Simultaneous clustering and optimization in function-on-scalar regression

Predicting a functional response from scalar predictors is challenging, especially with complex data structures. Traditional function-on-scalar regression (FOSR) methods emphasize smoothness or sparsity, but few address group structures in functional data. To address this gap, we introduce the network function-on-scalar Lasso (NFL), an innovative FOSR model that integrates simultaneous clustering and optimization (SCO) principles. The NFL model introduces a graph-structured sum-of-norms regularization to encourage similar functional responses for related observations (e.g., neighboring regions), while also performing sparse variable selection. An efficient semi-proximal alternating direction method of multipliers (ADMM) algorithm is developed for model estimation, scaling to high-dimensional functional data. We provide theoretical guarantees for the NFL estimator under regularity conditions, ensuring model accuracy and insight into its clustering consistency. Simulations and an environmental application predicting US county-level air quality trends demonstrate the NFL's superior prediction accuracy and ability to uncover meaningful group structures compared to existing methods.