A Multiscale Extension of Geographically Weighted Spline Nonparametric Regression: Model Formulation and Theoretical Properties

This study proposes a Multiscale Geographically Weighted Spline Nonparametric Regression (MS-GWSNR) model to simultaneously accommodate spatial heterogeneity, multiscale spatial variation, and nonlinear relationships within a unified regression framework. The proposed model extends Geographically Weighted Spline Nonparametric Regression (GWSNR) by incorporating component-specific bandwidths adapted from the Multiscale Geographically Weighted Regression (MGWR) approach. Parameter estimation is developed using a Weighted Least Squares (WLS) framework combined with a backfitting algorithm to address the absence of a closed-form estimator under multiple spatial weighting matrices. The theoretical properties of the estimator are derived through a smoothing matrix representation, including unbiasedness, variance, and Mean Squared Error (MSE). A numerical illustration using simulated spatial data with controlled multiscale heterogeneity shows that the proposed estimator converges stably within 12 iterations, with a final smoothing operator convergence value of 2.48 × 10−5. The estimation also yields satisfactory reconstruction accuracy with MAE of 0.0680 and RMSE of 0.0880. These results indicate that the proposed MS-GWSNR model can flexibly reconstruct nonlinear spatial relationships across different spatial scales while maintaining stable estimation performance.