## 3 System Model
### Acquisition of Belief Function
#### 4.2.2 Belief Function  
The domain of evidence theory is called recognition framework \(\Theta\), which includes a limited number of propositions and are recorded as \(\{A_{1},\ldots,A_{n}\}\). It corresponds to the basic event in probability theory, which is called focal element. In this paper, it corresponds to the forward neighboring nodes of each node. The propositions in \(\Theta\) are mutually exclusive. The calculation of belief function is referred to [26; 27], and this paper also makes the following relevant definitions as follows:  
Figure 3: Sinusoidal mapping of membership function.  
**Definition 1**.: _Suppose \(\Theta\) is the recognition framework. If the set function m: \(2^{\Theta}\in[0,1]\) (\(2^{\Theta}\) is the power set of \(\Theta\)) satisfy:_  
\[m(\Phi)=0 \tag{9}\]  
\[\sum_{j=1}^{n}m(A_{j})=1,A_{j}\in\Theta \tag{10}\]  
_where m is called the belief function allocation on recognition framework \(\Theta\). \(\Phi\) is the empty set. For \(A_{j}\in\Theta\), \(m(A_{j})\) is called the belief function value of \(A_{j}\). When \(m(A_{j})>0\), \(A_{j}\) is called the focal element on the allocation of belief function, that is, the reliability of a node selected as the next hop._  
**Definition 2**.: \[\alpha_{a}=\max(C_{a}(A_{j})),j\in FN(i)\] (11)  
\[\beta_{a}=\{[nW_{a}/\sum_{j=1}^{n}C_{a}(A_{j})-1]\}/(n-1),n=|FN(i)|,n\geq 2 \tag{12}\]  
\[R_{a}=(W_{a}\alpha_{a}\beta_{a})/(\sum_{j=1}^{n}W_{j}\alpha_{j}\beta_{j}),j\in FN (i) \tag{13}\]  
_where a is the three indexes (\(a=1,2,3\)), \(n\) is the number of forward neighbors of node \(i\). a and \(n\) are fixed values for the sensor network. \(W_{a}\) is the weighted coefficient of the index \(a\). In this paper, \(W_{1}=W_{2}=W_{3}=\frac{1}{3}\). \(C_{a}(A_{j})\) is the correlation coefficient of the attribute index \(a\) to each forward neighboring node. \(\alpha_{a}\), \(\beta_{a}\), \(R_{a}\) are denote the maximum correlation coefficient, the correlation allocation value and the reliability coefficient of the index \(a\)._  
In this paper, the membership function \(u_{aj}\) in fuzzy theory is used to replace the correlation coefficient \(C_{a}(A_{j})\)[26], both of which can be used to evaluate the trust degree belonging to the next hop based on the value of the indexes of the forward neighboring nodes. Thus, the correlation coefficient of the index \(a\) is \(C_{a}(A_{j})\gets u_{aj}\). Bringing \(C_{a}(A_{j})\) into above equations, thus \(\alpha_{a}\), \(\beta_{a}\), \(R_{a}\) of the forward neighboring node \(j\) at the index \(a\) can be obtained.  
Then, the belief function \(m_{a}(A_{j})\) of the index \(a\) to the proposition \(A_{j}\) corresponding to the forward neighbor node \(j\) can be obtained as follows:  
\[m_{a}(A_{j})=C_{a}(A_{j})/(\sum_{j=1}^{n}C_{a}(A_{j})+n\times K\times(1-R_{a}) \times(1-W_{a}\alpha_{a}\beta_{a})),n=|FN(i)| \tag{14}\]  
The function of uncertainty reliability \(\theta\) under index \(a\) is:  
\[m_{a}(\theta)=1-\sum_{j=1}^{n}m_{a}(A_{j}) \tag{15}\]  
It can be seen from Equations (14) and (15) that weighted coefficient \(W_{a}\) and the correction coefficient \(K\) have certain subjectivity, which is determined by field experience and sensor characteristics.