The model space in partial least squares regression

This paper documents an inconsistency caused by truncation in the conventional partial least squares (PLS) regression algorithm with one dependent variable. It is shown that the conventional PLS algorithm with orthogonal score vectors uses one model space to compute the regression vector but another model space to represent the reconstructed data. In comparison, the Bidiag2 bidiagonalization algorithm and the non-orthogonal PLS algorithm of Martens are consistent in using the same space for the regression vector and for the reconstructed data, while the SIMPLS algorithm has the same inconsistency as the conventional PLS algorithm. The magnitude of the difference depends on the degree of truncation of the model space: it is generally larger with greater truncations but is not a monotonic function of the truncation. A numerical example demonstrates implications for outlier detection. Consistent behavior upon truncation may be important when PLS regression is implemented as part of standardized analytical procedures. Copyright (C) 2007 John Wiley & Sons, Ltd.