Facing spatial massive data in science and society: Variable selection for spatial models

This work focuses on variable selection for spatial regression models, with locations on irregular lattices and errors according to Conditional or Simultaneous Auto-Regressive (CAR or SAR) models. The strategy is to whiten the residuals by estimating their spatial covariance matrix and then proceed by performing the standard L1-penalized regression LASSO for independent data on the transformed model. A result is stated that proves the sign consistency for general dependent errors provided that the transformed design matrix fulfills standard assumptions for the LASSO procedure and that the estimate of the residual covariance matrix is consistent. Then sufficient conditions on the weight matrix of the SAR or CAR model are given that ensure those conditions hold. A simulation study is driven that shows this method gives good result in terms of variables selection, while some underestimation of the coefficients is noted. It is compared to a strategy that estimates both the regression and the covariance parameters in a LARS procedure. Coefficients are better estimated with the Least Angle Regression (LARS) procedure but it gives in some cases much more false positive in the variable selection. The application is on the regression of income data in rural area of Uruguay on a set of covariates describing socio-economic characteristics of the households.