Variable selection in convex nonparametric least squares via structured Lasso: An application to the Swedish electricity distribution networks

Abstract: We study the problem of variable selection in convex nonparametric least squares (CNLS). Whereas the least absolute shrinkage and selection operator (Lasso) is a popular technique for least squares, its variable selection performance is unknown in CNLS problems. In this work, we investigate the performance of the Lasso estimator and find out it is usually unable to select variables efficiently. Exploiting the unique structure of the subgradients in CNLS, we develop a structured Lasso method by combining $\ell_1$-norm and $\ell_{\infty}$-norm. The relaxed version of the structured Lasso is proposed for achieving model sparsity and predictive performance simultaneously, where we can control the two effects--variable selection and model shrinkage--using separate tuning parameters. A Monte Carlo study is implemented to verify the finite sample performance of the proposed approaches. We also use real data from Swedish electricity distribution networks to illustrate the effects of the proposed variable selection techniques. The results from the simulation and application confirm that the proposed structured Lasso performs favorably, generally leading to sparser and more accurate predictive models, relative to the conventional Lasso methods in the literature.