Simultaneous coefficient penalization and model selection in geographically weighted regression: the geographically weighted lasso

In the field of spatial analysis, the interest of some researchers in modeling relationships between variables locally has led to the development of regression models with spatially varying coefficients. One such model that has been widely applied is geographically weighted regression (GWR). In the application of GWR, marginal inference on the spatial pattern of regression coefficients is often of interest, as is, less typically, prediction and estimation of the response variable. Empirical research and simulation studies have demonstrated that local correlation in explanatory variables can lead to estimated regression coefficients in GWR that are strongly correlated and, hence, problematic for inference on relationships between variables. The author introduces a penalized form of GWR, called the 'geographically weighted lasso' (GWL) which adds a constraint on the magnitude of the estimated regression coefficients to limit the effects of explanatory-variable correlation. The GWL also performs local model selection by potentially shrinking some of the estimated regression coefficients to zero in some locations of the study area. Two versions of the GWL are introduced: one designed to improve prediction of the response variable, and one more oriented toward constraining regression coefficients for inference. The results of applying the GWL to simulated and real datasets show that this method stabilizes regression coefficients in the presence of collinearity and produces lower prediction and estimation error of the response variable than does GWR and another constrained version of GWR-geographically weighted ridge regression.