Robustness properties of a robust PLS regression method

The presence of multicollinearity in regression data is no exception in real life examples. Instead of applying ordinary regression methods, biased regression techniques such as Principal Component Regression and Ridge Regression have been developed to cope with such data sets. In this paper we consider Partial Least Squares (PLS) regression by means of the SIMPLS algorithm. Because the SIMPLS algorithm is based on the empirical variance-covariance matrix of the data and on least squares regression, outliers have a damaging effect on the estimates. To reduce this pernicious effect of outliers, we propose to replace the empirical variance-covariance matrix in SIMPLS by a robust covariance estimator. We derive the influence function of the resulting PLS weight vectors and the regression estimates, and conclude that they will be bounded if the robust covariance estimator has a bounded influence function. Also the breakdown value is inherited from the robust estimator. We illustrate the results using the MCD estimator and the reweighted MCD estimator (RMCD) for low-dimensional data sets. Also some empirical properties are provided for a high-dimensional data set.