Skip to main content
SpringerLink
Log in

On Bezdek-Type Possibilistic Clustering for Spherical Data, Its Kernelization, and Spectral Clustering Approach

  • Conference paper
  • First Online:
Modeling Decisions for Artificial Intelligence (MDAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9880))

  • 1268 Accesses

Abstract

In this study, a Bezdek-type fuzzified possibilistic clustering algorithm for spherical data (bPCS), its kernelization (K-bPCS), and spectral clustering approach (sK-bPCS) are proposed. First, we propose the bPCS by setting a fuzzification parameter of the Tsallis entropy-based possibilistic clustering optimization problem for spherical data (tPCS) to infinity, and by modifying the cosine correlation-based dissimilarity between objects and cluster centers. Next, we kernelize bPCS to obtain K-bPCS, which can be applied to non-spherical data with the help of a given kernel, e.g., a Gaussian kernel. Furthermore, we propose a spectral clustering approach to K-bPCS called sK-bPCS, which aims to solve the initialization problem of bPCS and K-bPCS. Furthermore, we demonstrate that this spectral clustering approach is equivalent to kernelized principal component analysis (K-PCA). The validity of the proposed methods is verified through numerical examples.

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

Access this chapter

Log in via an institution
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

References

  1. Bezdek, J.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981)

    Book  MATH  Google Scholar 

  2. MacQueen, J.B.: Some methods of classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297 (1967)

    Google Scholar 

  3. Miyamoto, S., Mukaidono, M.: Fuzzy \(c\)-means as a regularization and maximum entropy approach. In: Proceedings of the 7th International Fuzzy Systems Association World Congress (IFSA 1997), vol. 2, pp. 86–92 (1997)

    Google Scholar 

  4. Ménard, M., Courboulay, V., Dardignac, P.: Possibilistic and probabilistic fuzzy clustering: unification within the framework of the non-extensive thermostatistics. Pattern Recogn. 36, 1325–1342 (2003)

    Article  MATH  Google Scholar 

  5. Krishnapuram, R., Keller, J.M.: A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst. 1, 98–110 (1993)

    Article  Google Scholar 

  6. Krishnapuram, R., Keller, J.M.: The possibilistic \(c\)-means algorithm: insights and recommendations. IEEE Trans. Fuzzy Syst. 4, 393–396 (1996)

    Article  Google Scholar 

  7. Strehl, A., Ghosh, J., Mooney, R.: Impact of similarity measures on web-page clustering. In: Proceedings of the AAAI2000, pp. 58–64 (2000)

    Google Scholar 

  8. Dhillon, I.S., Modha, D.S.: Concept decompositions for large sparse text data using clustering. Mach. Learn. 42, 143–175 (2001)

    Article  MATH  Google Scholar 

  9. Miyamoto, S., Mizutani, K.: Fuzzy multiset model and methods of nonlinear document clustering for information retrieval. In: Torra, V., Narukawa, Y. (eds.) MDAI 2004. LNCS (LNAI), vol. 3131, pp. 273–283. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  10. Mizutani, K., Inokuchi, R., Miyamoto, S.: Algorithms of nonlinear document clustering based on fuzzy set model. Int. J. Intell. Syst. 23(2), 176–198 (2008)

    Article  MATH  Google Scholar 

  11. Kanzawa, Y.: A maximizing model of Bezdek-like spherical fuzzy \(c\)-means. Int. J. Intell. Syst. 19(5), 662–669 (2015)

    Google Scholar 

  12. Kanzawa, Y.: On kernelization for a maximizing model of bezdek-like spherical fuzzy c-means clustering. In: Torra, V., Narukawa, Y., Endo, Y. (eds.) MDAI 2014. LNCS, vol. 8825, pp. 108–121. Springer, Heidelberg (2014)

    Google Scholar 

  13. Kanzawa, Y.: Fuzzy clustering based on \(\alpha \)-divergence for spherical data and for categorical multivariate data. In: Proceedings of the FUZZ-IEEE2015, #15091 (2015)

    Google Scholar 

  14. Kanzawa, Y.: On possibilistic clustering methods based on Shannon/Tsallis-entropy for spherical data and categorical multivariate data. In: Torra, V., Narukawa, T. (eds.) MDAI 2015. LNCS, vol. 9321, pp. 115–128. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  15. Scholkopf, B., Smola, A., Muller, K.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)

    Article  Google Scholar 

  16. Hornik, K., Feinerer, I., Kober, M., Buchta, C.: Spherical k-means clustering. J. Stat. Softw. 50(10), 1–22 (2012)

    Article  Google Scholar 

  17. Zelnik-Manor, L., Perona, P.: Self-tuning spectral clustering. Adv. Neural Inf. Process. Syst. 17, 1601–1608 (2005)

    Google Scholar 

  18. Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2, 193–218 (1985)

    Article  MATH  Google Scholar 

  19. Kanzawa, Y.: Sequential cluster extraction using power-regularized possibilistic \(c\)-means. JACIII 19(1), 67–73 (2015)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shibaura Institute of Technology, Koto, Tokyo, 135-8548, Japan

    Yuchi Kanzawa

Authors
  1. Yuchi Kanzawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuchi Kanzawa .

Editor information

Editors and Affiliations

  1. University of Kövde , Skövde, Sweden

    Vicenç Torra

  2. Toho Gakuen , Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. Enginyeria de la Informacio I de les Com, Univ Autonoma de Barcelona, Bellaterra, Spain

    Guillermo Navarro-Arribas

  4. Universitat d'Andorra , Sant Julià de Lòria, Andorra

    Cristina Yañez

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Kanzawa, Y. (2016). On Bezdek-Type Possibilistic Clustering for Spherical Data, Its Kernelization, and Spectral Clustering Approach. In: Torra, V., Narukawa, Y., Navarro-Arribas, G., Yañez, C. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2016. Lecture Notes in Computer Science(), vol 9880. Springer, Cham. https://doi.org/10.1007/978-3-319-45656-0_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45656-0_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45655-3

  • Online ISBN: 978-3-319-45656-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation