Algorithm 3 Residual-Boosting Training and Evaluation: VS-DCFF

Require: Real residual-stage data $({\mathbf{X}}^{\mathrm{b}},\mathbf{r})$; fused augmented data $({\tilde{\mathbf{X}}}^{\mathrm{aug}},{\tilde{r}}^{\mathrm{aug}})$; grid $\Theta$: $D\in \{4,5,6,7\}$, $\eta \in \{0.05,0.03,0.02\}$; $k=5$ TimeSeriesSplit folds Ensure: Best model ${\mathcal{M}}^{*}$; metrics $\{\mathrm{RMSE},\mathrm{MAE},R^2\}$  Hyperparameter Selection (real rows only, nested per fold):  $\mathrm{best}\text{\_}\mathrm{val}\text{\_}\mathrm{rmse}\leftarrow \infty$;   ${\theta }^{*}\leftarrow \varnothing$  for each $\theta \in \Theta$ do      $\mathrm{fold}\text{\_}\mathrm{rmses}\leftarrow \left[\right]$      for each of $k=5$ TimeSeriesSplit folds on $({\mathbf{X}}^{\mathrm{b}},\mathbf{r})$ do          Refit $g_{\mathrm{lin}}$, residual target, MI-selected feature set, and BRR/GPR estimators on that fold’s training subset only; apply fitted transformations to that fold’s validation subset (no cross-fold information sharing)          Train XGBRegressor($\theta$, $M=300$) on fold-train; evaluate RMSE on fold-validation          Append to $\mathrm{fold}\text{\_}\mathrm{rmses}$      end for      $\mathrm{mean}\text{\_}\mathrm{val}\text{\_}\mathrm{rmse}\leftarrow \mathrm{mean}\left(\mathrm{fold}\text{\_}\mathrm{rmses}\right)$      if $\mathrm{mean}\text{\_}\mathrm{val}\text{\_}\mathrm{rmsebest}\text{\_}\mathrm{val}\text{\_}\mathrm{rmse}$ then          $\mathrm{best}\text{\_}\mathrm{val}\text{\_}\mathrm{rmse}\leftarrow \mathrm{mean}\text{\_}\mathrm{val}\text{\_}\mathrm{rmse}$; ${\theta }^{*}\leftarrow \theta$      end if  end for  Final Fit: Train ${\mathcal{M}}^{*}\leftarrow$ XGBRegressor(${\theta }^{*}$, $M=300$) on $({\tilde{\mathbf{X}}}^{\mathrm{aug}},{\tilde{r}}^{\mathrm{aug}})$  Evaluate: On held-out chronological test set, compute ${\widehat{y}}_t=g_{\mathrm{lin}}\left({\mathbf{x}}_t\right)+{\mathcal{M}}^{*}\left({\mathbf{x}}_t^{\mathrm{b}}\right)$ using Equation (21); compute RMSE, MAE, $R^2$ using Equations (22) –(24)  Output: Return ${\mathcal{M}}^{*}$, metrics