Interpreting PLSR and PCR Solutions via Moore-Penrose Generalized Inverse

This paper interprets partial least squares regression (PLSR) and principal component regression (PCR) solutions and data transformations in terms of Moore-Penrose generalized inverse. By finding a Moore-Penrose inverse of the matrix X ⁺ for the solution b = X ⁺ y in a rather backward way, matrix expressions for the transformed X matrices are provided and the way they alter the original X data is shown for the PLSR and PCR methods. A numerical example is given to illustrate how the transformation matrices and the transformed X matrices change as the number of components varies.