]  
Case 2:  
For supervised elements \(i\) to belong to cluster \(k\), \(m_{ik}=M^{\prime}\) and \(m_{ij}=M\) for \(j\
eq k\), Equation (11) becomes:  
\[ \begin{cases}M^{\prime}U_{ik}{}^{M^{\prime}-1}D_{ik}{}^{2}=\lambda_{i},&and \\ MU_{ij}{}^{M-1}D_{ij}{}^{2}=\lambda_{i},&j\
eq k\end{cases} \]  
or  
\[M^{\prime}U_{ik}{}^{M^{\prime}-1}D_{ik}{}^{2}=M^{\prime}U_{i1}{}^{M^{\prime}-1 }D_{i1}{}^{2}=\ldots=M^{\prime}U_{ij}{}^{M^{\prime}-1}D_{ij}{}^{2}=\ldots=M^{ \prime}U_{iC}{}^{M^{\prime}-1}D_{iC}{}^{2}\]  
Combining with \(\sum_{j=1}^{C}U_{ij}=1\), to calculate \(U_{ij}\), we need to solve the following:  
\[ \begin{cases}M^{\prime}U_{ik}{}^{M^{\prime}-1}D_{ik}{}^{2}=MU_{i1}{}^{M-1}D_{ i1}{}^{2}=\ldots=MU_{ij}{}^{M-1}D_{ij}{}^{2}=\ldots=MU_{iC}{}^{M-1}D_{iC}{}^{2},&j \
eq k\\ \sum\limits_{j=1}^{C}U_{ij}=1&for\ all\ i\end{cases}  \tag{16}\]  
The steps to solve Equation (16) are shown through Equations (17)-(20) below. To make the presentation of the derivation seamless, we first give the calculation formulas, and the proof that this solution can solve Equation (16) will be presented in Section 3.2 with Proposition 1. Specifically, we calculate \(d_{min}=\underset{j=1,\ldots,C}{min}\left\{D_{ij}\right\}\), then:  
\[d_{ij}=\frac{D_{ij}}{d_{\min}},\forall j=1,\ldots,C \tag{17}\]  
\[\text{Calculate }\mu_{ij}=\left(\frac{1}{Md_{ij}{}^{2}}\right)^{\frac{1}{M-1}} \text{ for all }\ j\
eq k \tag{18}\]  
\[\text{Calculate }\mu_{ik}\text{ which is a variable in }\frac{\mu_{ik}}{\left(\mu_{ik}+\sum_{j=1,j\
eq k}^{C}\mu_{ij} \right)^{\frac{M^{\prime}-M}{M^{\prime}-1}}}=\left(\frac{1}{M^{\prime}d_{ik}{} ^{2}}\right)^{\frac{1}{M^{\prime}-1}} \tag{19}\]  
\[\text{Then normalize }:\ U_{ij}=\frac{\mu_{ij}}{\sum_{l=1}^{C}\mu_{il}} \tag{20}\]  
From there, we have the sSMC-FCM algorithm as follows. **Summary of steps for the proposed sSMC-FCM algorithm:**  
**Input:** the dataset \(X=\{X_{1},\ldots,X_{N}\}\), the fuzzification coefficients \(M>1\) and \(M^{\prime}>M\), and the set of supervised elements \(Y\subset\{1,2,\ldots,N\}\times\{1,2,\ldots,C\}\). **Output:** the partition of \(X\) into \(C\) clusters.