# A Data-Driven Fault Tree for a Time Causality Analysis in an Aging System
## 4 Case Study
### The HPC Fault Prognosis Using the ITCA Model  
In what follows, the main results of the proposed four-phase ITCA methodology applied on the NASA turbofan engine dataset are presented and discussed to perform the HPC fault prognosis task. As per the first phase, the training dataset is ordered according to the timestamp variable and divided into six equal, unlabelled subsets, where each subset \(SS_{i}\)\(i=1..6\) depicts the period of time \(\Delta_{i}\)\(i=1..6\). The subsets are ordered in a timely manner, where \(SS_{1}\) represents the best normal state of the turbofan while \(SS_{6}\) depicts its worst or failure state. Consequently, five labelled datasets are concatenated from those 6 subsets as follows \(D_{i}\): \(SS_{i}\) versus \(SS_{6}\)\(i=1..5\). Each dataset has 86 labelled observations. Note that the dataset is divided by fixed width for simplicity. However, the expert can assign different width thresholds to produce non-equal data subsets. Meanwhile, the number of subsets is important to capture the evolution of the faults over time. This is a trade-off between time step resolution and ITCA construction time. Phase 2 and Phase 3 are iteratively repeated to construct the ITCA model. The coverage tolerance selection STS threshold used by the time-based searching algorithm (Stage 2 of Phase 2) is set to 10%. In addition, the coverage threshold is set to 90% to control redundant knowledge in the common trees at Stage 1 of Phase 3, when a new level is considered in the ITCA model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  
\end{table} 
Table 1: Variable descriptions of the HPC fault mode. Figure 8 depicts the final ITCA model of the HPC fault mode. It includes six levels of decomposition to reproduce the causality structure between the HPC fault and its root-causes over six periods of time. Note that each level of the ITCA model consists of three layers that represent the solutions, patterns, and conditions related to the fault event or to one of its causes. The first level includes only one common feasible solution \(S_{1}\) over the five defined time periods (\(\Delta_{1}\) to \(\Delta_{5}\)). \(S_{1}\) has only one pattern, \(P_{1}\), which includes only one condition: \(C_{1}:P30>\lambda_{1}\). The plot A1 of Figure 8 characterizes the degradation of the variable P30 over time. Note that the cut-point curve (blue line) bounds the trend of the variable P30 in time. Additionally, the plot A2 of Figure 8 shows the common feasible solution coverage and the overlap percentages over the five time periods. Regarding Level 2 of the ITCA model, the same interpretation above can be performed for the variable T50. It is clear that the ITCA model captures the trend of the involved variables based on the cut-point curves. At Level 3, two common feasible solutions, \(S_{3}\) and \(S_{4}\), are found by the time-based searching algorithm. These solutions respect the construction setting; \(S_{3}\) explains the cause (\(C_{2}:\ T50\ \leq\ \lambda_{2}\)) at the time periods \(\Delta_{1}\) and \(\Delta_{2}\), while \(S_{4}\) dominates the three other periods  
Figure 8: Obtained ITCA model of the HPC degradation mode. \(\Delta_{3}\) to \(\Delta_{5}\). \(S_{3}\) and \(S_{4}\) each have only one pattern and condition. The plots C2 and D1 of Figure 8 depict the bordering of the cut-point curves that represent the degradation trends of the variables T24 and NF, respectively. Meanwhile, the C1 and D2 plots show the solution coverage and overlap percentages over the corresponding time periods. \(S_{3}\) and \(S_{4}\) describe the full-time causality of the cause event (\(C_{2}:\ T50\ \leq\ \lambda_{2}\)) through the time-OR gate by toggling between the two feasible solutions. Hence, \(S_{3}\) explains the event causality at only \(\Delta_{1}\) and \(\Delta_{2}\), while \(S_{4}\) illustrates the causality of the same event at \(\Delta_{3}\), \(\Delta_{4}\), and \(\Delta_{5}\).