Geographically Weighted Elastic Net: A Variable-Selection and Modeling Method under the Spatially Nonstationary Condition

Abstract

This study develops a linear regression model to select local, low-collinear explanatory variables. This model combines two well-known models: geographically weighted regression (GWR) and elastic net (EN). The GWR model posits that the regression coefficients vary as a function of location and focuses on solving the problem of explaining the relationships under the spatially nonstationary condition, which a global model cannot solve. GWR cannot fulfill the task of variable selection, however, which is problematic when there are many explanatory variables with nonnegligible multicollinearity. On the other hand, the EN model is a member of the regulated regression family. EN can trim the number of explanatory variables and select the most important ones by adding penalty terms in its cost function, and it has been proven to be robust under the high-multicollinearity condition. The EN model is a global model, however, and does not consider the spatial nonstationarity. To overcome these deficiencies, we proposed the geographically weighted elastic net (GWEN) model. GWEN uses the kernel weights derived from GWR and applies EN locally to select variables for each geographical location. The result is a set of locally selected, low-collinear explanatory variables with spatially varying coefficients. We demonstrated the GWEN method on a data set relating population changes to a set of social, economic, and environmental variables in the Lower Mississippi River Basin. The results show that GWEN has the advantages of both the high prediction accuracy of GWR and the low multicollinearity among explanatory variables of EN.