Using Zonotopes for Overestimation-Free Interval Least-Squares - Some Geodetic Applications

authored by: Steffen Schön, Hansjörg Kutterer
Abstract: Interval methods are very convenient to describe the uncertainty of measurements and derived parameters in engineering sciences. In this paper, the geodetic determination of points in the 2d or 3d Euclidean space by least-squares estimation is studied. Up to now, two problems limited the applicability of interval mathematics. First, due to overestimation, interval boxes are too pessimistic uncertainty measures for point positions. Second, the shape of the interval boxes depends on the orientation of the geodetic coordinate system to parametrise the configuration. It is shown that both problems can be overcome by zonotopes which describe directly the factual range of the leastsquares problem with interval-valued observations. The advantages of this concept are discussed using typical geodetic network scenarios. In addition, the possible benefit for other engineering applications is motivated.
Organisation(s): Geodetic Institute
External Organisation(s): Graz University of Technology
Type: Article
Journal: Reliable Computing
Volume: 11
Pages: 137-155
No. of pages: 19
ISSN: 1385-3139
Publication date: 04.2005
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Software, Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1007/s11155-005-3034-4 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{1ebe7ddee4d24320bdda0e7825679d9c,
title = "Using Zonotopes for Overestimation-Free Interval Least-Squares - Some Geodetic Applications",
abstract = "Interval methods are very convenient to describe the uncertainty of measurements and derived parameters in engineering sciences. In this paper, the geodetic determination of points in the 2d or 3d Euclidean space by least-squares estimation is studied. Up to now, two problems limited the applicability of interval mathematics. First, due to overestimation, interval boxes are too pessimistic uncertainty measures for point positions. Second, the shape of the interval boxes depends on the orientation of the geodetic coordinate system to parametrise the configuration. It is shown that both problems can be overcome by zonotopes which describe directly the factual range of the leastsquares problem with interval-valued observations. The advantages of this concept are discussed using typical geodetic network scenarios. In addition, the possible benefit for other engineering applications is motivated.",
author = "Steffen Sch{\"o}n and Hansj{\"o}rg Kutterer",
note = "Funding information: This paper shows results and new ideas developed during the research project KU 1250/1-2 at the German Geodetic Research Institute (DGFI, Munich). This project was sponsored by the Deutsche Forschungsgemeinschaft (DFG) which is gratefully acknowledged by the authors. The authors warmly thank the DGFI director Hon.-Prof. Hermann Drewes for his support as well as the whole DGFI team for their help and fruitful discussions during the authors{\textquoteright} stay in Munich. The constructive comments of the reviewers are gratefully acknowledged; they helped to clarify some topics and to improve the presentation.",
year = "2005",
month = apr,
doi = "10.1007/s11155-005-3034-4",
language = "English",
volume = "11",
pages = "137--155",
journal = "Reliable Computing",
issn = "1385-3139",
publisher = "Springer Netherlands",
number = "2",
}