## 4 Interval Improved Fuzzy Partitions Fuzzy C-Means Clustering Algorithm
### Interval Improved Fuzzy Partitions Fuzzy C-Means under City Block Distance
The prototype \(\widetilde{\widetilde{\delta}}_{i}=(\widetilde{\delta}_{i}^{1},\ \ldots,\ \widetilde{ \delta}_{i}^{p})\) of the \(i\)th cluster, which minimizes the objective function \(J_{lifpfcmc}(\widetilde{G},\ \widetilde{U},\ X)\), has the bounds of the interval \(\widetilde{g}_{i}^{j}=\left[\widetilde{\alpha}_{i}^{j},\ \widetilde{\beta}_{i}^{j}\right]\) (\(j=1,\ \ldots,\ p\)), which was updated according to the following problem:  
\[\underset{\widetilde{\alpha}_{i}^{j}\widetilde{\beta}_{i}^{j}}{\text{Minimize}} \sum\limits_{i=1}^{c}\sum\limits_{k=1}^{n}\widetilde{u}_{ik}^{2}\sum\limits_{j =1}^{p}\Bigl{(}\left|a_{k}^{j}-\widetilde{\alpha}_{i}^{j}\right|+\left|b_{k}^ {j}-\widetilde{\beta}_{i}^{j}\right|\Bigr{)}. \tag{26}\]  
This yields two well-known minimization problems in \(L_{1}\) norm, such as:  
\[\underset{\widetilde{\alpha}_{i}^{j}}{\text{Minimize}}\sum\limits_{i=1}^{c} \sum\limits_{k=1}^{n}\widetilde{u}_{ik}^{2}\sum\limits_{j=1}^{p}\Bigl{|}a_{k} ^{j}-\widetilde{\alpha}_{i}^{j}\Bigr{|},\text{ and }\underset{\widetilde{\beta}_{i}^{j}}{\text{Minimize}}\sum \limits_{i=1}^{c}\sum\limits_{k=1}^{n}\widetilde{u}_{ik}^{2}\sum\limits_{j=1}^ {p}\Bigl{|}b_{k}^{j}-\widetilde{\beta}_{i}^{j}\Bigr{|}. \tag{27}\]  
In [PERSON] et al., 2006 [32], the solutions of minimized \(\sum\limits_{k=1}^{n}\left|h_{k}^{j}-z_{i}^{j}\right|\) for \(z_{i}^{j}\) have been shown as the median of set \(\{h_{k}^{j}\), for \(k=1,\ \ldots,\ n\}\). In [PERSON] et al., 2006 [32], there was no \(\widetilde{u}_{ik}^{2}\) in the objection function. However, in this paper, we needed to consider \(\widetilde{u}_{ik}^{2}\). Hence, we needed to consider that if \(\widetilde{u}_{ik}>1/C\), then \(\widetilde{u}_{ik}=1\), and if \(\widetilde{u}_{ik}\leq 1/C\), then \(\widetilde{u}_{ik}=0\). This criteria for the fuzzy membership degree represented by it is greater than the average of all groups, so the degree \(\widetilde{u}_{ik}\) is strengthened, making its grouping more obvious. As a result, the updated equations of \(\widetilde{\alpha}_{i}^{j}\) and \(\widetilde{\beta}_{i}^{j}\) are obtained as:  
\[\widetilde{\alpha}_{i}^{j}=\widetilde{\alpha}_{k}^{j}\text{ and }\widetilde{\beta}_{i}^{j}= \widetilde{b}_{k}^{j} \tag{28}\]  
where \(\ \widetilde{\alpha}_{k}^{j}\) and \(\widetilde{b}_{k}^{j}\) denote the medians of sets \(\{a_{k}^{j}\), for \(k=1,\ \ldots,\ n\) and \(\widetilde{u}_{ik}=1\}\) and \(\{b_{k}^{j}\), for \(k=1,\ \ldots,\ n\) and \(\widetilde{u}_{ik}=1\}\), respectively. The proposed IIFPFCM under city block distance (IIFPFCMC) clustering algorithm used the median instead of the average to update the center point of the group. Hence, the characteristic of the median is that if there are extreme values such as outliers, it can better centralize the trend, and the average will affect affected by extreme values. Hence, when the interval data have outliers, the proposed IIFPFCMC clustering algorithm can effectively avoid being affected at the center point of the update group; that is, the proposed IIFPFCMC used Equations (26)-(28) to update the center point of the cluster and used the median value instead of the mean value. Usually, the characteristic of the median is that if there are extreme values, it can better centralize the tendency. However, the average will be affected by the extreme values. Therefore, when the data have outliers, using the median calculation at the center point of the update group can effective avoid the influence of outliers. Usually, the proposed IIFPFCMC clustering algorithm directly using the median is easily affected by the center point of the initial value, resulting in the situation that the center point of the cluster appears in the same group. Therefore, we separated the initial group center points with IIFPFCME, and then used the algorithm of the median for the IIFPFCMC clustering algorithm, which makes it easier to have a good convergence effect with modified Equations (26)-(28) in this paper. Algorithm 1 presents the procedures of the proposed IIFPFCME clustering algorithm with Euclidean distance.