## 4 Interval Improved Fuzzy Partitions Fuzzy C-Means Clustering Algorithm
### Interval Improved Fuzzy Partitions Fuzzy C-Means under Euclidean Distance  
The new objective function, \(J_{ifpfem}\), for the IIFPFCM clustering algorithm is based on a squared Euclidean distance between the vectors of intervals given in Equation (8) and of which the constraints are given in Equation (9):  
\[ \begin{array}{ll}J_{ifpfem}(\overline{G},\ \overline{U},\ X)&=\sum\limits_{i=1}^{c} \sum\limits_{k=1}^{n}\overline{u}_{ik}^{2}\mathrm{D}_{\mathrm{L2}}\left( \overrightarrow{x}_{k},\overrightarrow{g}_{i}\right)-\sum\limits_{k=1}^{n} \overrightarrow{e}_{k}\sum\limits_{i=1}^{c}\left(\overline{u}_{ik}-\frac{1}{2} \right)^{2},\\ &=\sum\limits_{i=1}^{c}\sum\limits_{k=1}^{n}\overline{u}_{ik}^{2}\sum\limits_{j =1}^{p}\left[\left(a_{k}^{j}-\overline{a}_{i}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{g}_{i}^{j}\right)^{2}\right]-\sum\limits_{k=1}^{n}\overrightarrow{e}_ {k}\sum\limits_{i=1}^{c}\left(\overline{u}_{ik}-\frac{1}{2}\right)^{2},\end{array}  \tag{8}\]subject to \(\sum_{i=1}^{c}\overline{u}_{ik}=1\). In Equation (8), \(\overline{u}_{ik}\) is the fuzzy membership degree of pattern \(k\) in the \(i\)th cluster, and \(\overline{U}=[\overline{u}_{ik}]\) is a \(c\times n\) matrix; the \(\overline{\epsilon}_{k}\) is similar to the IFPFCM defined in Equation (10):  
\[\overline{\epsilon}_{k}=\min\left\{\sum_{j=1}^{p}\left[\left(a_{k}^{j}- \overline{\epsilon}_{i}^{j}\right)^{2}+\left(b_{k}^{j}-\overline{\beta}_{i}^{ j}\right)^{2}\right]-\eta|i\in\{1,\,2,\,\ldots,\,c\}\right\}\text{, }\eta\in(0,\,\infty). \tag{10}\]  
Using the Lagrange multipliers method for the objective function \(J_{ifpfcme}\), the nonlinear unconstraint optimization can be obtained as:  
\[L_{ifpfcme}=\sum_{i=1}^{c}\sum_{k=1}^{n}\overline{u}_{ik}^{2}\sum_{j=1}^{p} \left[\left(a_{k}^{j}-\overline{\epsilon}_{i}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{\beta}_{i}^{j}\right)^{2}\right]-\sum_{k=1}^{n}\overline{\epsilon}_ {k}\sum_{i=1}^{c}\left(\overline{u}_{ik}-\frac{1}{2}\right)^{2}-\sum_{k=1}^{ n}\lambda_{k}\left(\sum_{i=1}^{c}\overline{u}_{ik}-1\right). \tag{11}\]  
The \(\overline{u}_{ik}\) was considered for the minimum problem; to derive an updated equation for the fuzzy membership degree, \(\overline{u}_{ik}\) can be solved as follows:  
\[\frac{\partial L_{ifpfcme}}{\partial\overline{u}_{ik}}=0\Leftrightarrow 2 \overline{u}_{ik}\sum_{j=1}^{p}\left[\left(a_{k}^{j}-\overline{\epsilon}_{i}^{ j}\right)^{2}+\left(b_{k}^{j}-\overline{\beta}_{i}^{j}\right)^{2}\right]-2 \overline{\epsilon}_{k}\left(\overline{u}_{ik}-\frac{1}{2}\right)-\lambda_{k} =0, \tag{12}\]  
for \(i=1,\,\,\ldots,\,c\) and \(k=1,\,\,\ldots,\,n\). Thus, Equation (12) can be rewritten as:  
\[\overline{u}_{ik}=\frac{\lambda_{k}-\overline{\epsilon}_{k}}{2\left(\sum_{j= 1}^{p}\left[\left(a_{k}^{j}-\overline{a}_{i}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{\beta}_{i}^{j}\right)^{2}\right]-\overline{\epsilon}_{k}\right)}, \tag{13}\]  
for \(i=1,\,\,\ldots,\,c\) and \(k=1,\,\,\ldots,\,n\). Substituting Equation (13) into Equation (9) is obtained as:  
\[\sum_{s=1}^{c}\frac{\lambda_{k}-\overline{\epsilon}_{k}}{2\left(\sum_{j=1}^{ p}\left[\left(a_{k}^{j}-\overline{a}_{s}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{\beta}_{s}^{j}\right)^{2}\right]-\overline{\epsilon}_{k}\right)}=1. \tag{14}\]  
According to Equations (13) and (14), the updated equation for the fuzzy membership degree \(\overline{u}_{ik}\) is:  
\[\overline{u}_{ik}=\frac{1}{\sum\limits_{s=1}^{c}\left(\frac{\sum_{j=1}^{p} \left[\left(a_{k}^{j}-\overline{a}_{i}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{\beta}_{i}^{j}\right)^{2}\right]-\overline{\epsilon}_{k}}{\sum_{j=1}^ {c}\left[\left(a_{k}^{j}-\overline{a}_{s}^{j}\right)^{2}+\left(b_{k}^{j}- \overline{\beta}_{s}^{j}\right)^{2}\right]-\overline{\epsilon}_{k}\right)}}, \tag{15}\]  
for \(i=1,\,\,\ldots,\,c\) and \(k=1,\,\,\ldots,\,n\).