Collinearity and Total Least Squares

The least squares (LS) and total least squares (TLS) methods are commonly used to solve the overdetermined system of equations $Ax \approx b$. The main objective of this paper is to examine TLS when A is nearly rank deficient by outlining its differences and similarities to the well-known truncated LS method. It is shown that TLS may be viewed as a regularization technique much like truncated LS, even though the rank reduction depends on b. The sensitivity of LS and TLS approximate nullspaces to perturbations in the data is also examined. Some numerical simulations are included.