The PCR method that we described above involves identifying linear combinations of X that best represent the predictors. These combinations (directions) are identified in an unsupervised way, since the response Y is not used to help determine the principal component directions. That is, the response Y does not supervise the identification of the principal components, thus there is no guarantee that the directions that best explain the predictors also are the best for predicting the response (even though that is often assumed). Partial least squares (PLS) are a supervisedalternative to PCR. Like PCR, PLS is a dimension reduction method, which first identifies a new smaller set of features that are linear combinations of the original features, then fits a linear model via least squares to the new M features. Yet, unlike PCR, PLS makes use of the response variable in order to identify the new features.
