## 4 Classification Tree Learning
#### Classification Tree Model Based on Modified Parameters
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Description** & **Math. Expression** \\ \hline \(k_{1}\) & Number of failures of Component 1 & \(k_{1}\) \\ \(k_{2}\) & Number of failures of Component 2 & \(k_{2}\) \\ Abs-tau & Absolute Kendall’s \(\tau\) Value & \(|\tau|\) \\ R-C1-Con & Ratio of consistency of Component 1 & \(\frac{6}{6}/\frac{k_{1}}{k_{1}}\) \\ R-C2-Con & Ratio of consistency of Component 2 & \(\frac{c}{c+d}/\frac{k_{2}}{k_{1}}\) \\ D-Com-Con & Difference of components’ consistency & \(\left|\frac{a}{b}-\frac{k_{1}}{t}\right|+\left|\frac{c}{c+d}-\frac{k_{2}}{k_{1 }}\right|\) \\ D-Sys-Con & Difference of system consistency & \(\left|\frac{a}{b}\times\frac{c}{c+d}-\frac{k_{2}}{t}\right|\) \\ \(CV_{1}\) & Coefficient of variation for Component 1 & \(\sqrt{\frac{a}{b^{2}}}/\frac{a}{b}=\frac{1}{\sqrt{a}}\) \\ \(CV_{2}\) & Coefficient of variation for Component 2 & \(\sqrt{\frac{d}{c(c+d+1)}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Adjusted parameters. The classification tree displayed in Figure 3 has two main branches. The right branch has a relatively lower absolute [PERSON]'s tau value (\(|\tau|<0.46\)) and encompasses Rules 1-6, whereas the left branch has a relatively higher absolute Kendall's tau value (\(|\tau|\geq 0.46\)) and encompasses Rules 7-9 (where the components are highly correlated). It can be observed that half of the right branch is dominated by the DAI class and two-thirds of the left branch is dominated by the AA class. In what follows, we provide some discussion on some of the rules to assist in explaining the intuition behind the results obtained from the analysis (see Table 7 for some of the metrics used). When the absolute [PERSON]'s tau value is relatively lower (i.e., \(|\tau|<0.46\)), we can derive Rules 1-6 from the right hand branch of the classification tree, and, when the absolute [PERSON]'s tau value is relatively higher (i.e., \(|\tau|\geq 0.46\)), we can derive Rules 7-9 from the left hand branch. The different scenarios that can be observed are discussed below:  
1. **Rule 1:** If the absolute Kendall's tau value is low and strictly less than 0.2, then the optimal approach to use is DAI since this indicates a weak association between \begin{table}
\begin{tabular}{c l l} \hline \hline
**Rule Precursor** & **Method** \\ \hline
1 & \(|\tau|<0.2\) & DAI \\
2 & 0.2 \(\leq|\tau|<0.46\) **and \(CV_{2}<0.12\) and \(D-Com-Con\geq 0.8\)** & DAI \\
3 & 0.2 \(\leq|\tau|<0.46\) **and \(CV_{2}<0.12\) and \(D-Com-Con<0.8\)** & AA \\
4 & 0.2 \(\leq|\tau|<0.46\) **and \(CV_{2}\geq 0.12\) and \(k_{2}<12\)** & DA \\
5 & 0.2 \(\leq|\tau|<0.46\) and \(CV_{2}\geq 0.12\) and \(k_{2}\leq 12\) and \(D-Com-Con<0.94\)** & AA \\
6 & 0.2 \(\leq|\tau|<0.46\) and \(CV_{2}\geq 0.12\) and \(k_{2}\leq 12\) and \(D-Com-Con\geq 0.94\)** & DAI \\
7 & \(|\tau|\geq 0.46\) and \(k_{2}<26\) & DA \\
8 & \(|\tau|\geq 0.65\) and \(k_{2}\geq 26\) & DA \\
9 & 0.46 \(\leq|\tau|<0.65\) and \(k_{2}\geq 26\) & AA \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Extracted decision rules. \begin{table}
\begin{tabular}{l l} \hline \hline
**Abs-tau \(|\tau|\): Absolute Kendall’s \(\tau\) Value** \\ _low value_ & Indicates weak association between components (independence can be assumed) \\ _high value_ & Indicates high correlation between components (dependence can be assumed) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameter metrics for explaining the intuition behind results. components. Therefore, the dependence information can safely be ignored. Additionally, only one variable is in this decision rule hence a higher predictive capacity can generally be implied. 2. **Rules 2 and 3:** When the absolute Kendall's tau value is low such that it is not less than 0.2 but strictly less than 0.46 and \(CV_{2}\) is low, then DAI can be used provided that the overall component consistency is poor (i.e., there is high difference) since the dependence information can safely be ignored and the reliability analyst is certain about the failure probability of Component 2.