Alternative Expressions of Regression Vector for Principal Component Regression and Partial Least Squares Regression

Principal component regression (PCR) and partial least squares regression (PLSR) are examined. They usually appear in completely different computational forms in the literature. Since PLSR procedure is always given as an iterative algorithm, it is difficult to see its relationship with PCR. In this paper, alternative unifying expressions of the regression vectors for PCR and PLSR are derived using singular value decomposition of the predictor variables matrix. They are non-iterative and comparative expressions and enable us to gain some insight into the interrelationships between those methods.