Locally sparse spatially varying coefficient models with altitude effects

Abstract In several fields such as ecological, climate, and environmental studies, modeling three-dimensional (3D) spatial heterogeneity, including altitude, is essential for capturing complex geographic variation. Geographically and altitudinal weighted regression (GAWR) is widely used for this purpose, extending traditional geographically weighted regression into three dimensions. However, GAWR cannot perform local variable selection—it assumes all predictors are relevant at every location - which can make the resulting models less accurate and difficult to interpret when true local sparsity exists. To address this limitation, we propose $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -GAWR, a novel framework that integrates GAWR with an $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -norm penalty for strict location-specific variable selection. This framework simultaneously models 3D spatial effects, including altitude, while identifying regionally relevant predictors in a data-driven manner. Since optimizing the $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> penalty involves a computationally challenging NP-hard problem, we employ an iterative adaptive splicing algorithm to search for near-optimal solutions with stable convergence efficiently. Simulation studies demonstrate that $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> —GAWR outperforms conventional GAWR in estimation accuracy and interpretability by correctly identifying zero coefficients. Application to South Korean air temperature data shows that the proposed model effectively captures altitude-driven spatial variations and produces sparse, interpretable coefficient estimates, demonstrating its strength in analyzing complex 3D spatial heterogeneity.