Finding Y‐relevant part of X by use of PCR and PLSR model reduction methods

The paper is considering the following question: using principal component regression (PCR) or partial least squares regression (PLSR), how much data can be removed from X while retaining the original ability to predict Y ? Two model reduction methods using similarity transformations are discussed, one giving projections of original loadings onto the column space of the fitted response matrix $\hat {\bf Y}$ (essentially the orthogonal signal correction (OSC) methods), and one giving projections of original scores onto the column space of the coefficient matrix $\hat {\bf B}$ (essentially the net analyte signal (NAS) methods). The loading projection method gives model residuals that are orthogonal to Y and $\hat {\bf Y}$ , which is valuable in certain applications. The score projection method, on the other hand, gives model residuals that are orthogonal to $\hat {\bf B}$ , which is essential in other applications. It is shown that the reduced matrix X   Y Sfrom the score projection method is a subset of the reduced matrix X   Y Lfrom the loading projection method. It therefore has the smallest Frobenius norm, and thus the smallest total column variance, assuming centered data. Copyright © 2007 John Wiley & Sons, Ltd.