Bayesian geographically weighted regression and its application for local modeling of relationships between tree variables

Geographically weighted regression (GWR) has become popular in recent years to deal with spatial autocorrelation and heterogeneity in forestry and ecological data. However, researchers have realized that GWR has some limitations, such as correlated model coefficients across study areas, strong influence of outliers, weak data problem, etc. In this study, we applied Bayesian geographically weighted regression (BGWR) and a robust BGWR (rBGWR) to model the relationship between tree crown and diameter using observed tree data and simulated data to investigate model fitting and performance in order to overcome some limitations of GWR. Our results indicated that, for observed tree data, the rBGWR estimated tree crown more accurate than both BGWR and GWR. For the simulated data, 74.1% of the estimated slope coefficients by rBGWR and 73.4% of the estimated slope coefficients by BGWR were not significantly different (α = 0.05) from the corresponding simulated slope coefficients. The estimation of model coefficients by rBGWR was not sensitive to outliers, but the coefficient estimation by GWR was strongly affected by those outliers. The majority of the coefficient estimates by rBGWR and BGWR for weak observations were not significantly (α = 0.05) different from the simulated coefficients. Therefore, BGWR (including rBGWR) may be a better alternative to overcome some limitations of GWR. In addition, both BGWR and rBGWR were more powerful than GWR to detect the spatial areas with non-constant variance or spatial outliers.