The Generalized Total Least Squares Problem : Formulation, Algorithm and Properties

Abstract

The Total Least Squares (TLS) method has been devised as a more global fitting technique than the ordinary least squares technique for solving overdetermined sets of linear equations AX ≈ B when errors occur in all data. If the errors on the measurements A and B are uncorrelated with zero mean and equal variance, TLS is able to compute a strongly consistent estimate of the true solution of the corresponding unperturbed set A ₀ X = B ₀. In this paper the TLS computations are generalized in order to maintain consistency of the solution in the following cases: first of all, some columns of A may be error-free and secondly, the errors on the remaining data may be correlated and not equally sized. Hereto, a numerically reliable Generalized TLS algorithm GTLS, based on the Generalized Singular Value Decomposition (GSVD), is developed. Additionally, the equivalence between the GTLS solution and alternative expressions of consistent estimators, described in literature, is proven. These relations allow to deduce the main statistical properties of the GTLS solution.