An Analysis of the Total Least Squares Problem

Total Least Squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector $b(m \times 1)$ and in the data matrix $A(m \times n)$. The technique has been discussed by several authors, and amounts to fitting a “best” subspace to the points $(a_i^T ,b_i ),i = 1, \cdots ,m$, where $a_i^T $ is the ith row of A. In this paper a singular value decomposition analysis of the TLS problem is presented. The sensitivity of the TLS problem as well as its relationship to ordinary least squares regression is explored. An algorithm for solving the TLS problem is proposed that utilizes the singular value decomposition and which provides a measure of the underlying problem’s sensitivity.