Skip to main content

Processing of \(Z^+\)-numbers Using the k Nearest Neighbors Method

  • Conference paper
  • First Online:
Advances in Soft and Hard Computing (ACS 2018)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 889))

Included in the following conference series:

  • 376 Accesses

Abstract

The paper presents that with the application of \(Z^+\)-numbers arithmetic, the k nearest neighbors method can be adapted to various types of data. Both, the learning data and the input data may be in the form of the crisp number, interval, fuzzy or \(Z^+\)-number. The paper discusses the methods of performing arithmetic operations on uncertain data of various types and explains how to use them in the kNN method. Experiments show that the method works correctly and gives credible results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181, 2923–2932 (2011)

    Article  Google Scholar 

  2. Atkeson, C.G., Moore, A.W., Schaal, S.A.: Locally weighted learning. Artif. Intell. Rev. 11, 11–73 (1997)

    Article  Google Scholar 

  3. Cichosz, P.: Learning Systems. WNT Publishing House, Warsaw (2000). [in Polish]

    MATH  Google Scholar 

  4. Hand, D., Mannila, H., Smyth, P.: Principles of Data Mining. The MIT Press, Cambridge (2001)

    Google Scholar 

  5. Kordos, M., Blachnik, M., Strzempa, D.: Do we need whatever more than k-NN? In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2010. LNCS, vol. 6113, pp. 414–421. Springer, Heidelberg (2010)

    Google Scholar 

  6. KorzeƄ, M., Klęsk, P.: Sets of approximating functions with finite Vapnik-Czervonenkis dimension for nearest-neighbours algorithm. Pattern Recogn. Lett. 32, 1882–1893 (2011)

    Article  Google Scholar 

  7. PluciƄski, M.: Application of the information-gap theory for evaluation of nearest neighbours method robustness to data uncertainty. Przegląd Elektrotechniczny 88(10b), 272–275 (2012)

    Google Scholar 

  8. PluciƄski, M., Pietrzykowski, M.: Application of the \(k\) nearest neighbors method to fuzzy data processing. Przegląd Elektrotechniczny 93(1), 77–81 (2017)

    Google Scholar 

  9. Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.A.: The Arithmetic of Z-Numbers: Theory and Applications. World Scientific, Singapore (2015)

    Book  Google Scholar 

  10. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)

    Article  MathSciNet  Google Scholar 

  11. Grzegorzewski, P.: Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst. 97, 83–94 (1998)

    Article  MathSciNet  Google Scholar 

  12. Piegat, A.: Fuzzy Modeling and Control. Physica, Heidelberg (2001)

    Book  Google Scholar 

  13. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)

    Book  Google Scholar 

  14. Dutta, P., Boruah, H., Ali, T.: Fuzzy arithmetic with and without using \(\alpha \)-cut method: a comparative study. Int. J. Latest Trends Comput. 2(1), 99–107 (2011)

    Google Scholar 

  15. Hanss, M.: Applied Fuzzy Arithmetic. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  16. Kaufmann, A., Gupta, M.M.: Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York (1991)

    MATH  Google Scholar 

  17. Springer, M.D.: The Algebra of Random Variables. John Wiley & Sons, New York (1979)

    MATH  Google Scholar 

  18. Jaroszewicz, S., KorzeƄ, M.: Arithmetic operations on independent random variables: a numerical approach. SIAM J. Sci. Comput. 34(3), 1251–1265 (2012)

    Article  MathSciNet  Google Scholar 

  19. Diamond, P., Rosenfeld, A.: Metric spaces of fuzzy sets. Fuzzy Sets Syst. 35, 241–249 (1990)

    Article  MathSciNet  Google Scholar 

  20. Tang, W., Li, X., Zhao, R.: Metric spaces of fuzzy variables. Comput. Ind. Eng. 57, 1268–1273 (2009)

    Article  Google Scholar 

  21. Rachev S.T., Klebanov L., Stoyanov S.V., Fabozzi F.: The Methods of Distances in the Theory of Probability and Statistics. Springer Science and Business Media (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marcin PluciƄski .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

PluciƄski, M. (2019). Processing of \(Z^+\)-numbers Using the k Nearest Neighbors Method. In: Pejaƛ, J., El Fray, I., Hyla, T., Kacprzyk, J. (eds) Advances in Soft and Hard Computing. ACS 2018. Advances in Intelligent Systems and Computing, vol 889. Springer, Cham. https://doi.org/10.1007/978-3-030-03314-9_7

Download citation

Publish with us

Policies and ethics