Stochastic gradient geographical weighted regression (sgGWR): scalable bandwidth optimization for geographically weighted regression

Abstract

GWR (Geographical Weighted Regression) is a widely accepted regression method under spatial dependency. Since the calibration of GWR is computationally intensive, some efficient methods for calibration were proposed. However, these methods require extensive computation environments or limit the class of kernel functions, limiting the applicability of GWR. To improve the applicability, we propose sgGWR (stochastic gradient GWR), an optimization approach for GWR based on stochastic gradient, which stochastically approximates cross-validation errors and applies gradient-based optimization methods. To achieve this, we show the analytical derivate of the GWR cross-validation error. sgGWR can handle a broader class of kernels than that by the existing scalable method, and we can benefit from it even high-performance computers cannot be accessed. Therefore, sgGWR fills in the gap that existing scalable methods do not cover. We examine the performances of sgGWR and the existing methods by simulation studies. Additionally, we apply sgGWR for the land price analysis for Tokyo, Japan. As a result, a spatio-temporal version of GWR has the best prediction performance, and it captures the spatio-temporal heterogeneity of regression coefficients.