# An Evidential Reasoning Assessment Method Based on Multidimensional Fault Conclusion
## 3 An Assessment Method of Evidence Reasoning Based on Fault Conclusion
### Assessment Model of ER Based on Fault Conclusion  
Before conducting ER and evidence fusion, quantitative data must be transformed into belief distributions relative to reference levels. Assume the \(p\)-th reference value of evidence (also attribute) \(z_{j}\) is represented as \(z_{j}\) and \(g_{p+1,j}\geq g_{p,j}\), with \(g_{M,j}\) being the maximum and \(g_{1,j}\) the minimum reference value. Following the acquisition of the initial reference value, the quantitative data can be translated into a belief distribution in the following manner:  
\[T(z_{j})=\{(g_{p,j},\gamma_{p,j}),p=1,2,...,M\}, \tag{1}\]  
where \(\gamma_{p,j}=\frac{g_{p+1,j}-z_{j}}{g_{p+1,j}-g_{p,j}}\), \(\gamma_{p+1,j}=1-\gamma_{p,j}\), \(g_{p,j}\leq z_{j}\leq g_{p+1,j}\), \(\gamma_{q,j}=0\), \(q=1,2,...,M\), \(q\
eq p\), \(p+1\).  
Quantitative attributes are represented using the following distribution:  
\[T(z_{j})=\{(G_{p,j},\gamma_{p,j}),p=1,2,...,M\}, \tag{2}\]  
where \(\gamma_{p,j}\geq 0\), \(\sum\limits_{p=1}^{M}\gamma_{p,j}\leq 1\), and \(\gamma_{p,j}\) represent the belief that the \(z_{j}\) is assessed as the reference level \(G_{p,j}\). \(T(z_{j})\) denotes the belief distribution of the \(z_{j}\). For quantitative attribute \(z_{j}\), let \(n_{p,j}\) represent the basic belief value that the individual is assessed at \(G_{p}\). \(n_{G,j}\) denotes the basic belief value that is not allocated to \(G_{p}\), that is,  
\[n_{p,j}=\lambda_{j}\gamma_{p,j},p=1,2,...,M \tag{3}\]  
where \(\lambda_{j}\) is the weight of the \(z_{j}\), as well as \(0\leq\lambda\leq 1\), \(\sum\limits_{j=1}^{K}\lambda_{j}=1\).  
\[n_{G,j}=1-\sum\limits_{p=1}^{M}n_{p,j}=1-\lambda_{j}\sum\limits_{p=1}^{M} \gamma_{p,j} \tag{4}\]  
In the ER method, \(n_{G,j}\) can be divided into two components: \(\bar{n}_{G,j}\) and \(\bar{n}_{G,j}\), as follows:  
\[\bar{n}_{G,j}=\lambda_{j}\Bigg{(}1-\sum\limits_{p=1}^{M}\gamma_{p,j}\Bigg{)} \tag{5}\]  
\[\bar{n}_{G,j}=1-\lambda_{j} \tag{6}\]  
where \(\bar{n}_{G,j}\) represents the degree of incompleteness in assessing attribute \(z_{j}\). The belief distribution is deemed complete when there is no unassigned belief, indicated by a value of 0, \(\bar{n}_{G,j}=0\). \(\bar{n}_{G,j}\) signifies the remaining basic belief assignment.  
Using the ER method to aggregate the first \(j\) attributes and the \((j+1)\)-th attribute, the belief assignment value at assessment level \(G_{p}\) is  
\[\{G_{p}\}:\ n_{p,J(j+1)}=Q_{J(j+1)}\Big{[}n_{p,J(j)}n_{p,j+1}+n_{G,J(j)}n_{p,j +1}+n_{G,j+1}n_{p,J(j)}\Big{]} \tag{7}\]  
\[n_{G,J(j)}=\bar{n}_{G,J(j)}+\bar{n}_{G,J(j)} \tag{8}\]\[\{G\}:\;\bar{n}_{G,J(j+1)}=Q_{I(j+1)}\Big{[}\bar{n}_{G,J(j)}\bar{n}_{G,j+1}+\bar{n} _{G,J(j)}\bar{n}_{G,j+1}+\bar{n}_{G,j+1}\bar{n}_{G,J(j)}\Big{]} \tag{9}\]  
\[\{G\}:\;\bar{n}_{G,J(j+1)}=Q_{I(j+1)}\Big{[}\bar{n}_{G,J(j)}\bar{n}_{G,j+1}\Big{]} \tag{10}\]  
\[Q_{I(j+1)}=\left[1-\sum_{s=1}^{M}\sum_{\begin{subarray}{c}r=1\\ r\
eq s\end{subarray}}^{M}n_{r,j+1}n_{s,J(j)}\right]^{-1},\;j=1,\ldots,K-1 \tag{11}\]  
When integrating all attributes through the iterative algorithm, the belief is distributed as follows:  
\[\{G\}:\;\hat{\gamma}_{p}=\frac{n_{p,J(K)}}{1-\bar{n}_{G,J(K)}},\;(p=1,\ldots,M) \tag{12}\]  
\[\{G\}:\;\hat{\gamma}_{G}=\frac{\bar{n}_{G,J(K)}}{1-\bar{n}_{G,J(K)}} \tag{13}\]  
where \(\hat{\gamma}_{p}\) indicates the belief level at which an attribute is assessed at the reference level \(G_{p}\). \(G_{p}\) and \(\hat{\gamma}_{G}\) denote the belief allocated to the global level, with \([\gamma_{p},(\gamma_{p}+\gamma_{G})]\) representing the belief range.  
The overall assessment \(z\) can be illustrated through the following belief distribution:  
\[T(z)=\{(G_{p},\hat{\gamma}_{p}),p=1,...,M\} \tag{14}\]  
Next, the health status of the high-speed train running gear is calculated using the following formula:  
\[X(z)=\sum_{p=1}^{M}G_{p}\times\hat{\gamma}_{p} \tag{15}\]  
where \(M\) represents the number of levels in the assessment result, and \(X(z)\) represents the utility value of individual \(z\) after the iterative fusion of \(K\) attributes.