## 3 Materials and Methods
### Proposed Framework  
In Figure 3 the concept of MCS interdependencies toward PdM is shown. The goal of the presented framework is twofold. First, we want to identify interdependencies between components based on a data-driven solution that requires to perform three consecutive steps: visual inspection of interdependencies (see Figure 3-step 1), the modeling (see Figure 3-step 2), and the plausibility check based on SHAP approach [41]), which is a suitable XAI method to explain the effect of various features intuitively (see Figure 3-step 3). Combining these methods provides an opportunity to analyze the interdependencies into different levels of detail, starting with high-level analysis using visual inspection and then drilling down to more details by comparing predictive models and employing the SHAP approach. The acquired knowledge from this process can be used to extract and include interdependencies in a predictive model. Second, visual inspection and the SHAP approach can be used to explain predictive models later in the process. This could be used during the decision-making process to help the shop floor workers make reliable decisions. They will be able to track predictive decisions that are, for instance, directly affected by interdependencies. Visual inspection is considered as it provides an easy and fast way to recognize new knowledge [49], thus, providing an opportunity to save time and effort in complex tasks such as interdependencies analyzes. For this purpose, we employed a multi-line approach, which has shown promising results on visual recognition of interdependencies within MCS [9]. This approach allows visualization of data over time, where each line represents the information regarding a specific component (e.g., accumulated sum of performed maintenance actions for each component). In this context, we aim to see if the replacement in one component affects the other components, thus, indicating the possible existence of interdependencies between these components. The effect of interdependencies in predictive results is quantified by comparing predictive models, including interdependencies against predictive models that excluded the information (i.e., features) about interdependencies. However, interdependencies are not trivial information to acquire; therefore, we have to be careful while we define the features that relate to interdependencies. In our case, the _time from the last replacement_ of components (e.g., usage time) is considered as an essential feature to encode interdependencies. This feature has information over time in use for a specific component. Therefore, the effect of this feature on the prediction of the failures for other components is an indicator that the interdependencies of the components can be found. To analyze and compare the distribution of data, we used a box-and-whiskers chart, as well as statistical approaches, such as the chi-square test [50]. Whereby the chi-square test as a non-parametric approach includes its robustness concerning data distribution and ease of computation. Moreover, a box-and-whiskers chart provides a faster and more reliable way to compare data distribution visually. \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{3}{c}{**Random Forest Classifier**} \\
**Failure** & **Precision** & **Recall** & **F1-Score** & **Support** \\ \hline None & 1.00 & 1.00 & 1.00 & 47,997 \\ comp1 & 0.96 & 0.89 & 0.93 & 238 \\ comp2 & 1.00 & 0.94 & 0.97 & 381 \\ comp3 & 0.98 & 0.92 & 0.95 & 167 \\ comp4 & 1.00 & 0.93 & 0.96 & **227** \\
**macro avg** & **0.99** & **0.94** & **0.96** & **49,010** \\ \hline \end{tabular}
\end{table} 
Table 3: Replicated predictive results using RF classifier. Results show that RF classification reach superior results in terms of classifying the failed components. As the definition of interdependencies (feature engineering) is complex and sensitive, we consider XAI approaches, such as global contribution analyzes based on the SHAP approach to assure and explain the presence of interdependencies between components.