## 3 System Optimization Model
### Similarity Criterion and Node's Classification  
In this section, we will discuss how to obtain the redundancy degree of a node with respect to its sensing area being covered by neighboring sensors. Then, a centralized approximation algorithm based on the partition of the target area is designed to obtain the approximate minimum set of nodes, which can retain the sufficient coverage of the target region and ensure the connectivity of the network. Many methods have been put forward to prolong the lifetime of networks. The most common techniques include: data fusion coding [46], balancing network energy consumption by routing protocol [47], eliminating the redundant data [48], etc. In practical applications, especially in environmental monitoring, a large number of sensor nodes is usually deployed randomly in the target area for ensuring that the monitoring region has sufficient coverage. However, the problem of node redundancy may be impractical to take into account and a lot of redundant nodes will be generated. The approach of searching for these redundant nodes and making them sleep intermittently may reduce the redundant data in network and channel congestion. Further, an appropriate scheduling mechanism can balance the energy consumption of the sensor nodes, and improve the energy efficiency. The fuzzy clustering technique adopts a fuzzy matrix to classify objects, and it is particularly effective when the boundaries between clusters of data are ambiguous. Different classification results can be obtained for different levels of confidence, and this forms a dynamic clustering chart. For all nodes within a single grid, the normal cloud model is established according to the historical data. By using the correlation property of the normal distribution, the overlapping area of any two nodes can be solved. Firstly, the matrix \(SS=(s_{ij})_{n\times n}\) can be constructed as:  
\[SS=\left[ \begin{array}{cccc}s_{11}&s_{12}&\cdots&s_{1n}\\ s_{21}&s_{22}&\cdots&s_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ s_{n1}&s_{n2}&\cdots&s_{m}\end{array} \right] \tag{12}\]  
Moreover, in order to make the original data mathematically tractable for fuzzy clustering, the original data matrix is needed to be standardized and transformed into fuzzy matrix via proper data transformation. The normalization formula is given by:\[r_{ij}=\left\{ \begin{array}{ll}1,&(i=j)\\ \\ \exp\{-\sum\limits_{k=1}^{n}\left|s_{ik}-s_{jk}\right|\},\,(i\
eq j)\end{array} \right. \tag{13}\]  
Therefore, the fuzzy similar matrix can be obtained:  
\[R=\left[ \begin{array}{cccc}1&r_{12}&\cdots&r_{1n}\\ r_{21}&1&\cdots&r_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ r_{n1}&r_{n2}&\cdots&1\end{array} \right] \tag{14}\]  
According to the characteristics of the fuzzy similar matrix, there must be a minimum number of natural \(k\) (\(k\leq n\)) which the condition of transitive closure satisfies \(t(R)=R^{k}\) and \(R^{l}=R^{k}\) for all natural numbers \(l\) greater than \(k\). Hence, the transitive closure of \(R\) can be calculated by the square method. Then, it is needed to find the \(\lambda\)-truncation matrix of the fuzzy similar matrix, and then the classification of \(U\) can be given at the level \(\lambda\). In the fuzzy clustering analysis, different classification can be obtained for different \(\lambda\in\) [0,1]. Here, taking into account the characteristics of similarity of the data collected by adjacent sensor nodes of ESNs, \(F\)-statistics is employed to determine the optimal values. According to the original matrix \(SS\), let \(\overline{s}=\left\{\overline{s}_{1},\ldots,\overline{s}_{i},\ldots,\overline {s}_{n}\right\}\) denote the center vector of the population sample, \(\overline{s}_{i}=\sum_{j=1}^{n}s_{ji}/n\). Suppose that \(n\) sensor nodes in the grid are classified into different categories according to the value of \(\lambda\), the number of categories is \(t\) and \(\eta_{i}\) denotes the number of nodes in the \(j\)-th category, \(\sum_{j=1}^{t}\eta_{j}=n\).