## 2 The Design of Clustered Detection System
### The Fuzzy c-Means Clustering Algorithm  
After selecting CHs by using FLS, the fuzzy c-means algorithm for clustering is used for forming clusters. Compared with hard clustering, in which every local sensor only belongs to one cluster, FCM has a more accurate and natural description of data. Let \(\{x_{i},\ i=1,\ 2,\ \ldots,\ N\}\) be the set of N local sensors, and \(x_{i}\) represents the \(i\)th local sensor. M is the number of CHs computed by Equation (2) and is also the number of clusters in the system.  
The fuzzy c-means algorithm is an improvement of the c-means algorithm [28]. The c-means algorithm is based on minimizing the following objective function, which represents the mean-square error:  
\[\mathrm{min}J_{e}=\sum\limits_{i=1}^{M}\sum\limits_{\mathbf{y}\in C_{j}}||\mathbf{y}- \mathbf{m_{i}}||^{2} \tag{3}\]  
where \(C_{j}\) represents the cluster of \(j\), \(\mathbf{m_{i}}\) is the center of \(C_{i}\), and \(\mathbf{y}(\mathbf{y}\in C_{j})\) represents all local sensors in \(C_{j}\). Different from the c-means algorithm, the fuzzy c-means algorithm is based on minimizing the following objective function under one restriction function:  
\[\left\{\begin{array}{l}\mathrm{min}J_{f}=\sum\limits_{j=1}^{M}\sum\limits_{i= 1}^{N}\left[\mu_{j}(\mathbf{x}_{i})\right]^{b}||\mathbf{x}_{i}-\mathbf{m_{j}}||^{2}\\ \mathrm{st.}\sum\limits_{j=1}^{M}\mu_{j}(\mathbf{x}_{i})=1,\ \ i=1,\ 2,\ \ldots,\ N \end{array}\right. \tag{4}\]  
where \(\mathbf{m_{j}}\) is the center of cluster \(j\), \(\mu_{j}(\mathbf{x}_{i})\) is the membership of the \(i\)th local sensor in cluster \(j\), and \(\mathrm{b}\) is a constant that can control the degree of blurring of clustering results. If \(b\to\)1, clusters formed by the fuzzy c-means algorithm is similar to those formed by c-means. If \(b=\infty\), the fuzzy c-means algorithm will get a completely fuzzy solution, which means every node belongs to each cluster with equal probability, but it does not bear the meaning of clustering. Generally, let \(b=2\). In addition, the fuzzy c-means algorithm requires that the sum of the memberships of one node for each cluster is 1, which is illustrated by Equation (4).  
The solution of Equation (4) can be obtained by optimizing \(\mu_{j}(\mathbf{x}_{i})\) and update \(\mathbf{m_{j}}\) according to Equation (5) through an iterative method:  
\[\left\{\begin{array}{lcl}\mu_{j}(x_{i})&=&\frac{\left(1/||\mathbf{x}_{i}-\mathbf{m_{ j}}||^{2}\right)^{1/(b-1)}}{\sum\limits_{i=1}^{M}\left(1/||\mathbf{x}_{i}-\mathbf{m_{i}}||^{2} \right)^{1/(b-1)}}\\ m_{j}&=&\frac{\sum\
olimits_{i=1}^{M}\left[\mu_{j}(\mathbf{x}_{i})\right]^{b}\mathbf{ x}_{i}}{\sum\
olimits_{i=1}^{M}\left[\mu_{j}(\mathbf{x}_{i})\right]^{b}} \end{array}\right. \tag{5}\]  
The specific description of clusters formed by FCM is illustrated in Algorithm 1.  
```
Input:\(x_{i}\), \(i=1,2,\ldots,n\); M; \(b\); \(\mu_{j}(x_{i})\); max_iteration_num  M: the number of clusters and obtained by Equation (2). \(b\): \(b=2\). \(\mu_{j}(x_{i})\), \(j=1,2\ldots,\) M, and \(i=1,2\ldots,\) N: the initialized membership values.  max_iteration_num: the maximum number of iterations. Output:\(\mu_{j}^{k}(x_{i})\), which is the membership of each node belonging to each cluster for the \(k\)th iteration. Iteration Process:while\(\left\{\left(\mu_{j}^{k}(x_{i})-\mu_{j}^{k-1}(x_{i})\right)>e\;\;and\;\;k<\text{ max\_iteration\_num}\right\}\)do \(m_{j}=\frac{\sum_{i=1}^{n}\left[\mu_{j}(x_{i})\right]^{n}x_{i}}{\sum_{i=1}^{n} \left[\mu_{j}(x_{i})\right]^{n}}\qquad\forall\;j=1,\;2,\;\ldots,\;M\) \(\mu_{j}(x_{i})=\frac{\left(1/\left|x_{i}-m_{j}\right|\right)^{1/(b-1)}}{\sum_{k =1}^{M}\left(1/\left|\mu_{j}-m_{k}\right|\right)^{2/(b-1)}}\qquad\forall\;i=1, \;2,\;\ldots,\;n;\;j=1,\;2,\;\ldots,\;M\) endwhile
```  
**Algorithm 1** Clusters formation by FCM