Robust Nonparametric Regression via Sparsity Control With Application to Load Curve Data Cleansing

Abstract—Nonparametric methods are widely applicable to statistical inference problems, since they rely on a few modeling assumptions. In this context, the fresh look advocated here perme-ates benefits from variable selection and compressive sampling, to robustify nonparametric regression against outliers—that is, data markedly deviating from the postulated models. A variational counterpart to least-trimmed squares regression is shown closely related to an-(pseudo)norm-regularized estimator, that en-courages sparsity in a vector explicitly modeling the outliers. This connection suggests efficient solvers based on convex relaxation, which lead naturally to a variational M-type estimator equivalent to the least-absolute shrinkage and selection operator (Lasso). Outliers are identified by judiciously tuning regularization pa-rameters, which amounts to controlling the sparsity of the outlier vector along the whole robustification path of Lasso solutions. Re-duced bias and enhanced generalization capability are attractive features of an improved estimator obtained after replacing the-(pseudo)norm with a nonconvex surrogate. The novel robust spline-based smoother is adopted to cleanse load curve data, a key task aiding operational decisions in the envisioned smart grid system. Computer simulations and tests on real load curve data corroborate the effectiveness of the novel sparsity-controlling robust estimators. Index Terms—Lasso, load curve cleansing, nonparametric re-gression, outlier rejection, sparsity, splines. I.