Robust High-Dimensional Regression: Sparse Least Trimmed Squares

• Activity Number: 249
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Andreas Alfons*+ and Christophe Croux and Sarah Gelper
• Companies: K. U. Leuven and K. U. Leuven and Erasmus University Rotterdam
• Keywords: outliers ; penalized regression ; trimming ; breakdown point ; large p ; small n

Abstract

In practical applications such as gene expression or fMRI studies, there is an increasing availability of data sets with a large number of variables, frequently much larger than the number of observations. Besides avoiding computational problems in such situations, sparse regression allows for better prediction performance through variance reduction, while at the same time improving interpretability of the resulting models through a smaller number of predictors. A frequently used estimator for sparse regression is the least absolute shrinkage and selection operator (lasso), which adds an L1 penalty on the coefficients to the least squares objective function. However, the lasso is not robust against outliers in the data. By adding an L1 penalty on the coefficient estimates to the least trimmed squares (LTS) estimator, we introduce a robust and sparse estimator. We present a C-step algorithm for the computation of this sparse LTS estimator, and derive the breakdown point of sparse LTS and other lasso-type estimators. In addition, sparse LTS is compared to competing methods in simulation studies and a real data application.