Graph-constrained quantile regression: Unifying structured regularization and robust modeling for enhanced accuracy and interpretability

Quantile regression has gained substantial popularity in the forecasting domain due to its flexibility in accommodating arbitrary response variable distributions. However, existing models predominantly rely on regularized approaches like the quantile least absolute shrinkage and selection operator (quantile LASSO), ignoring the critical role of spatial geometric structures play in enhancing prediction accuracy. This study proposes a novel forecasting model that integrates quantile regression with graphical regularization to exploit structural dependencies among predictors. The proposed model obtains both robustness and graphical structure among the predictors. The graphical regularization framework enables simultaneous predictor selection and exploitation of their correlations, leveraging graph-based penalties to capture geometric patterns. To efficiently solve the regularized optimization problem, we develop a proximal alternating direction method of multipliers (PADMM) algorithm, and theoretically prove its convergence. In empirical study, we consider several datasets to demonstrate the superior forecasting performance via comparing with other state-of-the-art statistical and deep learning models. The Freidman test is also provided to support our finding statistically.