## 3 Materials and Methods
### Feature Importance Analysis  
After creating the simulated dataset that contains three types of fault patterns and the normal sensor fault samples, we performed some prior investigation to identify the significant features from a number of statistical features. To proceed, in the first step, we computed 16 statistical features from the dataset. The statistical features that are computed from the sensor signals are listed in Table 2.  
Once the features are calculated, the next step is to observe the importance of these features in classifying the faults on the basis of Shapley values. The concept of Shapley values originates in game theory, where these values help to determine the contribution of each player to the total gain in a collaborative game. In a machine learning paradigm considering each feature in the dataset as a player, [PERSON] and [PERSON] in their seminal paper proposed the SHapley Additive exPlanations (SHAP) method [51] for interpreting the features' importance in a model's predicting capability.  
The SHAP values are obtained by comparing the model's predictions, first by including the particular feature and again by excluding the feature. For a dataset consisting of \(\mathbb{N}\) features, the SHAP value of a particular feature, \(\phi_{i}\), can be expressed as  
\[\phi_{i}=\sum_{\mathbb{S}\subset\mathbb{N}-i}\frac{|\mathbb{S}|!(|\mathbb{N}|- |\mathbb{S}|-1)!}{|\mathbb{N}|!}[\mathcal{F}(\mathbb{S}\in\{i\})-\mathcal{F}( \mathbb{S})] \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Feature Name** & **Mathematical Expression** \\ \hline Minimum & \(min(y_{i})\) \\ Maximum & \(max(y_{i})\) \\ Mean & \(\mu=\frac{1}{N}\sum_{i=1}^{N}y_{i}\) \\ Standard deviation & \(\sigma=\sqrt{\frac{\sum_{i=1}^{N}y_{i}}{N}}\) \\ Kurtosis & \(\mathcal{K}=\frac{1}{N}\sum_{i=1}^{N}\frac{(y_{i}-y)^{4}}{\sigma^{4}}\) \\ Skewness & \(\mathcal{S}=\frac{1}{N}\sum_{i=1}^{N}\frac{(y_{i}-y)^{3}}{\sigma^{3}}\) \\ Root mean square (RMS) & \(y_{RMS}=\sqrt{1}\sum_{i=1}^{N}|y_{i}^{2}|\) \\ Crest factor & \(\mathcal{C}\mathcal{G}=\frac{\text{max}(y_{i})}{y_{RMS}}\) \\ Shape factor & \(\mathcal{S}\mathcal{G}=\frac{\text{max}(y_{i})}{y_{RMS}}\) \\ Impulse factor & \(\mathcal{S}\mathcal{G}=\frac{\text{max}(y_{i})}{y_{RMS}}\) \\ Clearance factor & \(\mathcal{C}\mathcal{L}\mathcal{G}=\frac{\text{max}(y_{i}-y)}{(\frac{1}{N}\sum _{i=1}^{N}|y_{i}|)^{2}}\) \\ Variance & \(\mathcal{S}^{2}=\frac{\sum_{i=1}^{N}(y_{i}-y)}{n-1}\) \\ Energy & \(\mathcal{E}=\sum_{i=1}^{N}\sum_{i=1}^{N}2^{N}\) \\ Power & \(\mathcal{P}=\lim_{y\to 0}\frac{\sum_{i=1}^{N}\sum_{i=0}^{N-1}}{|y_{i}(n)|^{2}}\) \\ Peak to rms & \(\frac{\text{max}}{y_{RMS}}-y_{min}\) \\ Range & \(y_{max}-y_{min}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical features.  
Here, \(\mathbb{S}\) represents a subset excluding feature \(i\), \(\mathcal{G}(\mathbb{S}\in\{i\})\) denotes the model's prediction considering feature \(i\), and \(\mathcal{G}(\mathbb{S})\) is the prediction when feature \(i\) is not present. Finally, the summation is carried out over all the possible subsets of \(\mathbb{S}\). These SHAP values can then be utilized to express a feature's importance. Evaluating the features at an early stage will be beneficial in fault detection and classification in later stages because it will be enough to consider only the highly important features instead of all the features.