Projection Matrices for Partial Least Squares Regression and Principal Component Regression

In this paper projection matrices for partial least squares regression(PLSR) and principal component regression(PCR) are considered with no assumption on the shape and rank of the predictor variable matrix X. It is shown how the original X matrices are transformed to new matrices and how several different forms of projection matrices are obtained for each of the methods. An interpretation is given to the procedure of the regressions using the transformed X matrices and projection matrices. This gives not only solutions for PLSR and PCR in the case of general X but some insight into the properties of the procedures.