Abstract
While the Errors-In-Variables (EIV) Model has been treated as a special case of the nonlinear Gauss- Helmert Model (GHM) for more than a century, it was only in 1980 that Golub and Van Loan showed how the Total Least-Squares (TLS) solution can be obtained from a certain minimum eigenvalue problem, assuming a particular relationship between the diagonal dispersion matrices for the observations involved in both the data vector and the data matrix. More general, but always nonsingular, dispersion matrices to generate the “properly weighted” TLS solution were only recently introduced by Schaffrin and Wieser, Fang, and Mahboub, among others. Here, the case of singular dispersion matrices is investigated, and algorithms are presented under a rank condition that indicates the existence of a unique TLS solution, thereby adding a new method to the existing literature on TLS adjustment. In contrast to more general “measurement error models,” the restriction to the EIV-Model still allows the derivation of (nonlinear) closed formulas for the weighted TLS solution. The practicality will be evidenced by an example from geodetic science, namely the over-determined similarity transformation between different coordinate estimates for a set of identical points.
References
Helmert, F. R., 1907, Adjustment Computations by the Method of Least Squares (in German), Teubner: Leipzig, Germany, second edition.Search in Google Scholar
Magnus, J. R. and Neudecker, H., 2007, Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley: Chichester, England, third edition.Search in Google Scholar
Mahboub, V., 2012, On weighted total least-squares for geodetic transformations, Journal of Geodesy, 86(5):359-367.10.1007/s00190-011-0524-5Search in Google Scholar
Markovsky, I., Rastello, M. L., Premoli, A., Kukush, A., and Van Huffel, S., 2006, The element-wise weighted total least-squares problem, Computational Statistics & Data Analysis, 50(1):181-209.10.1016/j.csda.2004.07.014Search in Google Scholar
Neitzel, F. and Schaffrin, B., 2013, On the Gauss-Helmert Model with a singular dispersionmatrix where BQ is of lesser row rank than B, Studia Geophysica et Geodaetica, revised in June 2013.Search in Google Scholar
Pearson, K., 1901, LIII. On lines and planes of closest fit to systems of points in space, PhilosophicalMagazine, Series 6, 2(11):559-572.10.1080/14786440109462720Search in Google Scholar
Schaffrin, B., Neitzel, F., Uzun, S., and Mahboub, V., 2012, Modifying Cadzow’s algorithm to generate the optimal TLS-solution for the structured EIV-model of a similarity transformation, Journal of Geodetic Science, 2(2):98-106.10.2478/v10156-011-0028-5Search in Google Scholar
Schaffrin, B. and Wieser, A., 2008, On weighted total least-squares adjustment for linear regression, Journal of Geodesy, 82(7):415-421.10.1007/s00190-007-0190-9Search in Google Scholar
Snow, K., 2012, Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Prior Information, PhD dissertation, Report No. 502, Div. of Geodetic Science, School of Earth Sciences, The Ohio State Univ., Columbus/ OH.Search in Google Scholar
Xu, P., Liu, J., and Shi, C., 2012, Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis, Journal of Geodesy, 86(8):661-675.10.1007/s00190-012-0552-9Search in Google Scholar
© by K. Snow
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