Multiscale cooperative optimization in multiscale geographically weighted regression models

Scale in multiscale geographically weighted regression (MGWR) directly impacts the accuracy of coefficient estimates and shapes the comprehensive evaluation of the intensity of spatially non-stationary relationships. Presently, MGWR primarily utilizes back-fitting for sequentially optimizing multiple scales (MGWR-BF). However, the set of individual optima obtained through sequential optimization may not necessarily represent the global optimum. To address this issue, this paper proposes a multi-scale cooperative optimization within MGWR (MGWR-GA) model. Specifically, MGWR-GA employs a genetic algorithm to simultaneously input potential scale combinations, each comprising P scales. Subsequently, it introduces a dedicated overall estimation algorithm designed for these P scales, ultimately determining the optimal scale combinations based on the AICc. Simulation experiments have shown that, at least for global stationarity, the scales obtained by MGWR-GA approximate the true values across twelve different test environments. Additionally, the coefficient estimation bias of MGWR-GA is lower than that of MGWR-BF, especially in low signal-to-noise ratio settings. Empirical experiments further confirm the effectiveness of MGWR-GA in identifying both globally stationary and locally non-stationary scales. Furthermore, MGWR-GA outperforms MGWR-BF in terms of goodness-of-fit, adjusted goodness-of-fit, AICc and spatial autocorrelation of residuals. These findings indicate that MGWR-GA can serve as a valuable tool for modeling spatially non-stationary relationships.