Outlier-Resistant 𝐿𝟏 Orthogonal Regression via the Reformulation-Linearization Technique

Abstract

Assessing the linear relationship between a set of continuous predictors and a continuous response is a well-studied problem in statistics and data mining. 𝐿2-based methods such as ordinary least squares and orthogonal regression can be used to determine this relationship. However, both of these methods become impaired when influential values are present. This problem becomes compounded when outliers confound standard diagnostics. This work proposes an 𝐿1-norm orthogonal regression method (𝐿1OR) formulated as a nonconvex optimization problem. Solution strategies for finding globally optimal solutions are presented. Simulation studies are conducted to assess the resistance of the method to outliers and the consistency of the method. The method is also applied to real-world data arising from an environmental science application.