Details About the Thesis

Usually, we understand the classification as a partitioning of some collection of objects in disjoint subsets or classes, the objects from a class having common proprieties that make them different from the members of other classes. A fuzzy clustering is a generalization of this by the fact that the classes are not subsets of the collection, but fuzzy subsets similar with those defined by L.A. Zadeh [7]. In other words, the classes are mappings that gives each object a number between 0 and 1, named membership of the object to that class. Similar objects are identified by the fact that they have high memberships to the same class. We suppose that the memberships are chosen so that the sum of these for each point should be 1. In this sense, the fuzzy clustering is, also, a partitioning of a set of objects.

A fuzzy clustering algorithm is a "recipe" to produce a fuzzy clustering. The basis for building these algorithms are a quantitative measure of the similarity between objects and the criteria to relate the similarity with the membership degree. The criteria are often specified as functions of membership degrees, whose minimum or maximum define "good" classification. The algorithm becomes a numerical procedure for finding those membership degrees that optimizes the objective function.

The present work aims to approach the problem of the prototypes in fuzzy clustering.

The first chapter presents fundamental theoretical notions from the theory of fuzzy sets, notions absolutely necessary to be able to read the paper in good conditions.

The second chapter presents the problem of the hierarchical fuzzy clustering algorithms, with the particularities related to the use of different types of prototypes.

Here are presented the fuzzy hierarchical clustering with spherical and linear prototypes, and then some generalizations, namely clustering using ellipsoidal prototypes, adaptive prototypes and a non-metric clustering method using classical sets prototypes.

The third chapter presents the computer programs that implement the presented algorithms, details that are necessary in order to understand how these programs work.

The programs presented here perform the following tasks:

• reduction of characteristics to two dimensions in order to print the data;
• reduction of characteristics in order to eliminate the "bad" characteristics;
• fuzzy hierarchical clustering using the prototypes named above.

The fourth chapter presents a series of conclusions related to the use of the presented prototypes.

The paper contains two appendices, one with the sources of the programs and the other with the most interesting results obtained by me at the clustering of different sets of points chosen didactically.

Selected References

1. James C. Bezdek, C. Coray, R. Gunderson and J. Watson, Detection and characterization of cluster substructure, SIAM Journal on Applied Mathematics 40, 2 (1981), 339-371
2. D. Dumitrescu, Hierarchical pattern classification, Fuzzy Sets and Systems, 28 (1988), 145-162
3. D. Dumitrescu, Classification Theory (Romanian), Babes-Bolyai University of Cluj-Napoca, 1991
4. D. Dumitrescu, Horia F. Pop, A bibliography for fuzzy clustering and related fields, Studia Universitatis Babes-Bolyai, Series Math. 35, 3 (1990), 13-24
5. Horia F. Pop, Hierarchical fuzzy clustering with adaptive prototypes, National Students Conference on Computer Science, Iasi, April 1991
6. Michael P. Windham, Geometrical Fuzzy Clustering Algorithms, Preprint, Utah State University, 1983
7. Lotfi A. Zadeh, Fuzzy sets, Information and Control, 8 (1985), 338-353
8. W. Zangwill, Non-Linear Programming: A Unified Approach, Englewood Cliffs, NJ, Prentice Hall, 1969