## The Annals of Applied Probability

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

### A self-organizing cluster process

Robert M. Burton and William G. Faris

#### Abstract

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

**Source**

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

**Dates**

First available: 24 October 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1035463330

**Mathematical Reviews number (MathSciNet)**

MR1422984

**Digital Object Identifier**

doi:10.1214/aoap/1035463330

**Zentralblatt MATH identifier**

0870.60063

**Subjects**

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]

**Keywords**

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

#### Citation

Burton, Robert M.; Faris, William G. A self-organizing cluster process. The Annals of Applied Probability 6 (1996), no. 4, 1232--1247. doi:10.1214/aoap/1035463330. http://projecteuclid.org/euclid.aoap/1035463330.