## 4 The Proposed PUL-DUCG Model
### Reasoning Algorithm  
In this section, we use the EDAS method to conduct knowledge reasoning in the proposed PUL-DUCG model, which can locate the maximal probability event. The detailed reasoning steps of the PUL-DUCG are expressed below:  
**Step 3:** Compute the occurrence probabilities of target child events.  
In this study, the occurrence probability of a target child event is computed by  
\[\Pr\Big{\{}H_{kj}\mid E\Big{\}}=\Pr\Big{\{}H_{kj}E\Big{\}}/\Pr\{E\} \tag{17}\]  
where \(H_{kj}\) and \(E\) represent \(B\)- type events and \(X\)-type events, respectively.  
Suppose that we are interested in the events of \(B_{1,2}\mid X_{2,2}X_{3,2}\) and \(B_{4,2}\mid X_{2,2}X_{3,2}\) in Figure 3. Namely, we want to know which is more likely to cause the event of \(X_{2,2}X_{3,2}\) between \(B_{1,2}\) and \(B_{4,2}\). Thus, we can compute the probabilities of \(B_{1,2}\mid X_{2,2}X_{3,2}\) and \(B_{4,2}\mid X_{2,2}X_{3,2}\) by  
\[\Pr\{B_{1,2}\mid X_{2,2}X_{3,2}\}=\Pr\{B_{1,2}X_{2,2}X_{3,2}\}/\Pr\{X_{2,2}X_{ 3,2}\} \tag{18}\]  
**Step 4:** Obtain the average probability vector \(\widetilde{P}\).  
The average probability for each target child event \(\widetilde{p}_{j}\) can be derived by  
\[\widetilde{p}_{j}=\frac{\sum\limits_{i=1}^{m}\widetilde{p}_{ij}}{m},\;j=1,2, \ldots,q \tag{19}\]  
As a result, the average probability vector is established as \(\widetilde{P}=\big{[}\widetilde{p}_{1},\widetilde{p}_{2},\ldots,\widetilde{p }_{q}\big{]}\).  
**Step 5:** Determine the positive distance matrix \(D^{+}\) and the negative distance matrix \(D^{-}\) of probabilities to the average probability vector \(\widetilde{P}\).  
The positive distance matrix \(D^{+}=\left[d_{ij}^{+}\right]_{p\times q}\) can be obtained by  
\[d_{ij}^{+}=\left\{\begin{array}{ll}\frac{d\big{(}\widetilde{p}_{ij}\widetilde {p}_{j}\big{)}}{\epsilon(\widetilde{p}_{j})}&if\;\widetilde{p}_{ij}>\widetilde {p}_{j},\\ 0&if\;\widetilde{p}_{ij}\leq\widetilde{p}_{j}.\end{array}\right. \tag{20}\]  
Figure 3: The PUL-DUCG for root cause analysis of an abnormal superheat degree.  
The negative distance matrix \(D^{-}=\left[d^{-}_{ij}\right]_{p\times q}\) can be derived by  
\[d^{-}_{ij}=\left\{\begin{array}{ll}\frac{d\left(\widetilde{p}_{ij}\widetilde{p }_{j}\right)}{E\left(\widetilde{p}_{j}\right)}&if\ \widetilde{p}_{ij}\geq\widetilde{p}_{j},\\ 0&if\ \widetilde{p}_{ij}<\widetilde{p}_{j}.\end{array}\right. \tag{21}\]  
**Step 6:** Calculate the weighted sum of the probability assessment positive distance and negative distance for each target child event.  
The weighted sum of the probability assessment positive distance (\(SP_{i}\)) is calculated by  
\[SP_{i}=\sum_{j=1}^{q}w_{j}d^{+}_{ij}. \tag{22}\]  
In a similar way, the weighted sum of the probability assessment negative distance (\(SN_{i}\)) is obtained by  
\[SN_{i}=\sum_{j=1}^{q}w_{j}d^{-}_{ij}. \tag{23}\]  
**Step 7:** Normalize the values of \(SP_{i}\) and \(SN_{i}\) for each event.  
The normalized values of \(SP_{i}\) and \(SN_{i}\) for each event can be obtained by  
\[\overline{SP_{i}}=\frac{SP_{i}}{\underset{i}{\max}SP_{i}}, \tag{24}\]  
\[\overline{SN_{i}}=1-\frac{SN_{i}}{\underset{i}{\max}SN_{i}}. \tag{25}\]  
**Step 8:** Obtain the ranking order of target child events.  
The appraisal score for the \(i\)th event \(AS_{i}\) can be determined by  
\[AS_{i}=\frac{\left(\overline{SP_{i}}+\overline{SN_{i}}\right)}{2},\ i=1,2, \ldots,p \tag{26}\]  
where \(0\leq AS_{i}\leq 1\).  
The greater the appraisal score \(AS_{i}\), the higher occurrence probability of the target child events. Therefore, all the child events can be ranked based on the descending order of their appraisal score \(AS_{i}(i=1,2,\ldots,p)\).