A Hyperbolic Fuzzt k-Means Clustering and Algorithm for Neural Networks

  • Norio Watenable
  • Tadashi Imaizumi
  • Toshiko Kikuchi
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


A new fuzzy k-means clustering algorithm is proposed by introducing crisp regions of clusters. Boundaries of the regions are determined by hyperbolas and membership values are given by one or zero in each region. The area between crisp regions is a fuzzy region, where membership values are proportional to distances to crisp regions. Though the traditional hard k-means is a limit of the usual fuzzy k-means, results of the latter are fuzzy and then are not the same as results of the former. On the other hand a new method can produce the same results as those by the hard k-means. An algorithm for neural networks is given and a numerical example is illustrated.


Voronoi Diagram Fuzzy Cluster Classification Function Fuzzy Method Learning Neural Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BEZDEK, J.C. (1981): Pattern Recognition with Fuzzy Objective Function Algoritms. Plenum, New York.Google Scholar
  2. KOHONEN, T. (1995): Self-Organizing Maps, Springer.Google Scholar
  3. RUMELHART, D.E. and et al. (1986): Parallel Distributed Processing, The MIT Press, Massachusetts.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 2000

Authors and Affiliations

  • Norio Watenable
    • 1
  • Tadashi Imaizumi
    • 2
  • Toshiko Kikuchi
    • 3
  1. 1.Department of Industrial and Systems EngineeringChuo UniversityBunkyo-ku, TokyoJapan
  2. 2.Tama UniversityTama-shi, TokyoJapan
  3. 3.Faculty of EconomicsTohoku Gakuin UniversityMiyagiJapan

Personalised recommendations