A cost-effective algorithm for calibrating multiscale geographically weighted regression models

The multiscale geographically weighted regression (MGWR) model is a useful extension of the geographically weighted regression (GWR) model. MGWR, however, is a kind of Nadaraya–Watson kernel smoother, which usually leads to inaccurate estimates for the regression function and suffers from the boundary effect. Moreover, the widely used calibration technique for the MGWR with a back-fitting estimator (MGWR-BF) is computationally demanding, preventing it from being applied to large-scale data. To overcome these problems, we proposed a local linear-fitting-based MGWR (MGWR-LL) by introducing a local spatially varying coefficient model in which coefficients of different variables could be characterised as linear functions of spatial coordinates with different degrees of smoothness. Then the model was calibrated with a two-step least-squared estimated algorithm. Both simulated and actual data were implemented to validate the performance of the proposed method. The results consistently showed that the MGWR-LL automatically corrected for the boundary effect and improved the accuracy in most cases, not only in the goodness-of-fit measure but also in reducing the bias of the coefficient estimates. Moreover, the MGWR-LL significantly outperformed the MGWR-BF in computational cost, especially for larger-scale data. These results demonstrated that the proposed method can be a useful tool for the MGWR calibration.