## 2 Materials and Methods
### Clustering Algorithm
#### 2.4.2 Supervised Algorithms  
For the analysis of the data, a decision tree model is used, which compares performances with its more robust variants of random forest and gradient boosting for the prediction of each of the variables alone and together. The data are randomly split into training (80%) and test (20%) data for the model, and these are optimized by \"hyper-parameters tuning\" and \"cross validation\" techniques. First, the model parameters are adjusted, looking for the range where they reach the lowest errors (based on the root-mean-square error), and then, through an experimental design \"GridSearchCV\", a set of simulations is created where the model parameters are permuted with different random splits of the data, selecting the best model by weighted ranking of the models obtained in each training set. Finally, multiple metrics are used to evaluate and compare regression models in the fit on both training and test data, using the coefficient of determination (\(R^{2}\)). With respect to the model errors, the mean square error (\(MSE\)), root-mean-square error (\(RMSE\)), mean absolute error (\(MAE\)), median absolute error (\(MedAE\)), and root-mean-square logarithmic error (\(RMSLE\)) are used as evaluation indicators (see Equations (4)-(9)). Both the \(RMSE\) and \(MSE\) were selected because these quadratic errors are sensitive to outliers and penalize larger errors more. Since the data analyzed are sparse data, the presence of outliers or large errors can be especially significant. In addition, \(RMSE\) and \(MSE\) are standard metrics in the machine learning literature, which facilitates comparison of our results with other studies. Unlike quadratic errors, \(MAE\) and \(MedAE\) are robust metrics that are not strongly influenced by outliers. This is crucial when dealing with sparse data, where a single outlier can have a significant impact on model performance. \(MedAE\), being the median of absolute errors, offers an even more robust and focused perspective than \(MAE\). As for \(RMSLE\) specifically used to address the variable scale of the data, it is useful when errors in prediction are not penalized uniformly across the scale of the target variable, which is relevant in datasets with a wide range of values, such as those under analysis. The combination of these metrics allows us to address both the magnitude and distribution of prediction errors, thus providing a more complete picture of model performance. Each metric was selected to evaluate a specific aspect of the prediction error, which is critical for a robust and complete assessment. \[R^{2}=1-\frac{\sum_{i=1}^{n}\left(\hat{y_{i}}-\overline{y_{i}}\right)^{2}}{ \sum_{i=1}^{n}\left(y_{i}-\overline{y_{i}}\right)^{2}} \tag{4}\]\[MSE=\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{y_{i}}\right)^{2}}{n} \tag{5}\]  
\[RMSE=\sqrt{MSE}=\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{y_{i}}\right)^{2}}{n}} \tag{6}\]  
\[MAE=\frac{\sum_{i=1}^{n}\lvert y_{i}-\hat{y_{i}}\rvert}{n} \tag{7}\]  
\[MedAE=Median(\lvert y_{1}-\hat{y_{1}}\rvert,\lvert y_{2}-\hat{y_{2}}\rvert, \ldots,\lvert y_{n}-\hat{y_{n}}\rvert) \tag{8}\]  
\[RMSE=\sqrt{\frac{1}{n}\frac{n}{\sum_{i=1}^{n}\left(\log\left(y_{i}+1\right)- \log\left(\hat{y_{i}}+1\right)\right)^{2}}} \tag{9}\]  
such that \(y\) represents the actual value, \(\hat{y}\) represents the predicted value, \(\overline{y}\) represents the mean value of the dependent variable, and \(n\) represents the number of observations. Decision Tree Regression  
The decision tree is a machine learning algorithm used to make decisions based on multiple conditions or characteristics. It is based on the idea of dividing a dataset into smaller, homogeneous subsets according to certain criteria. The algorithm starts with a single root node that represents the entire dataset. Then, a feature is selected to divide the set into two subsets, maximizing the homogeneity of the data within each subset.