Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth

By Burkhard Schaffrin, K. Snow. 2010

58 citations recorded by the source corpus.

Abstract In this contribution a variation of Golub/Hansen/O’Leary’s Total Least-Squares (TLS) regularization technique is introduced, based on the Hybrid APproximation Solution (HAPS) within a nonlinear Gauss–Helmert Model. By applying a traditional Lagrange approach to a series of iteratively linearized Gauss–Helmert Models, a new iterative scheme has been found that, in practice, can generate the Tykhonov regularized TLS solution, provided that some care is taken to do the updates properly. The algorithm actually parallels the standard TLS approach as recommended in some of the geodetic literature, but unfortunately all too often in combination with erroneous updates that would still show convergence, although not necessarily to the (unregularized) TLS solution. Here, a key feature is that both standard and regularized TLS solutions result from the same computational framework, unlike the existing algorithms for Tykhonov-type TLS regularization. The new algorithm is then applied to a problem from archeology. There, both the radius and the center-point coordinates of a circle have to be determined, of which only a small part of the arc had been surveyed in-situ, thereby giving rise to an ill-conditioned set of equations. According to the archaeologists involved, this circular arc served as the starting line of a racetrack in the ancient Greek stadium of Corinth, ca. 500 BC. The present study compares previous estimates of the circle parameters with the newly developed “Regularized TLS Solution of Tykhonov type.”

Subfields: Applied Mathematics, Artificial Intelligence, Environmental Engineering

View Paper: https://www.semanticscholar.org/paper/f1f55cca780041610971c39d74729ebb71f634a2

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In Applied Mathematics

1. X. Fang
2. Frank Neitzel
3. Guobin Chang
4. Alireza Amiri-Simkooei
5. S. Jazaeri

In Artificial Intelligence

1. X. Fang
2. Frank Neitzel
3. Guobin Chang

In Environmental Engineering

1. X. Fang
2. Frank Neitzel
3. Guobin Chang

Citations in the corpus

• Variance components in errors-in-variables models: estimability, stability and bias analysis (2014, unknown): “Schaffrin and Snow (2010) reformulated the functional EIV model (1) as a system of nonlinear Gauss-Helmert condition equations.”
• Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation (2010, methodology): “This adjustment has been named “total least-squares” (TLS) by Golub and van Loan (1980) although it is known to be the standard LS-method as applied to the errors-in-variables (EIV) model, see Schaffrin and Snow (2010), for instance.”
• Weighted total least squares: necessary and sufficient conditions, fixed and random parameters (2013, methodology): “The evidence in this study reinforces the argument of Neitzel (2010) and Schaffrin and Snow (2010) that the TLS adjustment does not refer to a new adjustment method but rather to an adjustment for the model that contains the uncertain model matrix.”