Fuzzy Sets in Information Retrieval and Cluster Analysis
The present monograph intends to establish a solid link among three fields: fuzzy set theory, information retrieval, and cluster analysis. Fuzzy set theory supplies new concepts and methods for the other two fields, and provides a common frame work withi
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THEORY AND DECISION LIBRARY General Editors: W. Leinfellner and G. Eberlein Series A: Philosophy and Methodology of the Social Sciences Editors: W. Leinfellner (Technical University of Vienna) G. Eberlein (Technical University of Munich) Series B: Mathematical and Statistical Methods Editor: H. Skala (University ofPaderborn) Series C: Game Theory, Mathematical Programming and Operations Research Editor: S. H. Tijs (University of Nijmegen) Series D: System Theory, Knowledge Engineering and Problem Solving Editor: W. Janko (University of Economics, Vienna)
SERIES D: SYSTEM THEORY, KNOWLEDGE ENGINEERING AND
PROBLEM SOLVING
Editor: W. Janko (Vienna) Volume 4
Editorial Board G. Feichtinger (Vienna), H. T. Nguyen (Las Cruces), N. B. Nicolau (Palma de Mallorca), o. Opitz (Augsburg), H. J. Skala (paderborn), M. Sugeno (Yokohama).
Scope This series focuses on the design and description of organisations and systems with application to the social sciences. Formal treatment of the subjects is encouraged. Systems theory, information systems, system analysis, interrelated structures, program systems and expert systems are considered to be a theme within the series. The fundamental basics of such concepts including computational and algorithmic aspects and the investigation of the empirical behaviour of systems and organisations will be an essential part of this library. The study of problems related to the interface of systems and organisations to their environment is supported. Interdisciplinary considerations are welcome. The publication of recent and original results will be favoured.
For a list of titles published in this series, see final page.
FUZZY SETS IN INFORMATION RETRIEVAL AND CLUSTER ANALYSIS
by SADAAKI MIYAMOTO University ojTsukuba, Japan
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data Mi yamoto. Sadaak i. 1950Fuzzy sets in information retrieval and cluster ana1ysis I Sadaaki Miyamoto. p. cm. -- a, Z E S}. We always use a weak a-cut in this monograph. The distinction between these two types of a-cut is not important here, since we deal only with finite sets. Example 2.4. Let S and A be the same as those in Example 2.3. Assume that 0'=0.5. Then
C{0.5)A
= 0/1 + 1/2 + 1/3 + 1/4 + 0/5
C{0.5)AG = 1/1 + 1/2 + 0/3 + 0/4 + 1/5. As noted before, the a-cut is a fundamental operation which relates a fuzzy set and the one-parameter family of crisp sets. To show that a fuzzy set can be regarded as the family of its a-cuts, the following expression called resolution identity is used: A = UaC(a)A. a
While an a-cut transforms a fuzzy set into a crisp set, another operator represented by L(a) transforms a fuzzy set A into another fuzzy set L(aHA]:
_ {I'A{Z) if I'A{Z) ~ a I'L(a)[A]{Z) - 0 if I'A{Z) < a. L{aHA] is called a level fuzzy set or an a-level fuzzy set {Radecki,1977a)j a level fuzzy set L{a)[A] is also abbreviated as L(a)A. An identity that is similar to the resolution identity holds for level fuzzy sets: A = UL{aHA]. a
Figures 2.5 and 2.6 illustrate relationships
