Rough IPFCM Clustering Algorithm and Its Application on Smart Phones with Euclidean Distance

Abstract

New interval clustering technology for symbolic data analysis (SDA) on smart phones is shown to be beneficial for mobile computing devices for smart data analysis in this paper. A new interval clustering method that combined the rough set with interval possibilistic fuzzy C-means (IPFCM) algorithm under Euclidean distance is proposed and implemented on smart phones. Symbolic clustering algorithms (SCAs) have been widely used for pattern recognition, data mining, artificial intelligence, etc. In general, the SCA is unsupervised classification that is divided into groups according to symbolic data sets. However, the traditional interval fuzzy C-means (IFCM) clustering method still has noisy and data overlapping problems associated with these symbolic interval data. Hence, a new rough set with the interval possibilistic fuzzy C-means (RIPFCM) clustering algorithm with Euclidean distance was proposed to address the symbolic interval data (SID). That is, the proposed method can perform better than the traditional IFCM clustering algorithm for SID clustering in noisy environments and with data overlapping problems. The new RIPFCM algorithm under the Euclidean distance method was proposed to deal with SID on new applications in smart phones. Consequently, this method shows the expansion of the smart phone’s computing power and its future application in new SDA.

1. Introduction

In general, smart phones have better computing ability and connectivity to other devices than tradition phones. Their computing ability is similar to a personal computer in their ability to easily install and remove apps. Moreover, it is easy to use the internet via Wi-Fi and establish wireless connections via blue tooth and mobile broadband [1]. On the other hand, smart phones are small in size and easy and convenient to carry. In recent years, some popular mobile operating systems use Google’s Android systems, Apple’s iOS and Microsoft’s Windows mobile systems [2]. The Windows mobile system (WMS) are developed to streamline the operating system for mobile products. That is, the WMS is designed to be as close as possible to the desktop version of Windows operating systems. At the same time, the WMS can, in combination with hardware, form the Windows mobile platform [3]. The .NET Compact Framework (CF) has two components, namely Common Language Runtime and .NET CF class library. At the same time, the .NET CF also has support Visual Basic .NET and C# language. Hence, Microsoft’s WMS is easy to connect to Microsoft’s Azure under C#. On the other hand, Azure app service mobile apps (AASMA) represents the development of integrating a rich set of features into mobile apps. It can accelerate the development of mobile applications. At the same time, it is easy to add data storage in the cloud, allows enterprise or social authentication, enables push notifications and offline syncing, and can scale to millions of devices simultaneously [1]. This paper uses C#, which is compatible on AASMA for program development [4]. Hence, in this paper, we use C# language for the proposed interval clustering method to develop Windows mobile software applications. The era of the “touchscreen generation economy” is coming, and professional data analysis can be done at any time at the touch of a finger. Therefore, the development of AI methods on mobile platforms is currently a hot research field [5,6]. Kumari et al. [7] proposed the evaluation of machine learning and web-based processes for damage score estimation of existing buildings. Işık et al. [8] proposed precision irrigation systems using sensor network technology integrated with iOS/Android applications. Işık et al. [9] proposed the application of iOS/Android–based assessment and monitoring systems for building inventory under seismic impact. Harirchian et al. [10] proposed a prototype for the machine learning–based earthquake hazard safety assessment of structures by using a smartphone app.

In general, new interval clustering technology using the symbolic data analysis (SDA) method on smart phones is beneficial to mobile computing devices for smart data analysis. However, few studies have been published on this topic [11]. Hence, combining the rough set with the interval possibilistic fuzzy C-means (RIPFCM) algorithm under Euclidean distance is proposed and is implemented on smart phones in this paper. Symbolic clustering algorithms have been widely used for pattern recognition, data mining, artificial intelligence, etc. Therefore, this paper proposes a new interval fuzzy C-means clustering method to address symbolic interval data and the expansion of smart phones’ computing power for new symbolic data analysis. That is, SDA and the expansion of smart phones’ computing power is an important research topic in mobile computing and data analysis. The symbolic interval clustering algorithm involves unsupervised interval clustering that is divided into groups according to symbolic data sets. That is, analysis of interval data under fuzzy analysis requires a Type II fuzzy set [12] or interval fuzzy C-means (IFCM) clustering method [13]. Carvalho [13] extended the fuzzy C-means (FCM) clustering method and proposed an IFCM clustering method for symbolic interval data (SID) to conduct SDA. Jeng et al. [14,15] proposed an interval competitive agglomeration clustering algorithm. In general, the IFCM clustering method was proposed to deal with SID [13]. However, outliers and noisy problems are still associated with these SID. Chuang et al. [16] proposed a Hausdorff distance measure based an interval fuzzy possibilistic C-means (IFPCM) clustering algorithm for clustering SID. Jeng et al. [17] proposed an interval possibilistic FCM (IPFCM) clustering method for clustering SID. Because traditional IFPCM and IPFCM methods still have data overlapping problems for these SID, a novel IFCM clustering method that is roughly based on an interval possibilistic FCM (RIPFCM) clustering algorithm with Euclidean distance was proposed. The proposed method can outperform the IFCM clustering algorithm for SID clustering in noisy environments and for data overlapping problems. That is, we combined the rough set with the IPFCM algorithm under Euclidean distance for SID. We also extended the application of smart phones using machine learning and fuzzy clustering for SID. Finally, the contribution of the proposed RIPFCM clustering algorithm used the derived fuzzy membership degree (FMD) and possibilistic membership degree (PMD) formulas to enhance the proposed clustering method in SID that has noisy and data overlapping problems. An additional contribution is the successful implementation of the proposed clustering method on smart phones for new applications in mobile computing and data analysis.

The organization of the rest of the paper is as follows. In Section 2, a rough interval possibilistic fuzzy C-means clustering algorithm is proposed and discussed. The experimental results are shown in Section 3. Finally, the conclusions are summarized in Section 4.

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, Zhao and Zho [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 $\underset{\_}{A}(g_i)$ and fuzzy boundary $B(g_i)$, as shown in Figure 1. This is, if the datasets of $x_k$ for $k=1, \dots , n$ belong to crisp lower approximation with $\underset{\_}{A}(g_i)$ for $i=1, \dots , c$, 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 $\underset{\_}{A}(g_i)$ and fuzzy boundary $B(g_i)$ given in Equation (1) and the constraints are given in Equation (2).

$$J_{ripfcm}\left(G,U,T,X\right)=\{\begin{array}{c}w\times A_1+\tilde{w}\times B_1 if \underset{\_}{A}\left(g_i\right)\ne \varnothing , B\left(g_i\right)\ne \varnothing \\ A_1, if \underset{\_}{A}\left(g_i\right)\ne \varnothing , B\left(g_i\right)=\varnothing \\ B_1, if \underset{\_}{A}\left(g_i\right)=\varnothing , B\left(g_i\right)\ne \varnothing \end{array}$$

(1)

$$A_1={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{\varphi }_E({\overrightarrow{x}}_k,{\overrightarrow{g}}_i)+{\displaystyle \sum }_{i=1}^c{r}_i{\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n{\left(1-t_{ik}\right)}^{\eta }$$

$$={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{\displaystyle \sum }_{j=1}^p{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2+{\displaystyle \sum }_{i=1}^c{r}_i{\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n{\left(1-t_{ik}\right)}^{\eta }$$

$$={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n\left(q_1+q_2\right){\displaystyle \sum }_{j=1}^p{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2$$

$$B_1={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{\varphi }_E({\overrightarrow{x}}_k,{\overrightarrow{g}}_i)+{\displaystyle \sum }_{i=1}^c{r}_i{\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n{\left(1-t_{ik}\right)}^{\eta }$$

$$={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{\displaystyle \sum }_{j=1}^p{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2+{\displaystyle \sum }_{i=1}^c{r}_i{\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n{\left(1-t_{ik}\right)}^{\eta }$$

$$\mathrm{subject} \mathrm{to} {\displaystyle \sum _{i=1}^c{u}_{ik}=1} .$$

(2)

In Equation (1), $A_1$ is the crisp lower approximation, $B_1$ is fuzzy boundary, $w$ and $\tilde{w}$ are the weight of crisp lower approximation and fuzzy boundary, $\mathrm{w}+\tilde{w}=1,$ $\varnothing$ is the empty set, $u_{ik}$ is the FMD of pattern k in the ith cluster, $U=[u_{ik}]$ is a $c\times n$ matrix, $m\in \left[1,+\infty \right]$ is a fuzzy parameter that controls the FMD for each pattern k, $t_{ik}$ is the PMD of pattern k in the ith cluster, $T=[t_{ik}]$ is a $c\times n$ matrix, $\eta \in \left[1,+\infty \right]$ 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).

$$r_i=\frac{{\displaystyle \sum _{k=1}^n{t}_{ik}^{\eta }{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}}}{{\displaystyle \sum _{k=1}^n{t}_{ik}^{\eta }}}.$$

(3)

In this paper, the Euclidean distance measure is used as an interval measure and defined as below:

$${\varphi }_E({\overrightarrow{x}}_k,{\overrightarrow{g}}_i)={\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}$$

(4)

In Equation (4), ${\varphi }_E({\overrightarrow{x}}_k,{\overrightarrow{g}}_i)$ 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 ${\overrightarrow{x}}_k=(x_k^1,\dots ,x_k^p)$ where ${\overrightarrow{x}}_k=\left[a_k^j,b_k^j\right]$ with $a_k^j\le b_k^j.$ The prototype ${\overrightarrow{g}}_i$ can be also represented as a vector of intervals $(g_i^1,\dots ,g_i^j,\dots ,g_i^p)$ where ${\overrightarrow{g}}_i^j=\left[{\alpha }_i^j,{\beta }_i^j\right]$ with ${\alpha }_i^j\le {\beta }_i^j$ [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:

$$F_{ripfcm}={\displaystyle \sum _{i=1}^c{\displaystyle \sum _{k=1}^n\left(q_1{u}_{ik}^m+q_2{t}_{ik}^{\eta }\right){\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}}+}{\displaystyle \sum _{i=1}^c{r}_i{\displaystyle \sum _{k=1}^n{(1-t_{ik})}^{\eta }}-{\displaystyle \sum _{k=1}^n{\lambda }_k\left({\displaystyle \sum _{i=1}^c{u}_{ik}}-1\right)}}.$$

(5)

The FMD $u_{ik}$ is considered for the minimum problem that $\frac{\partial F_{ripfcm}}{\partial u_{ik}}$ is equal to zero. Hence, to obtain an updated equation for the FMD $u_{ik}$, Equation (5) can be rewritten as follows:

$$\begin{gathered} m{q}_1{u}_{ik}^{m-1}{\displaystyle \sum }_{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]-{\lambda }_k=0 \\ \mathrm{for} i=1, \dots , c \mathrm{and} k=1, \dots , n \end{gathered}$$

(6)

Therefore, the solution of Equation (6) for FMD $u_{ik}$ can be rewritten as follows:

$$u_{ik}={\left(\frac{{\lambda }_k}{m{q}_1{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}}\right)}^{\frac{1}{m-1}},$$

(7)

In order to simplify FMD $u_{ik}$, Equation (7) can be substituted into Equation (2), as follows:

$${\displaystyle \sum _{s=1}^c{\left(\frac{{\lambda }_k}{m{q}_1{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_s^j\right)}^2+{\left(b_k^j-{\beta }_s^j\right)}^2\right]}}\right)}^{\frac{1}{m-1}}=1}.$$

(8)

According to Equations (7) and (8), the updated equation for the FMD $u_{ik}$ is obtained as follows:

$$u_{ik}={\left({\displaystyle \sum _{s=1}^c{\left(\frac{{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}}{{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_s^j\right)}^2+{\left(b_k^j-{\beta }_s^j\right)}^2\right]}}\right)}^{1/\left(m-1\right)}}\right)}^{-1},$$

(9)

The PMD $t_{ik}$ is also considered for the minimum problem that $\frac{\partial F_{ripfcm}}{\partial t_{ik}}$ is equal to zero. Hence, to obtain an updated equation for the PMD $t_{ik}$, Equation (5) can be rewritten as follows:

$$\eta q_2{t}_{ik}^{\eta -1}{\displaystyle \sum }_{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]-\eta r_i{\left(1-t_{ik}\right)}^{\eta -1}=0$$

(10)

The solution of Equation (10) for PMD $t_{ik}$ can be rewritten as

$$t_{ik}=\frac{1}{1+{\left(\frac{q_2{\displaystyle \sum _{j=1}^p\left[{\left(a_k^j-{\alpha }_i^j\right)}^2+{\left(b_k^j-{\beta }_i^j\right)}^2\right]}}{r_i}\right)}^{\frac{1}{\eta -1}}}.$$

(11)

The prototype ${\overrightarrow{g}}_i=(g_i^1,\dots ,g_i^p)$ of the $i$th cluster, which minimizes the objective function $J_{ripfcm},$ has the bounds of the interval ${\overrightarrow{g}}_i^j=\left[{\alpha }_i^j,{\beta }_i^j\right]$(j = 1, …, p). Hence, the updated formula for the prototype ${\overrightarrow{g}}_i^j$ under the proposed method is as follows:

$${{\overrightarrow{g}}_i^j}_{ripfcm}=\{\begin{array}{c}w\times C_1+\tilde{w}\times D_1 if \underset{\_}{A}\left(g_i\right)\ne \varnothing , B\left(g_i\right)\ne \varnothing \\ C_1, if \underset{\_}{A}\left(g_i\right)\ne \varnothing , B\left(g_i\right)=\varnothing \\ D_1, if \underset{\_}{A}\left(g_i\right)=\varnothing , B\left(g_i\right)\ne \varnothing \end{array}$$

(12)

$$C_1\Rightarrow \{\begin{array}{c}\frac{\partial A_1}{\partial {\alpha }_i^j}=0\Rightarrow -2(q_1+q_2){\displaystyle {\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n}\left(a_k^j-{\alpha }_i^j\right)=0\Rightarrow {\alpha }_i^j=\frac{1}{\left|\underset{\_}{A}\left(g_i\right)\right|}{\displaystyle {\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n}{a}_k^j\\ \frac{\partial A_1}{\partial {\beta }_i^j}=0\Rightarrow -2(q_1+q_2){\displaystyle {\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n}\left(b_k^j-{\beta }_i^j\right)=0\Rightarrow {\beta }_i^j=\frac{1}{\left|\underset{\_}{A}\left(g_i\right)\right|}{\displaystyle {\displaystyle \sum }_{k=x_k\in \underset{\_}{A}\left(g_i\right)}^n}{b}_k^j\end{array}$$

$$D_1\Rightarrow \{\begin{array}{c}\frac{\partial B_1}{\partial {\alpha }_i^j}=0\Rightarrow -2{\displaystyle {\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n}\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}\left(a_k^j-{\alpha }_i^j\right)=0\Rightarrow {\alpha }_i^j=D_1^{{\alpha }_i^j}\\ \frac{\partial B_1}{\partial {\beta }_i^j}=0\Rightarrow -2{\displaystyle {\displaystyle \sum }_{k=x_k\in B\left(g_i\right)}^n}\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}\left(b_k^j-{\beta }_i^j\right)=0\Rightarrow {\beta }_i^j=D_1^{{\beta }_i^j}\end{array}$$

where $D_1^{{\alpha }_i^j}=\frac{{{\displaystyle \sum }}_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{a}_k^j}{{{\displaystyle \sum }}_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}}$ and $D_1^{{\beta }_i^j}=\frac{{{\displaystyle \sum }}_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}{b}_k^j}{{{\displaystyle \sum }}_{k=x_k\in B\left(g_i\right)}^n\left\{q_1{\left(u_{ik}\right)}^m+q_2{\left(t_{ik}\right)}^{\eta }\right\}}.$

Let ${\mu }_{ik}=\left(q_1{u}_{ik}+q_2{t}_{ik}\right)$ and then let ${\mu }_{ik}$ and ${\mu }_{sk}$ be the highest and second highest memberships of $x_k$ for $k=1, \dots , n$, respectively. If $\left(u_{ik}-u_{sk}\right)>\delta ,$ then $x_k\in \underset{\_}{A}\left(g_i\right),$ as well as $x_k\in \overline{A}\left(g_i\right);$ otherwise, $x_k\in \overline{A}\left(g_i\right),$ and $x_k\in \overline{A}\left(g_s\right),$ where $\delta$ is the weight of the rough degree. Algorithm 1 shows the procedure of the proposed RIPFCM clustering algorithm with Euclidean distance.

Algorithm 1 The procedure of the proposed RIPFCM clustering algorithm with Euclidean distance is shown in the following steps.

Step 1: Initialization for fix c between $2\le \mathrm{c}<n;$ fix m between $1<m<\infty ;$ fix $\eta$ between $1<\eta <\infty ;$     fix $\epsilon >0;$ fix $\delta$ between $0\le \delta \le \left(q_1+q_2\right);$ fix w and $\tilde{w}$, $w+\tilde{w}=1;$     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 ${\mu }_{ik}=\left(q_1{u}_{ik}+q_2{t}_{ik}\right)$ for $i=1, \dots , c$ and $k=1, \dots , n;$ then, let ${\mu }_{ik}$ and ${\mu }_{sk}$ be the highest and second highest memberships of $x_k$ for $k=1, \dots , n$.          If $\left(u_{ik}-u_{sk}\right)>\delta ,$ then $x_k\in \underset{\_}{A}\left(g_i\right),$ as well as $x_k\in \overline{A}\left(g_i\right);$          otherwise, $x_k\in \overline{A}\left(g_i\right),$ and $x_k\in \overline{A}\left(g_s\right).$ That is,          if the datasets of $x_k$ for $k=1, \dots , n$ belong to the crisp lower approximation with $\underset{\_}{A}\left(g_i\right)$ for $i=1, \dots , c$ 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 ${\overrightarrow{g}}_i$ using Equation (12). Step 5: Update the FMD $u_{ik}$ and the PMD $t_{ik}$ using Equations (9) and (10). Step 6: if ($\left|J_{ripfcm}^{l-1}-J_{ripfcm}^l\right|\le \epsilon$ or $l>L$), then Stop.      Otherwise, $l=l+1$ and go to Step 3.

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:

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^c{\left(v_i-\widetilde{v_i}\right)}^2}{c}},$$

(13)

where $v_i$ is the true cluster center and $\widetilde{v_i}$ 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 22th 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. Case 1 used three clusters in SID, and SID had no overlapping problems. In Case 1, SID had outliers and noise. Because different data sets were used, Case 1 was divided into Case 1a and Case 1b. Case 1a considered the SID with outliers. Figure 3 shows the SID with outliers in Case 1a.

In order to build the interval data sets, they were herein defined as $\left(\left[z1-{\gamma }_1/2,z1+{\gamma }_1/2\right],\left[z2-{\gamma }_2/2,z2+{\gamma }_2/2\right]\right)$, from which ${\gamma }_1$ and ${\gamma }_2$ were drawn randomly from the intervals specified in [14,15], respectively. The name and version of the software developed in this paper is Mobile RIPFCM V1.0. The mobile platform displayed in Figure 4 has three subpages, namely the main windows page in (a), the initial results page in (b) and the final results page in (c). The main windows page shows the information on parameters, the initial results page shows the initial state of this case, and the final results page shows the results after the iteration of Algorithm 1. IC is represented as the iterated counter in Figure 4. OFV is represented as the value of the object function in Figure 4. The design of the smart phones shown in Figures 6, 8, 13–16 is the same. The three colors of the center were red, green, and blue on the initial results page. The final results page also used the different colors to represent the results of each class of SID. The parameters of the RIPFCM clustering approach on the Windows mobile phone are $m=2$, $\mathsf{\eta }=2$, $q_1=0.3$, $q_2=25,$ and $\mathsf{\delta }=1.5,$ the number of iterations is 300, and the number of clusters (nc) is 3 for Case 1a, as shown in Figure 4a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 4b, and the final results are shown in Figure 4c for Case 1a.

Case 1b considered the SID with outliers and noise. Figure 5 shows the SID with outliers and noise for Case 1b.

The parameters of the RIPFCM approach on the Windows mobile phone are $m=1.4$, $\mathsf{\eta }=1.7$, $q_1=0.3$, $q_2=18,$ and $\mathsf{\delta }=1.6,$ the number of iterations is 300, and the nc is 3 for Case 1b with noise and outliers, as shown in Figure 6a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 6b, and the final results are shown in Figure 6c for Case 1b with noise and outliers.

Case 2 used three clusters in SID, and the SID had overlapping problems. In Case 2, the SID had outliers and noise. Case 2, similar to Case 1, was divided into Case 2a and Case 2b in accordance with the different data sets. Case 2a represents SID with outliers. Figure 7 shows the SID with outliers for Case 2a.

The parameters of the RIPFCM approach on the Windows mobile phone are $m=2$, $\mathsf{\eta }=1.5$, $q_1=0.5$, $q_2=25,$ and $\mathsf{\delta }=0.9,$ the number of iterations is 300, and the nc is 3 for Case 2a, as shown in Figure 8a. The initial state for the proposed mobile RIPFCM approach is shown n Figure 8b, and the final results are shown in Figure 8c for Case 2a.

Case 2b considered SID with outliers and noise, and the SID had overlapping problems. Figure 9 shows the SID with outliers and noise for Case 2b. In this case, a numerical example also illustrated the IFCM, IPFCM and RIPFCM under Euclidean distance.

We used Matlab to simulate this case. The number of iterations was 300. The results of IFCM with parameter $m=2$ for Case 2b are shown on Figure 10. The results of IPFCM with parameters $m=1.9$, $\mathsf{\eta }=2$, $q_1=1,$ and $q_2=25$ for Case 2b are shown on Figure 11. The results of RIPFCM with parameters $m=2$, $\mathsf{\eta }=1.3$, $q_1=0.3$, $q_2=22,$ and $\mathsf{\delta }=3.5$ for Case 2b are shown on Figure 12. The RMSE results with IFCM, IPFCM, and RIPFCM under Euclidean distance are shown in

Table 1. Comparison of RMSE for IDS4 with different approaches.

	IFCM [9]	IPFCM [13]	RIPFCM
RMSE	10.2120	4.6490	1.1871

.

In this case, the parameters of the RIPFCM approach on the Windows mobile phone are $m=2$, $\mathsf{\eta }=1.3$, $q_1=0.3$, $q_2=22,$ and $\mathsf{\delta }=3.5,$ the number of iterations is 300, and the nc is 3 for Case 2b with noise and outliers, as shown in Figure 13a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 13b, and the final results are shown in Figure 13c for Case 2b with noise and outliers.

Case 3 used six clusters in SID, and the SID had no overlapping problem. In Case 3, the SID had outliers and noise. Case 3a considered SID with outliers. The parameters of the RIPFCM approach on the Windows mobile phone are $m=1.2$, $\mathsf{\eta }=1.5$, $q_1=1$, $q_2=10,$ and $\mathsf{\delta }=0.5,$ the number of iterations is 300, and the nc is 6 for Case 3a, as shown in Figure 14a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 14b, and the final results are shown in Figure 14c for Case 3a.

Case 3b considered SID with outliers and noise. The parameters of the RIPFCM approach on the Windows mobile phone are $m=1.2$, $\mathsf{\eta }=2$, $q_1=1$, $q_2=8,$ and $\mathsf{\delta }=1.5,$ the number of iterations is 300, and the nc is 6 for Case 3b with noise and outliers, as shown in Figure 15a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 15b, and the final results are shown in Figure 15c for Case 3b with noise and outliers.

Case 4 used six clusters in SID, and the SID had overlapping problems. In Case 4, the SID consisted of a data set with outliers and noise for data overlapping problems. The parameters of the RIPFCM approach on the Windows mobile phone are $m=1.1$, $\mathsf{\eta }=1.4$, $q_1=0.8$, $q_2=11,$ and $\mathsf{\delta }=1.2,$ the number of iterations is 300, and the nc is 6 for Case 4 with noise and outliers, as shown in Figure 16a. The initial state for the proposed mobile RIPFCM approach is shown in Figure 16b, and the final results are shown in Figure 16c for Case 4 with noise and outliers.

Finally, Table 1 shows the comparisons of RMSE under four cases with different approaches. The validity of the results for the proposed method were verified.

From the simulation results, Table 1 and

Table 2. Comparison of RMSE under four cases with different approaches.

RMSE	IFCM [9]	IPFCM [13]	Proposed Method
RMSE of Case 1a	2.9519	1.4371	1.0590
RMSE of Case 1b	4.0212	1.1363	0.7626
RMSE of Case 2a	6.3297	4.7523	1.5127
RMSE of Case 2b	10.2120	4.6490	1.1871
RMSE of Case 3a	1.6043	1.3731	1.0983
RMSE of Case 3b	---	1.0457	1.0457
RMSE of Case 4	---	1.8449	1.3827

--- means poor convergence.

show that the proposed RIPFCM clustering algorithm has better performance than the IPFCM and IFCM clustering algorithm for the SID clustering with noise, outliers, and overlapping problems. At the same time, this also shows the expansion of smart phones’ computing power and a possible future application in new symbolic data analysis. Hence, the proposed method was also implemented on a smart phone and extended the application of the smart phone in machine learning, fuzzy clustering, and non-single-valued data analysis. In order to establish the number of clusters to be used in each case, we used the IFCM clustering method for different numbers of clusters. Then, we used the most suitable results to test the proposed method in order to enhance the accuracy of the performance.

4. Conclusions

The new RIPFCM clustering algorithm with Euclidean distance is based on the integration of the rough set and the IPFCM clustering algorithm; it was applied to the analysis of SID as developed in this paper. The proposed new RIPFCM clustering algorithm not only can efficiently handle the overlapping partition problem but also shows better performance than the IPFCM clustering algorithm in the handing of noise problems for SID. In addition, we also combined the rough set to reduce the iteration number and increase the convergence of the IPFCM clustering algorithm. At the same time, the proposed new method also was implemented on a smart phone to address SID. This implementation showed the expansion of smart phones’ computing power. Consequently, the proposed new method extended the application of smart phones in machine learning, fuzzy clustering, and non-single-valued data analysis and its future application in new SDA. Because the proposed method has more parameters, it has better accuracy performance with RMSE. However, its limitation is that it requires adjustment of the additional parameters, and thus the calculation speed of the proposed method will be slower.