The Fuzzy C+2 Means: Solving the Extended Ambiguity Reject in Clustering. Le Rejet D'Ambiguïté dans les Fc+2M: Extension à Toutes les Classes Composées

The Fuzzy C+2 Means: Solving the Extended Ambiguity Reject in Clustering

Le Rejet D'Ambiguïté dans les Fc+2M: Extension à Toutes les Classes Composées

Michel Ménard

Avenue Marillac 17042 La Rochelle Cedex 1, Fronce

Page: 
121-144
|
Received: 
2 April 1998
|
Accepted: 
N/A
|
Published: 
30 April 1999
| Citation

OPEN ACCESS

Abstract: 

We address in this paper a clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes. The method that we propose is formulated as a constrained minimization problem, whose solution depends on a cost function in which rejection options are introduced . Two types of rejection have been included : the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes. To compute these rejections, we propose an extension of the fuzzy c-means (FcM) algorithm of Bezdec ([2]). This algorithm is called the fuzzy c+2-means (Fc+2M). These measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes. The advantages of our method are (1) the degree of membership to the rejection classes for a pattern xk are learned in the iterative clustering problem; (2) the partial ambiguity rejections introduce a discounting process between the classical FcM membership functions in order to avoid the memberships to be spread across the classes ; (3) the membership functions are more immune to noise and correspond more closely to the notion of compatibility . 

Résumé 

Cet article concerne le problème de la classification dont l'objectif consiste à calculer une partition d'un ensemble de formes en classes disjointes. La méthode que nous proposons s'écrit sous la forme d'un problème de minimisation d'un critère de classification dont la solution dépend d'une fonction de contrainte dans laquelle la notion de rejet est introduite . Deux types de rejet ont été inclus : le rejet d'ambiguité qui concerne les individus situés sur la frontière de deux ou plusieurs classes et le rejet de distance gérant ceux qui sont loin de toutes les classes. La notion de rejet permet de gérer l'incertitude due à l'imprécision et à la définition incomplète des classes. Pour. calculer ces rejets, nous proposons une extension de l'algorithme des c -moyennes floues (FcM) de Bezdec [2]. Cet algorithme est appeléc+2-moyennes floues (Fc+2 M). Ces avantages sont :(1) le degré d'appartenance aux classes de rejet d'un individu est déterminé par un calcul itératif durant la phase de coalescence; (2) l'algorithme inclut une modélisation de l'hésitation ou de l'ambigu ité, et des fonctions d'appartenance sont affectées à tous les sous-ensembles de classes de 211 plutôt qu'aux éléments de c2 seulement (S2 étant l'ensemble des classes). De plus, le degré d'appartenance à la classe de rejet d'ambiguité totale d'un individu est calculé explicitement; (3) en introduisant les rejets d'ambigui'té partielle, nous créons une discontinuité entre les fonctions d'appartenance affectées aux classes de 52, ce qui permet de rendre ces fonctions d'appartenance indépendantes . 

Keywords: 

Pattern recognition, fuzzy sets, clustering, fuzzy c-means, distance rejection, ambiguity rejection .

Mots clés 

Reconnaissance des formes, théorie des ensembles flous, coalescence flou, c-moyennes floues, rejet d'ambiguité et rejet de distance.

1. Introduction
2. Coalescence Floue avec Rejet D'Ambiguité
3. Le Rejet D'Ambiguïté et de Distance Simultanément
4. Exemples Numériques
5. Conclusion
6. Annexe
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