Skip to main content
SpringerLink
Log in

A new pseudometric on a subclass of Riesz spaces based on similarity measures

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

The Bunke–Shearer metric is one of many distance measures used in data analysis for studying similarities between finite sets. In this work, we generalize it to a new pseudometric for a certain subclass of Riesz spaces. We exemplify some particular cases of this general pseudometric in the contexts of measure spaces, Euclidean spaces, and fuzzy sets.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bunke, H., Shearer, K.: A graph distance metric based on the maximal common subgraph. Pattern Recognit. Lett. 19(3), 255–259 (1998). https://doi.org/10.1016/S0167-8655(97)00179-7

    Article  MATH  Google Scholar 

  3. Cha, S.H.: Comprehensive survey on distance/similarity measures between probability density functions. Int. J. Math. Models Methods Appl. Sci. 1(4), 300–307 (2007)

    MathSciNet  Google Scholar 

  4. Chen, S.M.: A new approach to handling fuzzy decision-making problems. IEEE Trans. Syst. Man Cybern. 18(6), 1012–1016 (1988). https://doi.org/10.1109/21.23100

    Article  MATH  Google Scholar 

  5. Deza, M.M., Deza, E.: Encyclopedia of Distances, 4th edn. Springer, Berlin (2016)

    Book  Google Scholar 

  6. Dress, A., Lokot, T., Pustyl’nikov, L.: A new scale-invariant geometry on l1 spaces. Appl. Math. Lett. 17(7), 815–820 (2004). https://doi.org/10.1016/j.aml.2004.06.011

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, J., Xie, W.: Some notes on similarity measure and proximity measure. Fuzzy Sets Syst. 101(3), 403–412 (1999). https://doi.org/10.1016/S0165-0114(97)00108-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Horadam, K., Nyblom, M.: Distances between sets based on set commonality. Discret. Appl. Math. 167, 310–314 (2014). https://doi.org/10.1016/j.dam.2013.10.037

    Article  MathSciNet  MATH  Google Scholar 

  9. Levandowsky, M., Winter, D.: Distance between sets. Nature 234, 34–35 (1971). https://doi.org/10.1038/234034a0

    Article  Google Scholar 

  10. Lipkus, A.H.: A proof of the triangle inequality for the Tanimoto distance. J. Math. Chem. 26, 263–265 (1999). https://doi.org/10.1023/A:1019154432472

    Article  MATH  Google Scholar 

  11. Luxemburg, W., Zaanen, A.: Riesz Spaces, vol. 1. North-Holland Pub. Co, New York (1971)

    MATH  Google Scholar 

  12. Nieto, J., Torres, A., Vázquez-Trasande, M.: A metric space to study differences between polynucleotides. Appl. Math. Lett. 16(8), 1289–1294 (2003). https://doi.org/10.1016/S0893-9659(03)90131-5

    Article  MathSciNet  MATH  Google Scholar 

  13. Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces, 2nd edn. Springer, New York (1999)

    Book  Google Scholar 

  14. Solaiman, B., Bossé, E.: Possibility Theory for the Design of Information Fusion Systems, 1st edn, pp. 86, 108–109. Springer, Basel (2019)

  15. Zimmermann, H.: Fuzzy Set Theory and Its Applications, 3rd edn. Kluwer Academic Publishers, New York (1996)

    Book  Google Scholar 

Download references

Acknowledgements

The author would like to express utmost gratitude to Dr. Maite Fernández-Unzueta and Dr. José-Carlos Gómez-Larrañaga from Centro de Investigación en Matemáticas (CIMAT), Mexico for their careful reading, suggestions and full support during the writing of the paper.

Author information

Authors and Affiliations

  1. School of Science, National Autonomous University of Mexico, Circuito Exterior S/N C.U. Coyoacán, 04510, Mexico City, Mexico

    Carlos-Eduardo García-Reyes

Authors
  1. Carlos-Eduardo García-Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos-Eduardo García-Reyes.

Ethics declarations

Conflict of interest

The author declares that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García-Reyes, CE. A new pseudometric on a subclass of Riesz spaces based on similarity measures. Bol. Soc. Mat. Mex. 27, 50 (2021). https://doi.org/10.1007/s40590-021-00356-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40590-021-00356-w

Keywords

Mathematics Subject Classification

Search

Navigation