# 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
\[=\sum_{i=1}^{c}\sum_{k=-x_{k}\in B(g_{i})}^{n}\{q_{1}(u_{ik})^{m}+q_{2}(t _{ik})^{\eta}\}\sum_{j=1}^{p}\left(a_{k}^{j}-a_{i}^{j}\right)^{2}+\left(b_{k}^{j }-\beta_{i}^{j}\right)^{2}+\sum_{i=1}^{c}r_{i}\sum_{k=x_{k}\in B(g_{i})}^{n}(1-t _{ik})^{\eta}\] \[\text{subject to }\sum_{i=1}^{c}u_{ik}=1\;. \tag{2}\]  
In Equation (1), \(A_{1}\) is the crisp lower approximation, \(B_{1}\) is fuzzy boundary, \(w\) and \(\widetilde{w}\) are the weight of crisp lower approximation and fuzzy boundary, \(\mathrm{w}+\widetilde{w}=1\), \(\varnothing\) is the empty set, \(u_{ik}\) is the FMD of pattern \(k\) in the \(i\)th cluster, \(U=[u_{ik}]\) is a \(c\times n\) matrix, \(m\in[1,+\infty]\) is a fuzzy parameter that controls the FMD for each pattern \(k\), \(t_{ik}\) is the PMD of pattern \(k\) in the \(i\)th cluster, \(T=[t_{ik}]\) is a \(c\times n\) matrix, \(\eta\in[1,+\infty]\) is a parameter that controls the PMD for each pattern \(k\), and \(r_{i}\) is the similarity of the possibilistic fuzzy C-means defined in [17] and given in Equation (3). \[r_{i}=\frac{\sum_{k=1}^{n}t_{ik}^{\eta}\sum_{j=1}^{p}\left[\left(a_{k}^{j}-a_{ i}^{j}\right)^{2}+\left(b_{k}^{j}-\beta_{i}^{j}\right)^{2}\right]}{\sum_{k=1}^{n }t_{ik}^{\eta}}. \tag{3}\]  
In this paper, the Euclidean distance measure is used as an interval measure and defined as below:  
\[\phi_{E}(\overset{\rightarrow}{x}_{k},\overset{\rightarrow}{g}_{i})=\sum_{j =1}^{p}\left[\left(a_{k}^{j}-a_{i}^{j}\right)^{2}+\left(b_{k}^{j}-\beta_{i}^{ j}\right)^{2}\right] \tag{4}\]  
In Equation (4), \(\phi_{E}(\overset{\rightarrow}{x}_{k},\overset{\rightarrow}{g}_{i})\) is the square of a suitable Euclidean distance that measures the dissimilarity between the vectors of the symbolic interval-valued data. Each pattern \(k\) is represented as a vector of intervals \(\overset{\rightarrow}{x}_{k}=\left(x_{k}^{1},\ldots,x_{k}^{p}\right)\) where \(\overset{\rightarrow}{x}_{k}=\left[a_{k}^{j},b_{k}^{j}\right]\) with \(a_{k}^{j}\leq b_{k}^{j}\). The prototype \(\overset{\rightarrow}{g}_{i}\) can be also represented as a vector of intervals \((g_{1}^{1},\ldots,g_{1}^{j},\ldots,g_{i}^{p})\) where \(\overset{\rightarrow}{g}_{i}^{j}=\left[a_{i}^{j},\beta_{i}^{j}\right]\) with \(a_{i}^{j}\leq\beta_{i}^{j}\)[13]. The core concept of rough set is as specified in [23; 24], and the applications of rough set are as specified in [25; 26]. In this paper, the proposed RIPFCM clustering algorithm combines the rough set, interval Euclidean distance measure, and IPFCM clustering algorithm. The optimization problem of the proposed method can be addressed by minimizing the Lagrange multipliers method for the objective function \(J_{ripfcm}\). The new unconstrained optimization problem is obtained as:  
\[F_{ripfcm}=\sum_{i=1}^{c}\sum_{k=1}^{n}\left(q_{1}u_{ik}^{m}+q_{2}t_{ik}^{\eta} \right)\sum_{j=1}^{p}\left[\left(a_{k}^{j}-\alpha_{i}^{j}\right)^{2}+\left(b _{k}^{j}-\beta_{i}^{j}\right)^{2}\right]+\sum_{i=1}^{c}r_{i}\sum_{k=1}^{n} \left(1-t_{ik}\right)^{\eta}-\sum_{k=1}^{n}\lambda_{k}\left(\sum_{i=1}^{c}u_{ ik}-1\right). \tag{5}\]  
The FMD \(u_{ik}\) is considered for the minimum problem that \(\frac{\partial F_{ripfcm}}{\partial u_{ik}}\) is equal to zero.