On Total Least-Squares Adjustment with Constraints

Abstract

For calibration purposes, oftentimes various datasets are compared in such a way that observations enter the coefficient matrix of a Linear Model ("errors-in-variables"). In such a case, the Total Least-Squares approach would be appropriate that was pioneered by G. Golub and C. van Loan in the early eighties. In essence, rather than solving the usual normal equations system for the estimated parameters, the smallest singular values of a slightly extended system is set to be zero, and its eigenvector is re-scaled to provide the estimated parameter vector. The authors have recently presented their studies that show the potential of this technique to provide improved variograms for geostatistical Kriging applications.

Sometimes, however, stability or slow convergence problems may occur with the algorithm as designed so far. In order to increase the stability, additional parameters could be introduced to represent the functional model under investigation, but with a number of constraints that keep the original redundancy unchanged. In the end, the same Total Least-Squares Fit is supposed to result after fewer iterations from the newly developed scheme that, for the first time, allows the integration of constraints between the parameters, thus solving a case that was long considered "untreatable" by the original TLS algorithm.