# 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  
In general, fuzzy clustering and FCM clustering have been widely used in pattern recognition, data mining, machine learning, artificial intelligence, etc. [18; 19; 20], with single-valued data. In order to improve performance, it can be developed from the perspective of rough set and FCM clustering integration. Hence, [PERSON] and [PERSON] [21] proposed combining FCM clustering with fuzzy rough feature selection to conduct single-valued analysis. A rough set-based generalized FCM clustering algorithm was also proposed to address single-valued data [22]. In general, interval-valued data appear differently on single-valued data, representing uncertainty in the observed values. That is, the representation of interval-valued data is better able to handle the uncertainty of the data than single-valued data. Hence, the extended FCM clustering method which can deal with interval-valued data, is helpful for generalizing the applications of FCM clustering. In order to extend the interval-valued data analysis, a RIPFCM clustering algorithm with Euclidean distance data that extended [22] to interval-valued data analysis is proposed in this paper. The proposed RIPFCM clustering algorithm with Euclidean distance uses the concepts of the crisp lower approximation with \(\underline{A}(g_{i})\) and fuzzy boundary \(B(g_{i})\), as shown in Figure 1. This is, if the datasets of \(x_{k}\) for \(k=1,\ \ldots,\ n\) belong to crisp lower approximation with \(\underline{A}(g_{i})\) for \(i=1,\ \ldots,\ c\), the FMD \(u_{ik}\) and PMD \(t_{ik}\) are equal one; otherwise, the datasets belong to fuzzy boundary, and the FMD \(u_{ik}\) and PMD \(t_{ik}\) are between zero and one. Hence, the proposed RIPFCM clustering algorithm with Euclidean distance combines the rough set and IPFCM clustering algorithm with Euclidean distance in this paper. The new objective function \(J_{ripfcm}\) is based on a squared Euclidean distance between vectors of intervals; it includes the crisp lower approximation with \(\underline{A}(g_{i})\) and fuzzy boundary \(B(g_{i})\) given in Equation (1) and the constraints are given in Equation (2). \[J_{ripfcm}(G,U,T,X)=\left\{ \begin{array}{cl}w\times A_{1}+\widetilde{w}\times B _{1}\ if\ \underline{A}(g_{i})\
eq\varnothing,\ B(g_{i})\
eq\varnothing\\ A_{1},&if\ \underline{A}(g_{i})\
eq\varnothing,\ B(g_{i})=\varnothing\\ B_{1},&if\ \underline{A}(g_{i})=\varnothing,\ B(g_{i})\
eq\varnothing\\ \end{array} \right. \tag{1}\]  
\[A_{1}=\sum_{i=1}^{c}\sum_{k=x_{k}\in\underline{A}(g_{i})}^{n}\{q_{1}(u_{ik})^{ m}+q_{2}(t_{ik})^{\eta}\}\phi_{E}\left(\stackrel{{\rightarrow}}{{x}}_{k},\stackrel{{\rightarrow}}{{g}}_{i}\right)+\sum_{i=1}^{c}r_{i} \sum_{k=x_{k}\in\underline{A}(g_{i})}^{n}(1-t_{ik})^{\eta}\]  
\[=\sum_{i=1}^{c}\sum_{k=x_{k}\in\underline{A}(g_{i})}^{n}\{q_{1}(u_{ik})^{m}+q_{ 2}(t_{ik})^{\eta}\}\sum_{j=1}^{p}\left(d_{k}^{j}-\alpha_{i}^{j}\right)^{2}+ \left(b_{k}^{j}-\beta_{i}^{j}\right)^{2}+\sum_{i=1}^{c}r_{i}\sum_{k=x_{k}\in \underline{A}(g_{i})}^{n}(1-t_{ik})^{\eta}\]  
\[=\sum_{i=1}^{c}\sum_{k=x_{k}\in\underline{A}(g_{i})}^{n}(q_{1}+q_{2})\sum_{j= 1}^{p}\left(a_{k}^{j}-\alpha_{i}^{j}\right)^{2}+\left(b_{k}^{j}-\beta_{i}^{j} \right)^{2}\]  
\[B_{1}=\sum_{i=1}^{c}\sum_{k=x_{k}\in\underline{B}(g_{i})}^{n}\{q_{1}(u_{ik})^{ m}+q_{2}(t_{ik})^{\eta}\}\phi_{E}\left(\stackrel{{\rightarrow}}{{x}}_{k},\stackrel{{\rightarrow}}{{g}}_{i}\right)+\sum_{i=1}^{c}r_{i}\sum_ {k=x_{k}\in B(g_{i})}^{n}(1-t_{ik})^{\eta}\]  
Figure 1: The concept of the RIPFCM.