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Algorithms for L 1 and L p Fuzzy c-Means and Their Convergence

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Data Science, Classification, and Related Methods

Summary

Algorithms for L 1 and L p based fuzzy c-means are proposed. These algorithms calculate cluster centers in the general alternating algorithm of the fuzzy c-means. The algorithm for the L 1 space is based on a simple linear search on nodes of step functions derived from derivatives of components of the objective function for the fuzzy c-means, whereas the algorithm for the L p spaces use binary search on the nodes and then the interval to which the cluster center belong. Termination of the algorithms based on different criteria for the convergence is discussed. The algorithm for the L 1 space is proved to be convergent after a finite number of iterations. A numerical example is shown.

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References

  • Bezdek, J.C. (1981): Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum.

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Author information

Authors and Affiliations

  1. Institute of Information Sciences, Electronics University of Tsukuba, Ibaraki 305, Japan

    Sadaaki Miyamoto

  2. Development and Documentation Division Central Bureau of Statitics, Program Research, Indonesia

    Yudi Agusta

Authors
  1. Sadaaki Miyamoto
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  2. Yudi Agusta
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Editor information

Editors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

    Chikio Hayashi , Noboru Ohsumi  & Yasumasa Baba ,  & 

  2. School of Management, Science University of Tokyo, 500 Shimokiyoku, Kuki, Saitama 346, Japan

    Keiji Yajima

  3. Institut für Statistik, Rheinisch-Westfälische Technische Hochschule (RWTH), D-52056, Aachen, Germany

    Hans-Hermann Bock

  4. Faculty of Environmental Science & Technology, Okayama University, 2-1-1 Tsushima-naka, Okayama 700, Japan

    Yutaka Tanaka

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© 1998 Springer Japan

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Miyamoto, S., Agusta, Y. (1998). Algorithms for L 1 and L p Fuzzy c-Means and Their Convergence. In: Hayashi, C., Yajima, K., Bock, HH., Ohsumi, N., Tanaka, Y., Baba, Y. (eds) Data Science, Classification, and Related Methods. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Tokyo. https://doi.org/10.1007/978-4-431-65950-1_32

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  • DOI: https://doi.org/10.1007/978-4-431-65950-1_32

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-70208-5

  • Online ISBN: 978-4-431-65950-1

  • eBook Packages: Springer Book Archive

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