Modeling Spatial Anisotropic Relationships Using Gradient-Based Geographically Weighted Regression

Distance and direction play crucial roles in modeling the spatial nonstationarity relationship. Because Euclidean distance ignores the effect of direction, several modified geographically weighted regression (GWR) attempts have been made to model anisotropic relationships using various non-Euclidean distance metrics. These methods, however, adopt uniform parameters to define the non-Euclidean metrics over the whole study area, neglecting the varying numerical features existing in different regions. As a result, they fail to accurately depict spatial anisotropic relationships between variables. To address this issue, we propose a novel method called gradient-based geographically weighted regression (GGWR) that integrates the gradient of spatial relationships into GWR. Additionally, we introduce an l0-norm regularization technique to achieve the parameter estimation of GGWR. Both simulated and actual data sets were used to validate the proposed method, and the experimental results demonstrate that the gradient field of the spatial relationship obtained by GGWR can effectively characterize the direction and intensity of variable relationships at various locations. Moreover, GGWR outperforms other models, including GWR, directional geographically weighted regression, and Minkowski distance-based geographically weighted regression, in terms of fitting accuracy, coefficient estimation accuracy, and interpretation of coefficient symbols. These findings indicate that the GGWR can be a valuable tool for modeling spatial anisotropic relationships by leveraging the spatial relationship gradient field.