Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Probabilistic Measures and Integrals: How to Aggregate Imprecise Data

  • Conference paper
  • First Online:
Modeling Decisions for Artificial Intelligence (MDAI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12256))

  • 474 Accesses

Abstract

This paper develops the theory of probabilistic-valued measures and integrals as a suitable aggregation tool for dealing with certain types of imprecise information. The motivation comes from Moore’s interval mathematics, where the use of intervals in data processing is due to measurement inaccuracy errors. In case of rounding, the intervals can be considered in distribution function form linked to random variables uniformly distributed over the relevant intervals. We demonstrate how the convolution of distribution functions is taken into account, and integration with respect to probabilistic-valued measures is converted into convolving certain distribution functions. We also improve some existing features of the integral and investigate its convergence properties.

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-16-0337 and also cofinanced by bilateral call Slovak-Poland grant scheme No. SK-PL-18-0032 with the Polish National Agency for Academic Exchange PPN/BIL/2018/1/00049/U/00001.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Similar content being viewed by others

References

  1. Borzová-Molnárová, J., Halčinová, L., Hutník, O.: Probabilistic-valued decomposable set functions with respect to triangle functions. Inf. Sci. 295, 347–357 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bukovský, L.: The Structure of the Real Line. Monografie Matematyczne. Springer, Basel (2011). https://doi.org/10.1007/978-3-0348-0006-8

    Book  MATH  Google Scholar 

  3. Cobzaş, S.: Completeness with respect to the probabilistic Pompeiu-Hausdorff metric. Stud. Univ. Babeş-Bolyai, Math. 52(3), 43–65 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  5. Halčinová, L., Hutník, O.: An integral with respect to probabilistic-valued decomposable measures. Int. J. Approx. Reason. 55, 1469–1484 (2014)

    Article  MathSciNet  Google Scholar 

  6. Halčinová, L., Hutník, O., Mesiar, R.: On distance distribution functions-valued submeasures related to aggregation functions. Fuzzy Sets Syst. 194(1), 15–30 (2012)

    Article  MathSciNet  Google Scholar 

  7. Halčinová, L., Hutník, O., Mesiar, R.: On some classes of distance distribution functions-valued submeasures. Nonlinear Anal. 74(5), 1545–1554 (2011)

    Article  MathSciNet  Google Scholar 

  8. Halmos, P.: Measure Theory. Van Nostrand, New York (1968)

    Google Scholar 

  9. Hutník, O., Mesiar, R.: On a certain class of submeasures based on triangular norms. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 17, 297–316 (2009)

    Article  MathSciNet  Google Scholar 

  10. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, Trends in Logic, Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  11. Jong, S.K., Sung, L.K., Yang, H.K., Yu, S.J.J.: Generalized convolution of uniform distributions. Appl. Math. Inform. 28, 1573–1581 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U.S.A. 28, 535–537 (1942)

    Article  MathSciNet  Google Scholar 

  13. Mood, A.M., Graybill, F.A., Boes, D.C.: Introduction to the Theory of Statistics. McGraw-Hill, New York (1974)

    MATH  Google Scholar 

  14. Moore, R.E., Bierbaum, F.: Methods and Applications of Interval Analysis. Studies in Applied and Numerical Mathematics. SIAM, Philadelphia (1979)

    Google Scholar 

  15. Pap, E.: Applications of decomposable measures on nonlinear differential equations. Novi Sad J. Math. 31, 89–98 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Publishing, New York (1983)

    MATH  Google Scholar 

  17. Šerstnev, A.N.: Random normed spaces: problems of completeness. Kazan. Gos. Univ. Uch. Zap. 122, 3–20 (1962)

    Google Scholar 

  18. Torra, V., Navarro-Arribas, G.: Probabilistic metric spaces for privacy by design machine learning algorithms: modeling database changes. In: Garcia-Alfaro, J., Herrera-Joancomartí, J., Livraga, G., Rios, R. (eds.) DPM/CBT -2018. LNCS, vol. 11025, pp. 422–430. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00305-0_30

    Chapter  Google Scholar 

  19. Weber, S.: \(\perp \)-decomposable measures and integrals for Archimedean t-conorm. J. Math. Anal. Appl. 101, 114–138 (1984)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, ul. Wólczańska 215, 90-924, Lodz, Poland

    Michał Boczek & Marek Kaluszka

  2. Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01, Košice, Slovakia

    Lenka Halčinová & Ondrej Hutník

Authors
  1. Michał Boczek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lenka Halčinová
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ondrej Hutník
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marek Kaluszka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lenka Halčinová .

Editor information

Editors and Affiliations

  1. Department of Computing Science, Umeå University, Umeå, Sweden

    Vicenç Torra

  2. Department of Management Science, Tamagawa University, Tokyo, Japan

    Yasuo Narukawa

  3. Department of Operations, Innovation and Data Sciences, ESADE, Sant Cugat, Spain

    Jordi Nin

  4. Department of Operations, Innovation and Data Sciences, ESADE, Sant Cugat, Spain

    Núria Agell

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Boczek, M., Halčinová, L., Hutník, O., Kaluszka, M. (2020). Probabilistic Measures and Integrals: How to Aggregate Imprecise Data. In: Torra, V., Narukawa, Y., Nin, J., Agell, N. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2020. Lecture Notes in Computer Science(), vol 12256. Springer, Cham. https://doi.org/10.1007/978-3-030-57524-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-57524-3_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-57523-6

  • Online ISBN: 978-3-030-57524-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation