# Intelligent Multi-Fault Diagnosis for a Simplified Aircraft Fuel System
## 3 Methodology
### Interpretability of the Trained Diagnostic Algorithms  
**LR**. In multinomial logistic regressions, model interpretation is based on the analysis of coefficients, representing each feature's contribution when comparing faulty classes to the normal class. A larger absolute coefficient value indicates a stronger influence of that feature on the prediction. Positive coefficients suggest a positive relationship between the feature and the likelihood of a specific fault class, while negative coefficients imply an inverse relationship. Table 8 displays selected coefficients from the multinomial logistic regression model for Classes 2 to 16. Each column corresponds to a fault class, while each row represents the coefficients for a specific feature. For instance, in the case of FOHE blockages (Class 5 to 8), the negative coefficient for p6 indicates a negative relationship between p6 and the fault's occurrence. This suggests that a lower downstream pressure at p6 increases the likelihood of FOHE blockages, aligning with engineering principles. Similarly, the coefficients for pump speed and flow rate f1 indicate strong correlations with internal leakages (Class 9 to 16). According to the model, internal leakages become more probable as pump speed rises and the system flow rate decreases. In Figure 13, the horizontal axis represents the 32 classes, while the vertical axis shows the results as percentages. The blue bars represent the consistency between the signs of the logistic regression (LR) parameters fitted for each class (as shown in Table 8, representing the LR's interpretation) and the corresponding engineering insights in Table 3. The grey \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Accuracy** & **MCC** & **Kappa** & **F-1** \\ \hline LR & 0.7877 & 0.7773 & 0.7757 & 0.7481 \\ DT & 0.7101 & 0.6955 & 0.6942 & 0.6351 \\ SVM & 0.7729 & 0.7622 & 0.7596 & 0.7398 \\ ANN & 0.8787 & 0.8727 & 0.8725 & 0.8538 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance on the validation dataset after balancing complexity. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
**Classes** & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline
**Intercept** & 6.54 & \(-1.3\) & \(-3.27\) & 8.03 & 8.56 & \(-4.65\) & 0.65 & 6.77 & 9.97 & 0.59 & 8.98 & 8.62 & 7.78 & 2.75 & 1.28 \\ \hline
**Pump speed** & \(-3.35\) & 2.24 & \(-1.5\) & 1.60 & \(-1.1\) & 0.69 & \(-3.5\) & 33.75 & 34.41 & 36.94 & 35.00 & 34.91 & 35.03 & 35.27 & 36.04 \\ \hline
**p1** & \(-1.76\) & 0.34 & \(-1.24\) & \(-3.17  
bars depict the F-1 scores of the LR on the validation dataset. The table below the graph summarises the distribution of the consistency levels across classes. This consistency varied significantly among classes and was low (\(\leq\)50%) in nearly half of them, resulting in an average interpretability of only 58% for the LR. Additionally, the LR's predictive performance, shown by the grey bars, was weaker than that of the other three algorithms, as indicated in Figure 11.  
**DT. ** Decision tree (DT) results are generally intuitive and straightforward to interpret. The decision rules for each class can be traced by following the branches and nodes from the root to the leaf nodes. However, as the tree structure expands, interpretation becomes increasingly complex. Despite this, specific rules can be extracted by focusing on individual paths from the root node to each leaf node. In this study, the DT algorithm required 1100 decision nodes across more than 20 layers, with over 900 leaf nodes to classify all 32 classes effectively. Table 9 provides an example of a rule extracted from the trained DT. This rule involves 23 decision points, meaning that the corresponding branch contains 24 nodes: one root node, 22 internal nodes, and one leaf node. The table outlines each node in this branch except for the leaf node. The second column specifies the features used by the DT at each decision point, while the fourth column details the decisions made at these nodes.