Abstract
In Geodesy, parameter estimation based on the least-squares principle is a common tool for the solution of data analysis problems. It is assumed, at least implicitly, that the unavoidable observation errors are exclusively stochastic with zero expectation. The corresponding variance-covariance matrix (vcm) of the estimated parameters is then computed from the observations’ vcm just by means of variance propagation. However, the complete error budget of the observation process comprises additional, non-stochastic types of observation errors like, e.g., imprecision. Imprecision summarizes effects due to the imperfect knowledge about the observation setup. Fuzzy set theory and fuzzy data analysis supply adequate techniques to model and to handle imprecision. Since in geodetic data analysis both stochasticity and imprecision of the observations may be relevant, approaches for their combination are needed. Techniques from fuzzy-theory are introduced in this paper for the handling of observation imprecision. The joint treatment of observation stochasticity and imprecision is discussed. Numerical examples are given.
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References
Bandemer H.; Näther W. (1992): Fuzzy Data Analysis. Kluwer Academic Publishers, Dordrecht.
Baarda W. (1979): Mathematical models. OEEPE Official Publ. No. 11, 73–101.
Bardossy A. (1990): Note on fuzzy regression. Fuzzy Sets and Systems 37(1990) 65–75.
Bardossy A.; Hagaman R.; Duckstein L.; Bogardi I. (1992): Fuzzy least squares regression: theory and applications. In: Kacprzyk J.; Fedrizzi M. (Eds.): Fuzzy regression analysis. Omnitech Press, Warsaw, and Physica, Heidelberg, 181–193.
Brunner F. K. (1991): Über die Grenze von Modellen. Österr. Z. Vermess.wesen Photogrammetrie 79 (1991): 9–20.
Dubois D.; Prade H. (1980): Fuzzy Sets and Systems. Academic Press, New York.
Grafarend E.W.; Schaffrin B. (1993): Ausgleichungsrechnung in linearen Modellen. BI Wissenschaftsverlag, Mannheim.
Kaufmann A.; Gupta M. M. (1991): Introduction to Fuzzy Arithmetic — Theory and Applications. Van Nostrand Reinhold, New York.
Klir G.; Wierman M. (1998): Uncertainty-Based Information. Physica, Heidelberg.
Koch K.-R. (1999): Parameter Estimation and Hypothesis Testing in Linear Models. (2nd Ed.) Springer, Berlin.
Körner R. (1997): Linear models with random fuzzy variables. Dissertation, Fakultät für Mathematik und Informatik, Technische Universität Freiberg.
Kruse R.; Gebhardt J.; Klawonn F. (1994): Foundations of Fuzzy Systems. Wiley, Chichester.
Kruse R.; Meyer K. D. (1987): Statistics with vague data. D. Reidel, Dordrecht.
Kutterer H. (1994): Intervallmathematische Behandlung endlicher Unschärfen linearer Ausgleichungsmodelle. DGK C 423, München.
Kwakernaak H. (1978): Fuzzy random variables — I, definitions and theorems. Information Sciences 15 (1978): 1–29.
Näther W. (1997): Linear statistical inference for random fuzzy data. Statistics 29 (1997): 221–240.
Puri M. L.; Ralescu D. A. (1986): Fuzzy random variables. Journal of Mathematical Analysis and Applications 114, 409–422.
Viertl R. (1996): Statistical Methods for Non-Precise Data. CRC Press, Boca Raton New York London Tokyo.
Zadeh L. A. (1965): Fuzzy sets. Information Control 8 (1965): 338–353.
Zadeh L. A. (1971): Similarity relations and fuzzy orderings. Information Sciences 3 (1971): 177–200.
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Kutterer, H. (2001). Some Considerations on Fuzzy Least-Squares. In: Sideris, M.G. (eds) Gravity, Geoid and Geodynamics 2000. International Association of Geodesy Symposia, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04827-6_12
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DOI: https://doi.org/10.1007/978-3-662-04827-6_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07634-3
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