Properties of cyclic subspace regression

Various properties of the regression vector β ^ kl produced by cyclic subspace regression with regard to the meancentered linear regression equation y ˜ = X β + ɛ ˜ are put forth. In particular, the subspace associated with the creation of β ^ kl is shown to contain a basis that maximizes certain covariances with respect to P l y ˜ , the orthogonal projection of y ˜ onto a specific subspace of the range of X . This basis is constructed. Moreover, this paper shows how the maximum covariance values effect the β ^ kl . Several alternative representations of β ^ kl are also developed. These representations show that β ^ kl is a modified version of the l -factor principal components regression vector β ^ ll , with the modification occurring by a nonorthogonal projection. Additionally, these representations enable prediction properties associated with β ^ kl to be explicitly identified. Finally, methods for choosing factors are spelled out.