Performing similarity transformations using the error-in-variable model

Least-Squares (LS) adjustment method aims at estimating a vector of parameters ξ, from a linear model ( y = Aξ, +e) that includes an observation vector y, a vector of normally distributed errors e, and a matrix of variables A. However, in this linear model, also known as the Gauss-Markov model, the matrix of variables A is considered as fixed or error-free. This is not the case in many physical systems where errors exist in both the observation vector y, and the matrix of variables A. The Total Least-Squares (TLS) method uses a relatively new mathematical concept that was developed to solve estimation problems in so-called Error- In-all-Variables models. In this contribution, a novel application of the Total Least-Squares method for the Linear Conformal Coordinate Transformation is described. The unique structure of the data matrix A, where some variables appear twice, is also considered in a newly developed "Structured TLS procedure" for the Similarity Transformation. A practical coordinate transformation problem is presented to demonstrate this new technique, and a comparison is made between the standard (generalized) least squares approach and the TLS approach.