# Rough IPFCM Clustering Algorithm and Its Application on Smart Phones with Euclidean Distance
## 2 Proposed Rough Interval Possibilistic Fuzzy C-Means Clustering Algorithm with Euclidean Distance
\right.\]  
\[D_{1}\Rightarrow\left\{ \begin{array}{cc}\frac{\partial B_{1}}{\partial a_{i} ^{j}}=0\Rightarrow-2\sum\limits_{k=x_{1}\in\underline{B}(g_{i})}^{n}\left\{q _{1}(u_{ik})^{m}+q_{2}(t_{ik})^{q}\right\}\left(a_{k}^{j}-a_{i}^{j}\right)=0 \Rightarrow a_{i}^{j}=D_{1}^{\alpha_{i}^{j}}\\ \frac{\partial B_{1}}{\partial\beta_{i}^{j}}=0\Rightarrow-2\sum\limits_{k=x_{1 }\in\underline{B}(g_{i})}^{n}\left\{q_{1}(u_{ik})^{m}+q_{2}(t_{ik})^{q}\right\} \left(b_{k}^{j}-\beta_{i}^{j}\right)=0\Rightarrow\beta_{i}^{j}=D_{1}^{\beta_{i} ^{j}}\end{array} \right.\]  
where \(D_{1}^{\alpha_{i}^{j}}=\frac{\sum\limits_{k=x_{k}\in\underline{B}(g_{i})}^{n} \left\{q_{1}(u_{ik})^{m}+q_{2}(t_{ik})^{q}\right\}a_{k}^{j}}{\sum\limits_{k=x_ {k}\in\underline{A}(g_{i})}^{n}\left\{q_{1}(u_{ik})^{m}+q_{2}(t_{ik})^{q}\right\} }\) and \(D_{1}^{\beta_{i}^```
Step 1: Initialization for fix \(c\) between \(2\leq c<n\); fix \(m\) between \(1<m<\infty\); fix \(\eta\) between \(1<\eta<\infty\);  fix \(\varepsilon>0\); fix \(\delta\) between \(0\leq\delta\leq(q_{1}+q_{2})\); fix \(w\) and \(\widehat{w},w+\widehat{w}=1\);  set iteration counter to \(l=1\) and limit the number of iterations to \(L\). Initialize the FMD \(u_{ik}\); initialize the PMD \(t_{ik}\). Step 2: Estimate \(r_{i}\) using Equation (3). Step 3: Let \(\mu_{ik}=(q_{1}u_{ik}+q_{2}t_{ik})\) for \(i=1,\ \ldots,\ c\) and \(k=1,\ \ldots,\ n\); then, let \(\mu_{ik}\) and \(\mu_{sk}\) be the highest and second highest  memberships of \(x_{k}\) for \(k=1,\ \ldots,\ n\). If \((u_{ik}-u_{sk})>\delta\), then \(x_{k}\in\underline{A}(g_{i})\), as well as \(x_{k}\in\overline{A}(g_{i})\); otherwise, \(x_{k}\in\overline{A}(g_{i})\), and \(x_{k}\in\overline{A}(g_{s})\). That is, if the datasets of \(x_{k}\) for \(k=1,\ \ldots,\ n\) belong to the crisp lower approximation with \(\underline{A}(g_{i})\) for \(i=1,\ \ldots,\ c\)then FMD \(u_{ik}\) and  PMD \(t_{ik}\) equal one; otherwise,  they belong to the fuzzy boundary of FMD \(u_{ik}\) and PMD \(t_{ik}\) between zero and one. Step 4: Update the prototypes \(\widetilde{g}_{i}\) using Equation (12). Step 5: Update the FMD \(u_{ik}\) and the PMD \(t_{ik}\) using Equations (9) and (10). Step 6: if \(\left\{\left|J_{r_{r_{r}fcm}}^{l-1}-J_{r_{r_{r}fcm}}^{l}\right|\leq\varepsilon\ \text{ or }l>L\right\}\), then Stop. Otherwise, \(l=1+1\) and go to Step 3. ```

**Algorithm 1** The procedure of the proposed RIPFCM clustering algorithm with Euclidean distance is shown in the following steps. **Remark 1**. : _The proposed method combines the rough set and IPFCM. The rough set can easily deal with data that includes overlapping problems in the data set. The IPFCM can easily deal with data that includes noise and outlier problems in the data set. Step 3 of Algorithm 1 involves judgement under the rough set to determine which formula to use. Step 4 and 5 of Algorithm 1 uses the derived formula. When data have no overlapping problems, such as in cases 1 and 3, the data set focuses on noise and outliers, and fewer data are needed to use the rough set. Hence, less improvement of the proposed method can be seen, as shown in Table 2. When data have overlapping problems, such as in cases 2 and 4, the data set focuses on overlapping problems, and more data are needed to use the rough set. Hence, a large improvement of the proposed method can be seen, as shown in Table 2._

## 3 Experimental Results

Root mean squared error (\(RMSE\)) was used to calculate the true cluster center and corresponding cluster center, as follows:

\[RMSE=\sqrt{\frac{\sum_{i=1}^{c}(v_{i}-\widehat{v}_{i})^{2}}{c}}, \tag{13}\]

where \(v_{i}\) is the true cluster center and \(\widehat{v}_{i}\) is the corresponding cluster center. For further detail on the generation of SID, refer to [14; 15; 17]. Some preliminary results are published in the 22 th International Symposium on Advanced Intelligent Systems [11]. The full derivation and experimental results are presented in this paper. The proposed experimental architecture is shown in Figure 2. There are four cases using SID in this paper.