# Fault Inference of Electronic Equipment Based on Multi-State Fuzzy Bayesian Network
## 2 Module-Level FRM
### Fuzzy Membership Degree of Nodes in Fault State with Multiple Test Indicators  
The evaluation on fault state \(F\), \(F=\{f_{1},f_{2},\ \ldots,f_{m}\}\) of electronic equipment generally involves the set \(T=\{t_{1},t_{2},\ \ldots,t_{k}\}\) (\(k\) represents the total number of evaluation sets) including multiple test indicators. Under different fault states \(F\), the response is a different test index set \(T\), and the response function \(L_{f}\) is a matrix of \(k\times m\) order. Fuzziness is shown from test indicator set \(T\) to fault state set \(F\), that is, there is a fuzzy relationship \(R_{f}\). The element \(r_{ij}\) of \(R_{f}\) is the membership degree, which indicates the membership degree of an indicator to a certain fault; \(r(x)\) demonstrates the membership function for evaluating membership degree of the test indicator set to the set of fault states. \[ \begin{array}{l}F\stackrel{{ L_{f}}}{{\Rightarrow}}T,\ L_{f}= \left[\alpha_{ij}\right]_{k\times m}\\ T\stackrel{{ R_{f}}}{{\Rightarrow}}F,\ R_{f}=\left[r_{ij}\right]_{ m\times k}\end{array}  \tag{1}\]  
Due to random and normal distribution of parameters of electronic elements, the density function complying with fuzzy normal distribution--namely, Gaussian function--is selected to calculate membership degrees of each parameter. \[r_{ij}(x)=e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \tag{2}\]  
where, \(r_{ij}(x)\) is the membership degree of parameter \(x\) and \(\mu\) represents the expected value of distribution, that is, the value in this evaluating interval when membership degree is \(1\); \(\sigma\) indicates the width of the Gaussian function. The membership function of a certain eigenvalue is shown in the Figure 1. According to the simulation results and test experience, experts can provide the center eigenvalue is \(t_{0}\) in normal operating state and the minimum and maximum allowable indicators are \(t_{L}\) and \(t_{U}\). Therefore, the minimum and maximum indicators for judging a system in fault state are \(t_{LF}\) and \(t_{UF}\). The parameter relationship is demonstrated as above, namely \(t_{LF}<t_{L}<t_{0}<t_{U}<t_{UF}\). \(r_{1j}(t)\), \(r_{2j}(t)\), \(r_{3j}(t)\), \(r_{4j}(t)\), and \(r_{5j}(t)\) are membership degrees of test indicators subjected to open-circuit fault, fault induced by smaller parameters, normal state, fault induced by large parameter, and short-circuit fault. Figure 1: Membership degrees corresponding to eigenvalues. \[r_{1j}(t)=\begin{cases}1&t\leq t_{LF}\\ e^{-\frac{(t-t_{LF})^{2}}{2\sigma_{1}^{2}}}&t_{LF}<t\leq t_{L},\ \sigma_{1}=\frac{t_{L}-t_{LF}}{3}\\ r_{2j}(t)= \begin{cases}e^{-\frac{(t-t_{L})^{2}}{2\sigma_{21}^{2}}}&t_{LF}<t\leq t _{L},\sigma_{21}=\frac{t_{L}-t_{LF}}{3}\\ e^{-\frac{(t-t_{L})^{2}}{2\sigma_{22}^{2}}}&t_{L}<t\leq t_{0},\sigma_{22}= \frac{t_{0}-t_{L}}{3}\\ \end{cases}  \tag{3}\] \[r_{3j}(t)= \begin{cases}e^{-\frac{(t-t_{0})^{2}}{2\sigma_{31}^{2}}}&t_{ L}<t\leq t_{0},\sigma_{31}=\frac{t_{0}-t_{L}}{3}\\ e^{-\frac{(t-t_{0})^{2}}{2\sigma_{32}^{2}}}&t_{0}<t\leq t_{U},\sigma_{32}= \frac{t_{U}-t_{0}}{3}\\ \end{cases} \] \[r_{4j}(t)= \begin{cases}e^{-\frac{(t-t_{L})^{2}}{2\sigma_{4}^{2}}}&t_{ 0}<t\leq t_{U},\sigma_{41}=\frac{t_{U}-t_{0}}{3}\\ e^{-\frac{(t-t_{L})^{2}}{2\sigma_{4}^{2}}}&t_{U}<t\leq t_{UF},\sigma_{42}= \frac{t_{UF}-t_{U}}{3}\\ \end{cases} \] \[r_{5j}(t)= \begin{cases}e^{-\frac{(t-t_{UF})^{2}}{2\sigma_{5}^{2}}}&t_ {U}<t\leq t_{UF},\sigma_{5}=\frac{t_{UF}-t_{U}}{3}\\ 1&t>t_{UF}\\ \end{cases} \]  
\(r_{1j}(t)\), \(r_{2j}(t)\), \(r_{3j}(t)\), \(r_{4j}(t)\), and \(r_{5j}(t)\) indicate the membership degrees of test indicators in the inference of the five fault states. The weight of each test indicators, namely weigh of membership degree, which is given by expert experience, affects the total membership degree.