Robust estimation and variable selection for cellwise contaminated data

Abstract

Outliers are widespread in real-world datasets. Recognizing outliers and running robust analyses is still a challenging topic. Recently, there has been

increased attention on cellwise outliers. In contrast to traditional rowwise (observation-wise) outliers, cellwise outliers target individual cells within observations of a dataset, where only specific cells within each row may be contaminated. Several challenges need to be addressed in this field of research, such as outlier detection, robust covariance matrix estimation and robust (sparse) regression.

We introduce a Gaussian rank based Lasso estimator, which uses the Gaussian rank correlation to obtain an initial empirical covariance matrix among

the response and potential active predictors. We re-parameterise the classical linear regression model design matrix and the response vector to take advantage of these robustly estimated components

before applying the adaptive Lasso to obtain consistent variable selection results.

We also introduce cellwise regularized Lasso, a regularized regression method to address cellwise outliers, which employs a cellwise shrinkage procedure that shrinks outlying cells based on the magnitude of regression residuals and cell deviations.