# 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
Hence, a time-based searching algorithm is developed in the ITCA methodology to obtain all the similar feasible solutions over time. It is depicted in the following Algorithm 1. Figure 3 illustrates the proposed time-based searching algorithm using the above three concatenated datasets \(D_{1}\), \(D_{2}\) and \(D_{3}\) of the toy example (Figure 1). Applying Step 1 to Step 4, the algorithm finds a set of five feasible solutions that respect the STS threshold of 90%. To clearly understand this, we assume that each solution consists of only one pattern. From \(D_{1}\)  
```
Input. (i) (n-1)-labelled datasets corresponding to the defined time periods (\(\Delta\)); (ii) Set of generated patterns: P\({}_{gm}\)= {P\({}_{i}\),P\({}_{2}\),..., P\({}_{i}\)}; where (\(l\)) is the number of the discovered patterns; (iii) Overlap (OL) threshold; (iv) Solution tolerance selection (STS) threshold. For each (n-1)-labelled dataset that represents a defined time period (\(\Delta\)): Step 1. Select a start pattern (P\({}_{i}\)) and calculate its coverage. 1.1 Remove the overlapped patterns with P:based on the preset overlap threshold: 1. At number of combination (n) = 2; 2. Select the combination with P\({}_{i}\) that has the maximum coverage. 1.2 Repeat (the sub-steps 1.1-i and ii) until the number of combinations (n) = number of the discovered patterns; 1.3 Compare the selected combinations at each \(n\) and select the combination that maximizes the coverage with a minimal number of patterns. Step 2. Select another pattern (P\({}_{i}\)) as a start point and repeat 1.1, 1.2 and 1.3. Step 3. Repeat 2 until considering each pattern as a start point. Step 4. Compare the selected combinations over the start patterns P\({}_{i}\)and select the combination that includes the minimal number of patterns and its coverage value within the STS threshold. End Step 5. Compare the selected combinations that represent (n-1)-labelled datasets and select the combinations that maximize the similarity over the defined periods (\(\Delta\)), where each period (\(\Delta\)) is represented by only one combination. Output. Set of similar feasible solutions: Sol = {S\({}_{1}\), S\({}_{2}\),..., S\({}_{k}\)}; where (\(k\)) is the number of similar feasible solutions. ```  
**Algorithm 1**Time-based searching algorithm: _Search for similar feasible solutions over time._  
Figure 4 depicts the curve of the cut-point values that reflect the evolution of similar feasible solutions obtained over the three periods \(\Delta_{1}\), \(\Delta_{2}\), and \(\Delta_{3}\). Note that these periods are consecutive, and the cut-point curve may have a positive, negative, or constant trend over time depending on how the cut-point values change over time. * Stage 3: Construct a common logic tree over time. The similar feasible solutions obtained are visualized in a one-level fault tree through the condition, pattern, and solution layers. At the condition layer, all the involved conditions are connected to their respective patterns using the AND gate. At the pattern layer, all the patterns of the similar feasible solutions are connected to that solution using the OR gate. Similarly, at the solution layer, all the selected similar feasible solutions are connected to the fault event using the OR gate. * Stage 4: Assign the probabilities. The common logic tree is quantified using the probabilities of the solutions, patterns and conditions involved in similar feasible solutions obtained from the concatenated dataset individually. Let \(N_{k}\) and \(N_{T}\) be the number of observations covered by the condition \(C_{k}\) and the total number of observations in one concatenated dataset, respectively.