Abstract
Fuzzy techniques describe expert opinions. At first glance, we would therefore expect that the more accurately the corresponding membership functions describe the expert’s opinions, the better the corresponding results. In practice, however, contrary to these expectations, the simplest – and not very accurate – triangular membership functions often work the best. In this paper, on the example of the use of membership functions in F-transform techniques, we provide a possible theoretical explanation for this surprising empirical phenomenon.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Belohlavek, R., Dauben, J.W., Klir, G.J.: Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, New York (2017)
Brito, A.E., Kosheleva, O.: Interval + image = wavelet: for image processing under interval uncertainty, wavelets are optimal. Reliab. Comput. 4(3), 291–301 (1998)
Chui, C.: An Introduction to Wavelets. Academic Press, San Diego (1992)
Edwards, R.E.: Functional Analysis: Theory and Applications. Dover, New York (2011)
Jaulin, L., Kiefer, M., Dicrit, O., Walter, E.: Applied Interval Analysis. Springer, London (2001). https://doi.org/10.1007/978-1-4471-0249-6
Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, Upper Saddle River (1995)
Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, Burlington (2008)
Mayer, G.: Interval Analysis and Automatic Result Verification. De Gruyter, Berlin (2017)
Mendel, J.M.: Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-51370-6
Meyer, Y.: Wavelets Algorithms and Applications. SIAM, Philadelphia (1993)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logic. Chapman and Hall/CRC, Boca Raton (2006)
Novak, V., Perfilieva, I., Holcapek, M., Kreinovich, V.: Filtering out high frequencies in time series using F-transform. Inf. Sci. 274, 192–209 (2014)
Novak, V., Perfilieva, I., Kreinovich, V.: F-transform in the analysis of periodic signals. In: Inuiguchi, M., Kusunoki, Y., Seki, M. (eds.), Proceedings of the 15th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty CJS’2012, Osaka, Japan, 24–27 September 2012 (2012)
Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)
Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157, 993–1023 (2006)
Perfilieva, I.: F-transform. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 113–130. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-43505-2_7
Perfilieva, I., Danková, M., Bede, B.: Towards a higher degree F-transform. Fuzzy Sets Syst. 180(1), 3–19 (2011)
Perfilieva, I., Kreinovich, V., Novak, V.: F-transform in view of trend extraction. In: Inuiguchi, M., Kusunoki, Y., Seki, M. (eds.) Proceedings of the 15th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty CJS’2012, Osaka, Japan, 24–27 September 2012 (2012)
Rabinovich, S.G.: Measurement Errors and Uncertainty: Theory and Practice. Springer, Berlin (2005). https://doi.org/10.1007/0-387-29143-1
Sheskin, D.J.: Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC, Boca Raton (2011)
Vetterli, M., Kovacevic, J.: Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs (1995)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Acknowledgments
This work was supported in part by the US National Science Foundation grant HRD-1242122.
The authors are greatly thankful to the anonymous referees for valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Kosheleva, O., Kreinovich, V. (2018). Why Triangular Membership Functions are Often Efficient in F-transform Applications: Relation to Probabilistic and Interval Uncertainty and to Haar Wavelets. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 854. Springer, Cham. https://doi.org/10.1007/978-3-319-91476-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-91476-3_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91475-6
Online ISBN: 978-3-319-91476-3
eBook Packages: Computer ScienceComputer Science (R0)