🤖 AI Summary
This study addresses the limitations of traditional two-dimensional nonparametric regression methods, which often ignore boundary geometry on non-rectangular domains and struggle to accommodate spatially heterogeneous smoothness. The authors propose a novel approach that constructs locally adaptive splines based on constrained Delaunay triangulation and introduces a carefully designed adaptive prior within a Bayesian framework. This method achieves optimal spatial adaptivity to unknown local smoothness over irregular domains while strictly respecting domain boundaries. It is applicable to any triangulation satisfying mild shape-regularity conditions. Theoretical analysis establishes that the posterior contraction rate attains the oracle-optimal rate, and simulations demonstrate superior estimation accuracy and lower model complexity compared to existing methods.
📝 Abstract
Conventional nonparametric regression methods for two-dimensional non-rectangular domains often overlook domain geometry and allow smoothing across boundaries. In spatial and geostatistical applications, this assumption is frequently invalid because domain boundaries typically constrain interactions among observations. Accommodating spatially varying smoothness is also substantially more challenging than in the univariate setting, and most existing methods do not adequately capture this local structure of the target function. To address these challenges, we propose Bayesian triangulation splines, which constructs locally adaptive splines over a polygonal domain. The method employs constrained Delaunay triangulations to respect boundary geometry and adapt to heterogeneous smoothness. A carefully designed prior further improves empirical performance. Under a global Sobolev smoothness assumption, we show that the proposed method achieves the optimal posterior contraction rate and adapts to unknown smoothness. We also show that the method exhibits ideal spatial adaptation in the sense that it achieves the oracle rate for inhomogeneous or locally varying structural features. Crucially, this oracle guarantee is not specific to constrained Delaunay triangulations, but holds over any triangulation satisfying weak shape-regularity conditions. Simulation studies confirm that the proposed method outperforms existing approaches by achieving higher estimation accuracy while maintaining low model complexity.