Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation |
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| Authors: | Frank Neitzel |
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| Institution: | 1. University of Applied Sciences Mainz, Mainz, Germany
2. School of Earth Sciences, The Ohio State University, Columbus, OH, USA
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| Abstract: | In this contribution it is shown that the so-called “total least-squares estimate” (TLS) within an errors-in-variables (EIV)
model can be identified as a special case of the method of least-squares within the nonlinear Gauss–Helmert model. In contrast
to the EIV-model, the nonlinear GH-model does not impose any restrictions on the form of functional relationship between the
quantities involved in the model. Even more complex EIV-models, which require specific approaches like “generalized total
least-squares” (GTLS) or “structured total least-squares” (STLS), can be treated as nonlinear GH-models without any serious
problems. The example of a similarity transformation of planar coordinates shows that the “total least-squares solution” can
be obtained easily from a rigorous evaluation of the Gauss–Helmert model. In contrast to weighted TLS, weights can then be
introduced without further limitations. Using two numerical examples taken from the literature, these solutions are compared
with those obtained from certain specialized TLS approaches. |
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