Analysis and Properties of the Generalized Total Least Squares Problem $AX \approx B$ When Some or All Columns in A are Subject to Error

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 $AX \approx B$ when errors occur in all data. This method, introduced into numerical analysis by Golub and Van Loan, is strongly based on the Singular Value Decomposition (SVD). If the errors in 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 the statistical literature, these coefficients are called the parameters of a classical errors-in-variables model.In this paper, the TLS problem, as well as the TLS computations, are generalized in order to maintain consistency of the parameter estimates in a general errors-in-variables model; i.e., some of the columns of A may be known exactly and the covariance matrix of the errors in the rows of the remaining data matrix may be...