A robust partial least squares regression method with applications

Abstract

Partial least squares (PLS) regression is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper, we show that if the sample covariance matrix is properly robustified further robustification of the linear regression steps of the PLS algorithm becomes unnecessary. The robust estimate of the covariance matrix is computed by searching for outliers in univariate projections of the data on a combination of random directions (Stahel-Donoho) and specific directions obtained by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto [1]. It is shown that this procedure is fast to apply and provides better results than other methods proposed in the literature. Its performance is illustrated by Monte Carlo and by an example, where the algorithm is able to show features of the data which were undetected by previous methods.

On Visualizing Mixed-Type Data

“…For outliers detection in high-dimensional multivariate data, see also Peña and Prieto (2007). The latter approach has been successfully applied for outliers detection in a regression context (González, Peña, and Romera 2009).…”

A New Approach to Robust Partial Least Squares Regression Analysis

“…The application of this algorithm can be seen as a two step procedure: (1) the weights i w that define the new orthogonal regressor i t , are computed with Equations (2.7) and (2.8) by using the covariance matrix of the observations; (2) the regression coefficients i q are computed from a simple regression between the response, y and the regressor i t . As it is shown in Equation (2.9), these two steps depend only on the covariance matrix of the observations and it may be thought that if this matrix is properly robustified the procedure will be robust [3].…”

“…Procedures combining robust covariance matrices and robust regression methods have been proposed by Gil and Romera (1998), Hubert and Vanden Branden (2003). González et al (2009) also concentrated in the case of univariate response (PLS1) and showed that if the sample covariance matrix is properly robustified the PLS1 algorithm will be robust and, therefore, further robustification of the linear regression steps of the PLS1 algorithm is unnecessary [2,3,5].…”

“…In this study, similar to Gil and Romera (1998) and González et al (2009) studies, we also concentrate in the case of univariate response (PLS1) and we present a procedure which applies the standard PLS1 algorithm to a robust covariance matrix. In our study, we estimate the covariance matrix used in PLS1 algorithm robustly by using well-known S-estimators.…”

An Improved Generalized Predictive Control in a Robust Dynamic Partial Least Square Framework

“…To overcome this problem, several robust PLS methods are proposed in recent years. One of them is obtained by robustifying the sample covariance matrix between the input and output datasets based on SIMPLS algorithm [8,9]. Another is PLS calibration with outliers detection, which selects a subset of samples dataset randomly and gets the initial residual set and then detects and discards outliers in the subset [10].…”