Spatially Heterogeneous Additive Model (SHAM): A semiparametric framework for modeling simultaneously spatially varying processes and nonlinear relationships

Spatial heterogeneity, which is as fundamental as spatial dependence in defining spatial characteristics, presents substantial challenges for modeling spatial data. When addressing spatial heterogeneity within regression relationships, conventional methods such as Geographically Weighted Regression (GWR) typically assume a linear structure between the dependent and independent variables. This assumption can lead to biased or misleading inferences if it does not hold in practice. To better characterize spatial heterogeneity while relaxing the often restrictive linearity assumption, this paper introduces a class of spatially heterogeneous additive models by integrating additive models with spatial heterogeneity, the proposed framework effectively addresses these limitations. Notably, several existing methods for spatial heterogeneity analysis can adopt this framework to relax linear relationship constraints, thereby facilitating the simultaneous modeling of complex nonlinear relationships and spatial heterogeneity. We develop a spline-GWR technique to estimate the proposed model, and construct an 𝐹 - type test statistic to examine the significance of spatial heterogeneity. Numerical simulations are conducted to evaluate the finite-sample performance of the proposed estimators and test statistics, confirming their efficacy under a range of scenarios. Finally, the proposed methodology is applied to a real-world dataset for empirical analysis.