## 2 Materials and Methods
### Nonparametric Statistical Tests  
To account for the stochastic nature of the proposed pipeline, nonparametric statistical tests were employed to determine whether a model's performance is statistically superior to that of other models, based on the validation dataset metrics. The Friedman test [82] is a widely used nonparametric statistical test designed to detect differences in performance across multiple trials when the same subjects (datasets) are exposed to different treatments or conditions, such as predictions from various ML-based models. This test ranks the performance of each model per dataset and then compares the average ranks across models to assess whether statistically significant differences exist. In the case of tied ranks, average ranks are assigned.  
Given that \(r_{i}^{l}\) represents the rank of the \(j^{th}\) model among \(k\) models on the the \(i^{th}\) dataset out of \(N\) datasets, the Friedman test computes the average rank of each model using \(R_{j}=1/N\sum_{i}r_{i}^{j}\). Under the null hypothesis (\(p\geq 0.05\)), which assumes all models perform equivalently, the [PERSON] test statistic \(\chi_{F}^{2}\) is calculated as Equation (6):  
\[\chi_{F}^{2}=\frac{12N}{k(k+1)}\left(\sum_{j}R_{j}^{2}-\frac{k(k+1)^{2}}{4}\right) \tag{6}\]  
This statistic follows a Chi-square distribution with \(k-1\) degrees of freedom when \(N>10\) and \(k>5\). If the null hypothesis is rejected (\(p<0.05\)), it indicates a statistically significant difference in model performance. In such cases, [PERSON]'s post hoc test [83] is applied for pairwise model comparisons to identify which models perform significantly better. A statistically significant difference between two models is established if their average ranks differ by at least the critical difference (\(CD\)), which is computed as Equation (7):  
\[CD=q_{\alpha}\sqrt{\frac{k(k+1)}{6N}} \tag{7}\]  
where \(q_{\alpha}\) (typically consider \(q_{0.05}\)) is derived from the Studentized range distribution divided by \(\sqrt{2}\). By employing these statistical tests sequentially, it is possible to rank the models and determine whether a particular model consistently outperforms the others, identifying whether a clear \"champion\" model exists, based on statistically significant differences.  
For more information, the procedures described by [PERSON] [69] could be consulted.  
Once the final hyperparameter configurations for all models were determined, their validation performances (\(MAE\), \(R^{2}\), and \(RMSE\)) across 30 repetitions of the 10-fold cross-validation procedure were subjected to the [PERSON] test to verify whether the null hypothesis is rejected. If so, the Nemenyi test was applied for pairwise comparisons. The results of these pairwise comparisons allowed for the ranking of the models based on their win counts across different interactions, helping to determine the best-performing models--provided that their differences exceed the \(CD\) threshold. Pseudocode 4 outlines the nonparametric statistical test procedure.  
**Pseudocode 4.** Nonparametric statistical test procedure.  
**Input**: all \(Best\)\(Model_{i}\), \(Validation\)\(Metrics_{i}\in I\); all \(Stacking_{j}\), \(Validation\)\(Metrics_{j}\in J\);  
_Metrics_.  
\(Z=I\cup J\to z\in Z,z\in I\) or \(z\in J\).  
_Validation Metrics_\(z=Validation\)\(Metrics_{i}\cup Validation\)\(Metrics_{j}\).  
**Pairwise Testing Comparison:**  
Let \(m\in Metrics\) be the \(MAE\), \(R^{2}\), and \(RMSE\) metrics, and \(p_{m}\) be the probability value for the metric \(m\).  
**For \(m\in Metrics\)**:  
\(p_{m}=\)\(Friedman\)\(Test_{m}(Validation\)\(Metrics_{z},\forall z\in Z)\).  
**If \(p_{m}\leq 0.05\)**:  
**For \(z_{1},z_{2}\in Z\)**:  
\(Rank_{z_{1}(m)},Rank_{z_{2}(m)}=\)  
\(Nemenyi\)\(Test_{m}\Big{(}\)\(Validation\)\(Metrics_{z_{1}(m)},Validation\)\(Metrics_{z_{2}(m)}\Big{)}\).  
**end**  
_Nemenyi Best \(Solution_{m}=\)\(\underset{Rank_{z(m)}}{\text{argmax}}(z,\forall z\in Z)\)._  
**end**  
**end Output**: \(Nemenyi\)\(Best\)\(Solution_{m},\forall m\in Metrics\).