Algebraic relationships between classical regression and total least-squares estimation

Abstract

The total least-squares (TLS) technique, which is well known in numerical linear algebra and able to compute strongly consistent estimators of the parameters in a linear errors-in-variables model, is compared algebraically with the classical regression estimators. Using the singular-value decomposition and geometric concepts, algebraic equivalences and important relationships between the classical regression techniques and TLS estimation are established with special reference to problems of collinearity. The equivalence between principal-component and latent-root regression in collinearity problems is proven, and the difference between latent-root regression and TLS estimation is clarified.