Partial least median of squares regression

In modern data analysis, there is an increasing availability of datasets with numerous variables. Linear models that deal with abundant predictor variables often have poor performance because they tend to produce large variances. As well known, partial least squares (PLS) regression standouts because it is serviceable even if the number of variables far exceeds the number of samples. However, PLS, at its core, is a least-squares method based on latent space, which is spanned by the components extracted from the original predictors. Hence, it is sensitive to outliers. In this study, incorporating the idea of least median of squares, we propose a new robust PLS method, namely, partial least median of squares (PLMS) regression. Unlike most of the robust counterparts, we solve the PLMS problem via modern optimization rather than a heuristic process or a reweighting strategy. A classical PLS method and two of the most efficient robust PLS methods are compared with our method. Results on simulations and two real-world data sets demonstrate the effectiveness and robustness of our approach.