A scale-adaptive estimation for mixed geographically and temporally weighted regression models

Mixed geographically and temporally weighted regression (GTWR) models, a combination of linear and spatiotemporally varying coefficient models, have been demonstrated as an effective tool for spatiotemporal data analysis under global homogeneity and spatiotemporal heterogeneity. Simultaneously, multiscale estimation for GTWR models has also attracted wide attention due to its scale flexibility. However, most of the existing estimation methods for the mixed GTWR models still have the limitation that either all of regression relationships operate at the same spatiotemporal scale, or each of coefficients is estimated using back-fitting procedures that are very time-consuming. In order to improve the estimation accuracy and alleviate the computation burden, we propose a multiscale method with the adaptive bandwidth (short for scale-adaptive) for calibrating mixed GTWR (say mixed MGTWR) models. In the proposed multiscale estimation approach, a two-step method is used to estimate the constant coefficients and varying coefficients, and then each of the varying coefficients is again estimated by back-fitting procedures with different bandwidth sizes. In addition, we also address the calculation of “hat matrix” in the multiscale estimation for GTWR model and then derive the hat matrix of the complete MGTWR model. Simulation experiments assess the performance of the proposed scale-adaptive calibration method. The results show that the proposed method is much more efficient than existing estimation methods with regard to estimation accuracy and computation efficiency. Moreover, the proposed scale-adaptive method can also correctly reflect the inherent spatiotemporal operating scales of the explanatory variables. Finally, a real-world example demonstrates the applicability of the proposed scale-adaptive method.