The Annals of Applied Probability

A self-organizing cluster process

Robert M. Burton and William G. Faris

Full-text: Open access


The state of the self-organizing cluster process is a finite subset of points in a bounded region. This subset represents an evolving discrete approximation to a continuous probability distribution in the region. The dynamics of the process is determined by an independent sequence of random points in the region chosen according to the distribution. At each time step the random point attracts the nearest point in the finite set. In this way the subset learns to approximate its environment. It is shown that initial states approach each other exponentially fast for all time with probability one. Thus all memory of the initial state is lost; the environment alone determines future history.

Article information

Ann. Appl. Probab. Volume 6, Number 4 (1996), 1232-1247.

First available in Project Euclid: 24 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx] 68T10: Pattern recognition, speech recognition {For cluster analysis, see 62H30}
Secondary: 62J20: Diagnostics 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Self-organization Kohonen pattern recognition neural network cluster Markov chain random transformation


Burton, Robert M.; Faris, William G. A self-organizing cluster process. Ann. Appl. Probab. 6 (1996), no. 4, 1232--1247. doi:10.1214/aoap/1035463330.

Export citation


  • [1] Bouton, C. and Pag´es, G. (1993). Self-organization and a.s. convergence of the onedimensional Kohonen algorithm with nonuniformly distributed stimuli. Stochastic Process. Appl. 47 249-274.
  • [2] Bouton, C. and Pag´es, G. (1994). Convergence in distribution of the one-dimensional Kohonen algorithms when the stimuli are not uniform. Adv. in Appl. Probab. 26 80-103.
  • [3] Cottrell, M. and Fort, J.-C. (1987). ´Etude d'un processus d'auto-organization. Ann. Inst. H. Poincar´e Probab. Statist. 23 1-20.
  • [4] Dudley, R. M. (1989). Real Analy sis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • [5] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • [6] Erwin, E., Obermay er, K. and Schulten, K. (1992). Self-organizing maps: ordering, convergence properties, and energy functions. Biol. Cy bernet. 67 47-55.
  • [7] Kangas, J. A., Kohonen, T. K. and Laaksonen, J. T. (1990). Variants of self-organizing maps. IEEE Transactions on Neural Networks 1 93-99.
  • [8] Kohonen, T. (1989). Self-Organization and Associative Memory, 3rd ed. Springer, Berlin.
  • [9] Li, X., Gasteiger, J. and Zupan, J. (1993). On the topology distortion in self-organizing feature maps. Biol. Cy bernet. 70 189-198.
  • [10] Lo, Z.-P. and Bavarian, B. (1991). On the rate of convergence in topology preserving neural networks. Biol. Cy bernet. 65 55-63.
  • [11] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [12] Ritter, H., Martinetz, T. and Schulten, K. (1992). Neural Computation and SelfOrganizing Maps. Addison-Wesley, Reading, MA.
  • [13] Tolat, V. V. (1990). An analysis of Kohonen's self-organizing maps using a sy stem of energy functionals. Biol. Cy bernet. 64 155-164.