A Bayesian Implementation of the Multiscale Geographically Weighted Regression Model with INLA

The multiscale geographically weighted regression (MGWR) model is an important extension of the classical geographically weighted regression (GWR) model that can be used to explore the spatial nonstationarity of the regression relationship in spatial analysis, but also allows for different scales on conditional relationships between response and different predictors. A Bayesian version of the MGWR model is proposed to obtain estimates of the spatially varying coefficients and the bandwidths simultaneously. The hierarchical form of the Bayesian MGWR model has attractive features, including obtaining posterior estimates of the bandwidths and local parameters simultaneously, and their uncertainty can be easily measured. For Bayesian posterior inference, an efficient algorithm based on integrated nested Laplace approximation is introduced to provide a great alternative of the classical Markov chain Monte Carlo algorithm under the Bayesian framework. The performance of the proposed method is evaluated through simulation study, and it is shown that the proposed approach can correctly identify the differences between scales of parameter surfaces and also obtain precise posterior estimates. Finally, for illustration, this approach is used to analyze monthly housing cost data in the state of Georgia.