# 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
At Level 4, two other feasible solutions, \(S_{5}\) and \(S_{6}\), are found that explain the events \(C_{3}:\ T24\leq\lambda_{3}\)) and \(C_{4}:\ NF>\lambda_{4}\), respectively. At Level 5, only one common feasible solution \(S_{7}\) is found that explains both events' (\(C_{5}:\ PS30\leq\lambda_{5}\) and \(C_{6}:\ Pii\leq\lambda_{6}\)) causality over the five periods of time. This solution includes one pattern \(P_{7}\) with only one condition \(C_{7}:\ NRF\leq\lambda_{7}\). The same reasoning can be made with the only common feasible solution \(S_{8}\), which explains the condition \(C_{7}\) at the last level of the ITCA model using only one pattern \(P_{8}\) that consists of one root cause: \(C_{8}:\ BPR\leq\lambda_{8}\). The cut-point curve of Figure 8. H1 bounds the trend of \(C_{8}\). From the obtained logic tree of Figure 8, the ITCA model confirms the discussion above about the main root cause of the HPC fault mode. Effectively, the first level of the ITCA model identifies the variable P30 (total pressure at HPC outlet) as the only fault indicator of the HPC degradation over time. Therefore, P30 can be employed to predict the remaining useful time of the turbofan engine according to the HPC fault mode. At the second level, the variable T50 (total temperature at an LPT outlet) is discovered to explain the effect of the temperature of combustion on the total pressure at the HPC outlet. T50 refines the knowledge discovered about P30. The same reasoning continues until reaching the final Level, 6, where the ITCA model discovers the variable BPR (bypass ratio), which is identified by the expert as the main control element that affects the occurrence of the HPC fault mode over time. Therefore, the ITCA model provides the expert with more refined knowledge, outlining the effects of the root causes on the fault trend over time, which help him to achieve the prognosis task in an efficient way. The probabilities associated with the ITCA model are calculated using Equations (1) to (4) of Stage 4, Phase 2. They quantify the occurrence of similar feasible solutions, patterns, and associated conditions, period after period, at each level of the ITAC model. Figure 9 plots the probabilities of the eight discovered conditions over five periods of time. Note that the occurrence of each feasible solution is equal to the probability of its associated conditions due to the structure of the obtained logic tree. For example, plot A in Figure 9 represents the probability curve of \(S_{1}:P_{1}:C_{1}\) over the periods \(\Delta_{1}\) to \(\Delta_{5}\). The maximum probability value is equal to 0.16 at each period, since the original data are divided into six equal-size data subsets. Therefore, each subset represents 0.16 from the original data size. Note that each common feasible solution tries to maximize its class coverage, so that the associated condition probability value may not exceed that coverage value over the five periods. Based on the ITCA model and the calculation of probabilities, only one time causality rule can be derived over five investigated periods, as follows:  
\[\mathcal{P}(HPC(\Delta_{i}))=\mathcal{P}(C_{8}(\Delta_{i}))\ i=1\ldots 5 \tag{5}\]  
The time causality rule expresses the contribution of the root-cause on the occurrence of the HPC fault, period after period, according to the \(C_{8}\) cut-point curve. Each cut-point value provides the essential knowledge to sustain the turbofan for more or less time in each defined period interval through the maintenance action. For instance, the turbofan can spend more time in \(\Delta_{1}\) by making the \(C_{8}\) variable (BPR) value under the corresponding cut-point value for a set of time.