Abstract
In engineering practice, usually measurement errors are described by normal distributions. However, in some cases, the distribution is heavy-tailed and thus, not normal. In such situations, empirical evidence shows that the Student distributions are most adequate. The corresponding recommendation – based on empirical evidence – is included in the International Organization for Standardization guide. In this paper, we explain this empirical fact by showing that a natural fuzzy-logic-based formalization of commonsense requirements leads exactly to the Student’s distributions.
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References
Aczél, J.: Lectures on Functional Equations and Their Applications. Dover, New York (2006)
Aczél, J., Dhombres, H.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)
International Organization for Standardization (ISO), ISO/IEC Guide 98-3:2008, Uncertainty of Measurement – Part 3: Guide to the Expression of Uncertainty in Measurement, GUM 1995 (2008)
Jaynes, E.T., Bretthorst, G.L.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)
Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, Upper Saddle River (1995)
Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1983)
Nguyen, H.T., Kreinovich, V., Wojciechowski, P.: Strict archimedean t-norms and t-conorms are universal approximators. Int. J. Approx. Reason. 18(3–4), 239–249 (1998)
Nguyen, H.T., Kreinovich, V., Wu, B., Xiang, G.: Computing Statistics Under Interval and Fuzzy Uncertainty. Springer, Heidelberg (2012)
Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logic. Chapman and Hall/CRC, Boca Raton (2006)
Novitskii, P.V., Zograph, I.A.: Estimating the Measurement Errors. Energoatomizdat, Leningrad (1991). (in Russian)
Orlov, A.I.: How often are the observations normal? Ind. Lab. 57(7), 770–772 (1991)
Rabinovich, S.G.: Measurement Errors and Uncertainty: Theory and Practice. Springer, Berlin (2005)
Resnick, S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)
Sheskin, D.J.: Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC, Boca Raton (2011)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Acknowledgments
This work was performed when Vladik was a visiting researcher with the Geodetic Institute of the Leibniz University of Hannover, a visit supported by the German Science Foundation. This work was also supported in part by NSF grant HRD-1242122.
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Alkhatib, H., Kargoll, B., Neumann, I., Kreinovich, V. (2018). Normalization-Invariant Fuzzy Logic Operations Explain Empirical Success of Student Distributions in Describing Measurement Uncertainty. In: Melin, P., Castillo, O., Kacprzyk, J., Reformat, M., Melek, W. (eds) Fuzzy Logic in Intelligent System Design. NAFIPS 2017. Advances in Intelligent Systems and Computing, vol 648. Springer, Cham. https://doi.org/10.1007/978-3-319-67137-6_34
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DOI: https://doi.org/10.1007/978-3-319-67137-6_34
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