Brought to you by EarthWeb
IT Library Logo

Click Here!
Click Here!

Search the site:
 
EXPERT SEARCH -----
Programming Languages
Databases
Security
Web Services
Network Services
Middleware
Components
Operating Systems
User Interfaces
Groupware & Collaboration
Content Management
Productivity Applications
Hardware
Fun & Games

EarthWeb Direct EarthWeb Direct Fatbrain Auctions Support Source Answers

EarthWeb sites
Crossnodes
Datamation
Developer.com
DICE
EarthWeb.com
EarthWeb Direct
ERP Hub
Gamelan
GoCertify.com
HTMLGoodies
Intranet Journal
IT Knowledge
IT Library
JavaGoodies
JARS
JavaScripts.com
open source IT
RoadCoders
Y2K Info

Previous Table of Contents Next


2.3. FUZZY CLUSTERING

An intuitive approach to objective rule generation is based upon fuzzy clustering of input-output data. One simple and applicable idea, especially for systems with large numbers of input variables, was suggested by Sugeno and Yasukawa (1993). In this approach, one first clusters only the output space, which can be always considered as a single-dimensional space in fuzzy models. The fuzzy partition of the input space is specified at the next step by generating the projection of the output clusters into each input variable space separately. Using this method, the rule generation step could be separated from the input selection step, as will be discussed later.

The idea of fuzzy clustering is to divide the output data into fuzzy clusters that overlap each other. Therefore, the assignment of each data point to each cluster is defined by a membership grade in [0,1]. Formally, clustering unlabeled data X = {x1, x2, ..., xN} Rh, where N is the number of data vectors and h is the dimension of each data vector, is the assignment of c number of cluster labels to the vectors in X. c-Clusters of X are sets with (c,N) membership values {uik} that can be conveniently arranged as a (cxN) matrix U = [uik]. The problem of fuzzy clustering is to find the optimum membership matrix U for fuzzy clustering in X. The most widely used objective function is the weighted within-groups sum of squared errors function Jm, which is defined as the constrained optimization problem (Bezdek, 1981):

where 0 [less than or equal to] uik [less than or equal to] 1, i, k; k, [inverted sans serif], such that uik > 0; 0 < V = {v1, v2, ..., vc} is the vector of (unknown) cluster centers, and is any inner product norm.

A is a positive definite matrix that specifies the shape of the cluster. The common selection for the matrix A is the identity matrix, leading to the definition of Euclidean distance, and consequently to spherical clusters. There are, however, investigations where the matrix A is taken as the covariance matrix that generates models with elliptic clusters (Kosko, 1996).

In current literature, most fuzzy clustering studies are carried out by the Fuzzy C-Means (FCM) algorithm through an iterative optimization. However, there are three major difficulties with FCM clustering algorithm:

  1. Number of clusters (c) should be assigned a priori.
  2. No theoretical basis has been established for an optimal choice of weighting exponent "m."
  3. Conditions of the FCM algorithm are only necessary conditions for local extrema of Jm.

That is, different choices of initial V0 might lead to different local extrema. Therefore, it is desirable to modify the FCM algorithm.


Previous Table of Contents Next

footer nav
Use of this site is subject certain Terms & Conditions.
Copyright (c) 1996-1999 EarthWeb, Inc.. All rights reserved. Reproduction in whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Please read our privacy policy for details.