## 2 The FCS and WFCS Algorithms
### WFCS Algorithm  
#### 2.2.1 The Principle of WFCS  
Aiming at clustering data more reasonably, we introduce feature weight into the FCS. Firstly, we define the feature weighting fuzzy within-cluster matrix \(S_{\mathit{WFW}}\) and between-cluster matrix \(S_{\mathit{WFB}}\) as follows:  
\[S_{\mathit{WFW}}= \sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{s} \mu_{y}^{m}\omega_{k}^{\alpha}\|x_{j\text{\text{\textminus}}}-a_{ik}\|^{2} \tag{7}\] \[S_{\mathit{WFB}}= \sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{s} \eta_{i}\mu_{y}^{m}\omega_{k}^{\alpha}\|a_{ik}-\overline{X_{k}}\|^{2} \tag{8}\]  
We extend the formula of \(\eta_{i}\) as:  
\[\eta_{i}= \frac{\beta}{4}\frac{\min_{\mathit{ssi}}\omega^{\alpha}\|a_{i}-a_{i }\|^{2}}{\max_{t}\omega^{\alpha}\|a_{i}-\overline{X}\|^{2}} \tag{9}\]  
Based on Equation (7) and Equation (8), the objective function is shown as:  
\[J_{\mathit{WFCS}}= \sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{s} \mu_{y}^{m}\omega_{k}^{\alpha}\|x_{j\text{\textminus}}-a_{ik}\|^{2}-\sum \limits_{i=1}^{c}\sum\limits_{j=1}^{s}\sum\limits_{k=1}^{s}\eta_{i}\mu_{y}^{m }\omega_{k}^{\alpha}\|a_{ik}-\overline{X_{k}}\|^{2} \tag{10}\]  
Hence, WFCS can be formulated as an optimization problem which can be expressed as:  
Figure 1: Illustration of the crisp kernel.  
_Algorithms_**2015**, \(8\)  
\[\left\{\begin{aligned} &\min J_{{}_{WFCS}}\\ & s.t.\sum_{i=1}^{c}\mu_{y}=\mathrm{l},\sum_{k=1}^{s}\omega_{k}= \mathrm{l}\end{aligned}\right. \tag{11}\]  
Equation (11) can be solved via the Lagrange multiplier. The \(\mathrm{L}\) function can be given by:  
\[L=\sum_{i=1}^{c}\sum_{j=1}^{n}\sum_{k=1}^{s}\mu_{y}^{m}\omega_{k}^{{}^{\prime \prime}}\left\|x_{jk}-a_{ik}\right\|^{2}-\sum_{i=1}^{c}\sum_{j=1}^{n}\sum_{k=1}^ {s}\eta_{i}\mu_{y}^{m}\omega_{k}^{{}^{\prime\prime}}\left\|a_{ik}-\overline{X_ {k}}\right\|^{2}+\sum_{j=1}^{n}\lambda_{j}\left(\sum_{i=1}^{c}\mu_{y}-1\right) -\gamma\left(\sum_{k=1}^{s}\omega_{k}-1\right) \tag{12}\]  
Let the partial derivatives of \(\mathrm{L}\) function with respect to \(\mu_{y},\ \omega_{k},\ \lambda_{j}\) and \(\gamma\) equal to zero. Then we have:  
\[a_{ik}=\frac{\sum_{j=1}^{n}\mu_{y}^{m}\left(x_{jk}-\eta_{i}\overline{X_{k}} \right)}{\sum_{j=1}^{n}\mu_{y}^{m}\left(\mathrm{l}-\eta_{i}\right)} \tag{13}\]  
\[\omega_{k}=\frac{\left(\sum_{i=1}^{c}\sum_{j=1}^{n}\mu_{y}^{m} \left(\left\|x_{jk}-a_{ik}\right\|^{2}-\eta_{i}\left\|a_{ik}-\overline{X_{k}} \right\|^{2}\right)\right)^{\frac{1}{\mathrm{l}-\alpha}}}{\sum_{i=1}^{s}\left[ \sum_{i=1}^{c}\sum_{j=1}^{n}\mu_{y}^{m}\left(\left\|x_{ji}-a_{ii}\right\|^{2}- \eta_{i}\left\|a_{ii}-\overline{X_{i}}\right\|^{2}\right)\right]^{\frac{1}{ \mathrm{l}-\alpha}}} \tag{14}\]  
\[\mu_{y}=\frac{\left(\sum_{k=1}^{s}\omega_{k}^{\alpha}\left(\left\|x_{jk}-a_{ik }\right\|^{2}-\eta_{i}\left\|a_{ik}-\overline{X_{k}}\right\|^{2}\right)\right) ^{\frac{1}{\mathrm{l}-m}}}{\sum_{i=1}^{c}\left(\sum_{k=1}^{s}\omega_{k}^{ \alpha}\left(\left\|x_{jk}-a_{ik}\right\|^{2}-\eta_{i}\left\|a_{ik}-\overline{ X_{k}}\right\|^{2}\right)\right)^{\frac{1}{\mathrm{l}-\alpha}}} \tag{15}\]  
According to Equations (13)-(15), dataset \(X\) can be partitioned into \(c\) clusters by iteratively updating \(a\), \(\omega\) and \(\mu\).  
We note here that the objective functions of WFCS and ESSC include the within-cluster compactness and between-cluster separation. However, in ESSC the parameter \(\eta\) will be assigned a value at the beginning of the iteration procedure and will be fixed. Furthermore, if \(\eta\)=0, the ESSC will degrade to the entropy weighting clustering algorithm without the between-cluster information. However, in the proposed WFCS, \(\eta\) will be calculated automatically by the between-cluster information and will not be zero if the parameter \(\beta\
eq 0\).