Cyclic subspace regression with analysis of the hat matrix

Cyclic subspace regression (CSR) is a new approach to the complex multivari ate calibration problem. The simple algorithm produces solutions for princi pal component regression (PCR), partial least squares (PLS), least squares (LS), and other related intermediate regressions. This paper describes furt her analysis of CSR and shows that by using hat matrices, CSR regression ve ctors are formed from a summation of weighted eigenvectors where weights ar e determined from the hat matrix, singular values, and sample space eigenve ctors. Examination of CSR weights for PCR and PLS further documents differe nces and similarities and provides information to assist in determining pre diction rank for PCR and PLS. By redefining CSR in terms of weighted eigenv ectors, it can be shown when PLS and PCR produce essentially the same resul ts where minor differences stem from overfitting by PLS. Additionally, weig hts derived from the hat matrix show when PCR and PLS generate different re sults and why. Equations are shown for the sample space that reveal PLS to be a method based on oblique projections while PCR uses orthogonal projecti ons. The optimal intermediate CSR model can be identified as well. A near i nfrared data set is studied and illustrates principles involved. (C) 1999 E lsevier Science B.V. All rights reserved.