The HPPLS Procedure

Partial least squares (PLS) works by extracting one factor at a time. Let be the centered and scaled matrix of predictors, and let be the centered and scaled matrix of response values. The PLS method starts with a linear combination of the predictors, where is called a score vector and is its associated weight vector. The PLS method predicts both and by regression on :

The vectors and are called the X- and Y-loadings, respectively.

The specific linear combination is the one that has maximum covariance with some response linear combination . Another characterization is that the X-weight, , and the Y-weight, , are proportional to the first left and right singular vectors, respectively, of the covariance matrix or, equivalently, the first eigenvectors of and , respectively.

This accounts for how the first PLS factor is extracted. The second factor is extracted in the same way by replacing and with the X- and Y-residuals from the first factor:

These residuals are also called the deflated and blocks. The process of extracting a score vector and deflating the data matrices is repeated for as many extracted factors as are wanted.