# A Data-Driven Fault Tree for a Time Causality Analysis in an Aging System
## 3 The ITCA Methodology
### Phase 2: Build a One Level Fault Tree  
Phase 2 iteratively extracts all of the logic trees that differentiate the fault event (class \(n\)) from each class \(i\), \(i=1\),..., \((n-1)\) of the degraded observations individually. Each logic tree highlights the relevant variables that discriminate the observations of the failure state from the degraded ones, from one period to another. Then, the obtained logic trees are merged into one common fault tree, which identifies and isolates the variables that discriminate the failure state from the degraded ones over time. To do so, [PERSON] and [PERSON] [10] developed a four-stage methodology, named Interpretable Logic Tree Analysis (ILTA), to build a one-level fault tree from a two-class dataset (i.e., normal and failure classes). The methodology discovers the knowledge from the dataset (Stage 1); forms feasible solutions (Stage 2); constructs the fault tree (Stage 3); and finally quantifies the fault tree using [PERSON] theorem (Stage 4). Although such a methodology can be applied separately with each dataset \(D_{i}\), \(i=1\),..., \((n-1)\), the merged fault tree may be difficult to interpret due to the dependence of the datasets over time. To overcome this limitation, Stage 2 of the ILTA methodology needs to be improved. Nevertheless, for the convenience  
Figure 2: Data preparation phase. of the reader, we briefly recall the four stages of the ILTA methodology and highlight the improvements to Stage 2 in the following. * Stage 1: Discover knowledge. Discovering knowledge from a two-class dataset can be achieved through different pattern generation and extraction techniques, such as the logic analysis of data (LAD) [27] and prediction rule ensembles (PRE) [28]. The pattern is a conjunction of certain conditions that discriminate one class of observations from another class. Each condition includes a variable, an inequality sign, and a cut point value. Furthermore, the percentage of observations covered by a given pattern may characterize the knowledge expanse caught by that pattern. However, when the observations of the same class are covered by more than one pattern, an overlap between those patterns may occur, with a certain percentage leading to redundant knowledge. * Stage 2: Obtain similar feasible solutions. A solution is defined as a combination of certain patterns that cover the observations of the same class. Each solution can be characterized by its coverage (Cov) and overlap (OL) percentages. The feasible solution is a solution that respects certain criteria. In the ILTA methodology, only the feasible solution that maximizes the class Cov and minimizes the class OL is selected, which leads to maximizing the interpretability and minimizing the redundancy of the discovered knowledge. However, in the ITCA methodology, we need to search for all of the feasible solutions that respect not only the Cov and OL threshold percentages, but also with minimal number of patterns to capture the common knowledge at the same level over time. In other words, the minimal number of patterns having the maximum Cov and the minimum OL allows us to characterize the fault using global knowledge at the first levels of the tree. When this causality is represented in the tree and the related knowledge is removed from the dataset, the subsequent feasible solutions will reveal other knowledge that depicts sub-causalities not yet discovered and represented in the tree. As Stage 2 aims to select similar feasible solutions that characterize knowledge discovery over time, we seek the most frequent patterns over the predefined periods of time. In addition, the frequent pattern involves the same variable and inequality sign in the shared conditions, independent of the cut-point values. Therefore, the initial version of the burn-and-build algorithm proposed in [10] is improved to form a set of feasible solutions instead of only one for each period, using another decision criterion called the solution tolerance selection (STS) threshold.