## 4 Classification Tree Learning
#### Classification Tree Model Based on Modified Parameters  
To obtain a visualized representation of the rules that can be used to determine which approach should be employed under different values of parameters (i.e., different combinations), the Recursive Partitioning (rpart) package in R, which builds classification or regression models based on the concepts discussed in [32], was implemented. Rpart builds classification models of a very general structure based on a double stage procedure. Initially, a single variable which best splits the data into two groups is found and the data is separated. This process is applied separately to each sub-group in a recursive manner until the sub-groups either reach a minimum size or until no improvement can be made. This results in an undoubtedly too complex model. Finally, cross-validation procedure is performed in order to trim back the full tree. As mentioned above in this section, considering only the original parameters as the input factors (i.e., attributes) for the decision tree will not incorporate important qualitative knowledge into the model. In other words, we need modified parameters to build a more interpretable classification tree as well as to improve the prediction accuracy. Therefore, modified parameters for a model of products of a probability and a frequency for parallel systems are defined as input or attribute variables. Table 5 shows the list of the adjusted parameters and details. The classification tree can then be constructed by considering the adjusted parameters as input factors at 10% boundary value. Using 50,000 simulated cases, the tree that was constructed by utilizing rpart package in R is shown in Figure 3. In evaluating the performance of the classification tree model, we found the accuracy (61.4%) of the tree obtained utilizing the modified parameters as inputs to be higher than the accuracy (56.3%) of the classification tree obtained while using the initial input parameters. The classification tree model built by using modified parameters also performs significantly better than a random classifier (where a 1/3 accuracy is expected). It can be deduced from the classification tree that four of the nine parameters can be regarded as key factors influencing the magnitude of errors: Abs-tau, D-Com-Con, \(CV_{2}\), and \(k_{2}\) (see Table 5 for description). It is possible to investigate the range and combination of these influencing variables and determine how they produce predictions (of the magnitude of errors) since classification tree learning is a white-box model. In that regard, the reliability analyst has the ability to identify the conditions where the most accurate but costly DA analysis should be used. The optimal analytical approach under different scenarios can be determined from the terminal nodes in Figure 3 to be one of AA, DA, or DAI. A concise description of the extracted decision rules is provided in Table 6. Specifically, the left column contains the rule number, the middle column shows the pre-conditions of the rules, and the right column contains resulting optimal Bayesian analysis approach for that particular rule to be chosen.