# Sequential-Fault Diagnosis Strategy for High-Speed Train Traction Systems Based on Unreliable Tests
## 3 Methodology
### Evaluation Method of Test Reliability Based on the Cloud Model
(4) Construct a floating cloud evaluation matrix  
After obtaining the evaluation cloud model of each test under each index and the evaluation weight of each maintenance technician, the evaluation information for each maintenance technician can be represented using the floating cloud model. Assuming that there are \(k\) maintenance technicians participating in the evaluation, the corresponding weight of each maintenance technician is \(P_{k}\), and the cloud evaluation matrix \(C=(Ex,En,He)\) is calculated:  
\[\left\{ \begin{array}{l}En=\frac{P_{1}^{2}}{\sum\limits_{i=1}^{k}P_{i}^{2}}En _{1}+\frac{P_{1}^{2}}{\sum\limits_{i=1}^{k}P_{i}^{2}}En_{2}+\cdots+\frac{P_{ 1}^{2}}{\sum\limits_{i=1}^{k}P_{i}^{2}}En_{k}\\ Ex=\frac{P_{1}^{2}E_{11}+P_{2}E_{22}+\cdots+P_{k}E_{k}}{P_{1}+P_{2}+\cdots+P_{ k}}\\ He=\frac{P_{1}^{2}}{\sum\limits_{i=1}^{k}P_{i}^{2}}He_{1}+\frac{P_{1}^{2}}{\sum \limits_{i=1}^{k}P_{i}^{2}}He_{2}+\cdots+\frac{P_{1}^{2}}{\sum\limits_{i=1}^{k }P_{i}^{2}}He_{k}\\ \end{array} \right. \tag{4}\]  
Finally, the cloud evaluation matrix is obtained:  
\[C= \begin{bmatrix}B_{1}&B_{2}&\cdots&B_{m}\\ t_{1}&C_{11}=(Ex_{11},En_{11},He_{11})&C_{12}=(Ex_{12},En_{12},He_{12})&\cdots&C_ {1m}=(Ex_{1m},En_{1m},He_{1m})\\ t_{2}&C_{21}=(Ex_{21},En_{21},He_{21})&\cdots&\cdots&C_{2m}=(Ex_{2m},En_{2m},He_{ 2m})\\ \vdots&\vdots&\vdots&\vdots\\ t_{n}&C_{n1}=(Ex_{n1},En_{11},He_{n1})&C_{n2}=(Ex_{n2},En_{n2},He_{n2})&\cdots&C_ {nm}=(Ex_{nm},En_{nm},He_{nm})\\ \end{bmatrix}  \tag{5}\]  
where \(t=(t_{1},t_{2},\cdots,t_{n})\) represents the test set, and \(B=(B_{1},B_{2},\cdots,B_{n})\) represents the evaluation index set. (5) Determine the weight of the evaluation indicators  
When determining the index weight, we used the Analytic Hierarchy Process (AHP) method. The calculation steps are as follows:  
(1) Construct a judgment matrix The nine-scale method is a commonly used method that is suitable for determining the relative importance between indicators. The judgment matrix of pairwise comparisons can be expressed as:  
\[B= \begin{bmatrix}b_{11}&b_{12}&\cdots&b_{1n}\\ b_{21}&b_{22}&\cdots&b_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ b_{n1}&b_{n2}&\cdots&b_{nn}\end{bmatrix}  \tag{6}\]  
where \(b_{ij}(1\leq i\leq n,1\leq j\leq n)\) is represented the relative importance of index \(b_{i}\) to index \(b_{j}\). The elements in the judgment matrix have the following relationship, and there is a quantitative relationship between \(b_{ij}\) and \(b_{ji}\), which can be expressed as follows:  
\[\left\{ \begin{array}{l}b_{ij}\cdot b_{ji}=1\\ b_{ii}=1\\ b_{ij}>0\end{array} \right. \tag{7}\]  
For example, if the decision maker thinks that index \(b_{i}\) is much more important than the index \(b_{j}\), then \(b_{ij}=7\). (2) Weight calculation  
After obtaining the relative importance judgment matrix \(B\), we calculate its maximum eigenvalue \(\lambda_{\text{max}}\) and calculate its normalized eigenvector \(W=\left(\omega_{1},\omega_{2},\cdots,\omega_{n}\right)^{T}\). The value of each \(\omega_{i}\) in the vector is the subjective weight value corresponding to each index. Here, the root square method is used to calculate \(\lambda_{\text{max}}\) and \(W\). Firstly, the weight of each index is calculated. On the basis of obtaining the weight, \(\lambda_{\text{max}}\) is calculated to obtain the feature vector required for inspection. The steps are as follows:  
1 The product of each row \(b_{ij}\) of the judgment matrix \(B\) can be expressed as:  
\[M_{i}=\prod\limits_{i=1}^{n}b_{ij}(i,j=1,2,\cdots,n) \tag{8}\]  
2 The nth power root of \(M_{i}\) can be expressed as:  
\[\overline{W_{i}}=\sqrt[n]{b_{ij}}(i=1,2,\cdots,n) \tag{9}\]  
3 Vector \(\overline{W}=(\overline{W_{1}},\overline{W_{2}},\cdots,\overline{W_{n}})\) is normalized, which can be expressed as:  
\[\omega_{i}=\frac{\overline{W_{i}}}{\sum\limits_{j=1}^{n}\overline{W_{i}}}(i=1,2,\cdots,n) \tag{10}\]  
The obtained \(\omega_{i}\) is the weight of the five evaluation indexes of the test reliability.