Weak nonlinearity in a model which arises from the Helmert transformation

Jan Ševčík

Applications of Mathematics (2003)

  • Volume: 48, Issue: 3, page 161-174
  • ISSN: 0862-7940

Abstract

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Nowadays, the algorithm most frequently used for determination of the estimators of parameters which define a transformation between two coordinate systems (in this case the Helmert transformation) is derived under one unreal assumption of errorless measurement in the first system. As it is practically impossible to ensure errorless measurements, we can hardly believe that the results of this algorithm are “optimal”. In 1998, Kubáček and Kubáčková proposed an algorithm which takes errors in both systems into consideration. It seems to be closer to reality and at least in this sense better. However, a partial disadvantage of this algorithm is the necessity of linearization of the model which describes the problem of the given transformation. The defence of this simplification especially with respect to the bias of linear functions of the final estimators, or better to say the specification of conditions under which such a modification is statistically insignificant is the aim of this paper.

How to cite

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Ševčík, Jan. "Weak nonlinearity in a model which arises from the Helmert transformation." Applications of Mathematics 48.3 (2003): 161-174. <http://eudml.org/doc/33142>.

@article{Ševčík2003,
abstract = {Nowadays, the algorithm most frequently used for determination of the estimators of parameters which define a transformation between two coordinate systems (in this case the Helmert transformation) is derived under one unreal assumption of errorless measurement in the first system. As it is practically impossible to ensure errorless measurements, we can hardly believe that the results of this algorithm are “optimal”. In 1998, Kubáček and Kubáčková proposed an algorithm which takes errors in both systems into consideration. It seems to be closer to reality and at least in this sense better. However, a partial disadvantage of this algorithm is the necessity of linearization of the model which describes the problem of the given transformation. The defence of this simplification especially with respect to the bias of linear functions of the final estimators, or better to say the specification of conditions under which such a modification is statistically insignificant is the aim of this paper.},
author = {Ševčík, Jan},
journal = {Applications of Mathematics},
keywords = {Helmert transformation; linear regression model; nonlinearity measures; weak nonlinearity; Helmert transformation; linear regression model; nonlinearity measures; weak nonlinearity},
language = {eng},
number = {3},
pages = {161-174},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weak nonlinearity in a model which arises from the Helmert transformation},
url = {http://eudml.org/doc/33142},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Ševčík, Jan
TI - Weak nonlinearity in a model which arises from the Helmert transformation
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 3
SP - 161
EP - 174
AB - Nowadays, the algorithm most frequently used for determination of the estimators of parameters which define a transformation between two coordinate systems (in this case the Helmert transformation) is derived under one unreal assumption of errorless measurement in the first system. As it is practically impossible to ensure errorless measurements, we can hardly believe that the results of this algorithm are “optimal”. In 1998, Kubáček and Kubáčková proposed an algorithm which takes errors in both systems into consideration. It seems to be closer to reality and at least in this sense better. However, a partial disadvantage of this algorithm is the necessity of linearization of the model which describes the problem of the given transformation. The defence of this simplification especially with respect to the bias of linear functions of the final estimators, or better to say the specification of conditions under which such a modification is statistically insignificant is the aim of this paper.
LA - eng
KW - Helmert transformation; linear regression model; nonlinearity measures; weak nonlinearity; Helmert transformation; linear regression model; nonlinearity measures; weak nonlinearity
UR - http://eudml.org/doc/33142
ER -

References

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  1. Regression models with a weak nonlinearity, Preprint of the Department of Mathematical Analysis and Applied Mathematics, Faculty of Science, Palacký University, 1998. (1998) MR1843367
  2. Testing statistical hypotheses in deformation measurement; One generalization of the Scheffé theorem, Acta Univ. Palacki Olomuc., Fac. rer. nat., Mathematica 37 (1998), 81–88. (1998) MR1690476
  3. Generalization of the orthogonal regression on the case of the Helmert transformation, WDS ’99 Proceedings of Contributed Papers, part I, MATFYZPRESS, Praha, 1999. (1999) 

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