## 2 Materials and Methods
### Data Preprocessing and Variable Selection  
The EVasCu dataset comprising 389 samples was utilized to train the machine learning regression models. In the rare instances of missing data (less than 0.5%), values were imputed using the median of the corresponding variable to maintain dataset integrity. Additionally, to address the disparity in units and magnitudes among the different variables, a z-score normalization was applied, thus ensuring that each feature contributed equally to the analysis. This normalization mitigates the influence of outliers and enhances the algorithm's convergence speed during training.  
Next, the assessment of variable importance was conducted using a Random Forest model, which is recognized for its ability to handle large datasets and capture nonlinear interactions among variables [20]. To determine the importance of each feature, a permutation-based method was employed. This method involves randomly permuting the values of each variable and observing the impact on model performance. Specifically, after a variable is permuted, the model is re-evaluated, and the increase in Mean Squared Error (MSE), compared to the original model without permutation, is measured. To this respect, the MSE can be mathematically defined as follows:  
\[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2} \tag{1}\]  
where \(n\) is the number of observations, \(y_{i}\) is the actual value of the observation, and \(\hat{y_{i}}\) is the predicted value from the model.  
The importance of a feature is then quantified as the average increase in prediction error (\(\Delta\text{MSE}\)) that results when that feature is altered. This increase in error indicates the model's reliance on the variable for making accurate predictions and can be mathematically defined as follows:  
\[\Delta\text{MSE}=\text{MSE}_{\text{permuted}}-\text{MSE}_{\text{original}} \tag{2}\]  
where \(\text{MSE}_{original}\) is the MSE of the model predictions using the original, unpermuted data and \(\text{MSE}_{permuted}\) is the MSE obtained after permuting the values of a particular feature, thus simulating the loss of information provided by that feature. Finally, the difference \(\Delta\)MSE represents how much the prediction error increases when the model is deprived of the information provided by the specific feature. A larger increase signifies a greater importance of the feature for the model's accuracy.  
Finally, to complement the feature selection process and improve model interpretability, a SHAP-based analysis was performed using the trained Random Forest model. SHAP is a game-theoretic approach that assigns each feature a contribution value for individual predictions based on Shapley values from cooperative game theory [14]. Unlike traditional global importance metrics, SHAP provides both global and local interpretability, allowing a detailed understanding of how each variable influences the model's output on a per-subject basis [14]. In this study, the global importance of each feature was computed as the mean absolute SHAP value across all observations, effectively reflecting its average contribution to the prediction of aPWV.