The HPPLS Procedure

SIMPLS

Note that each extracted PLS factor is defined in terms of different X-variables bold upper X Subscript i. This leads to difficulties in comparing different scores, weights, and so on. The SIMPLS method of De Jong (1993) overcomes these difficulties by computing each score bold t Subscript i Baseline equals bold upper X bold r Subscript i in terms of the original (centered and scaled) predictors bold upper X. The SIMPLS X-weight vectors r Subscript i are similar to the eigenvectors of bold upper S bold upper S Superscript prime Baseline equals bold upper X prime bold upper Y bold upper Y prime bold upper X, but they satisfy a different orthogonality condition. The bold r 1 vector is just the first eigenvector bold e 1 (so that the first SIMPLS score is the same as the first PLS score). However, the second eigenvector maximizes

bold e prime 1 bold upper S bold upper S prime bold e 2 subject to bold e prime 1 bold e 2 equals 0

whereas the second SIMPLS weight bold r 2 maximizes

bold r prime 1 upper S upper S prime bold r 2 subject to bold r prime 1 bold upper X prime bold upper X bold r 2 equals bold t prime 1 bold t 2 equals 0

The SIMPLS scores are identical to the PLS scores for one response but slightly different for more than one response; see De Jong (1993) for details. The X- and Y-loadings are defined as in PLS, but because the scores are all defined in terms of bold upper X, it is easy to compute the overall model coefficients bold upper B:

StartLayout 1st Row 1st Column ModifyingAbove bold upper Y With caret 2nd Column equals 3rd Column sigma-summation Underscript i Endscripts bold t Subscript i Baseline bold c prime Subscript i 2nd Row 1st Column Blank 2nd Column equals 3rd Column sigma-summation Underscript i Endscripts bold upper X bold r Subscript i Baseline bold c prime Subscript i 3rd Row 1st Column Blank 2nd Column equals 3rd Column bold upper X bold upper B comma where bold upper B equals bold upper R bold upper C prime EndLayout
Last updated: December 09, 2022