Similarity and Compatibility in Fuzzy Set Theory

Transcription

1 Similarity and Compatibility in Fuzzy Set Theory

2 Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska Warsaw, Poland kacprzyk@ibspan.waw.pl = 2941 Further volumes of this series can be found at our homepage. Vol. 72. M. Mares Fuzzy Cooperative Games ISBN Vol. 73. Y. Yoshida (Ed.) Dynamical Aspects in Fuzzy Decision, 2001 ISBN Vol. 74. H.-N. Teodorescu, L.C. Jain and A. Kandel (Eds.) Hardware Implementation of Intelligent Systems, 2001 ISBN Vol. 75. V. Loia and S. Sessa (Eds.) Soft Computing Agents, 2001 ISBN Vol. 76. D. Ruan, J. Kacprzyk and M. Fedrizzi (Eds.) Soft Computing for Risk Evaluation and Management, 2001 ISBN Vol. 77. W. Liu Propositional, Probabilistic and Evidential Reasoning, 2001 ISBN Vol. 78. U. Seiffert and L. C. Jain (Eds.) Self-Organizing Neural Networks, 2002 ISBN Vol. 79. A. Osyczka Evolutionary Algorithms for Single and Multicriteria Design Optimization, 2002 ISBN Vol. 80. P. Wong, F. Aminzadeh and M. Nikravesh (Eds.) Soft Computing for Reservoir Characterization and Modeling, 2002 ISBN Vol. 81. V. Dimitrov and V. Korotkich (Eds.) Fuzzy Logic, 2002 ISBN Vol. 82. Ch. Carlsson and R. Fuller Fuz...ry Reasoning in Decision Making and Optimization, 2002 ISBN Vol. 83. S. Barro and R. Marin (Eds.) Fuzzy Logic in Medicine, 2002 ISBN Vol. 84. L.c. Jain and J. Kacprzyk (Eds.) New Learning Paradigms in Soft Computing, 2002 ISBN Vol. 85. D. Rutkowska Neuro-Fuzzy Architectures and Hybrid Learning, 2002 ISBN Vol. 86. M.B. Gorzalczany Computational Intelligence Systems and Applications, 2002 ISBN Vol. 87. C. Bertoluzza, M.A. Gil and D.A. Ralescu (Eds.) Statistical Modeling, Analysis and Management of Fuz...ry Data, 2002 ISBN Vol. 88. R.P. Srivastava and T.J. Mock (Eds.) Belief Functions in Business Decisions, 2002 ISBN Vol. 89. B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena and R.R. Yager (Eds.) Technologies for Constructing Intelligent Systems I ISBN Vol. 90. B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena and R.R. Yager (Eds.) Technologies for Constructing Intelligent Systems 2 ISBN Vol. 91. J.J. Buckley, E. Eslami and T. Feuring Fuz...ry Mathematics in Economics and Engineering, 2002 ISBN Vol. 92. P. P. Angelov Evolving Rule-Based Models, 2002 ISBN

3 Valerie V. Cross Thomas A. Sudkamp Similarity and Compatibility in Fuzzy Set Theory Assessment and Applications With 27 Figures and 46 Tables Springer-Verlag Berlin Heidelberg GmbH

4 Associate Professor Valerie V. Cross University of South Carolina Department of Computer Science and Engineering Columbia, SC USA Professor Thomas A. Sudkamp Wright State University Department of Computer Science 3640 Colonel Glenn Hwy Dayton, OH USA ISSN ISBN ISBN (ebook) DOI / Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Cross, Valerie V: Similarity and compatibility in fuzzy set theory: assessment and applications; with 46 tables I Valerie V. Cross; Thomas A. Sudkamp. - Heidelberg; New York: Physica-VerL, 2002 (Studies in fuzziness and soft computing; VoL 93) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Physica-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 2002 Originally published by Physica-Verlag Heidelberg in 2002 Softcover reprint ofthe hardcover 1 st edition 2002 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Hardcover Design: Erich Kirchner, Heidelberg

5 Dedication Valerie Cross: To Bill, Greg, Dan, and Doug, for their patience and love. Thomas Sudkamp: To my wife Janice and my daughter Beth-thanks, simply for being my wife and daughter.

6 Preface The topic of this book is the fundamental role of the assessment of compatibility in applications of fuzzy set theory and approximate reasoning. The term compatibility is used to encompass various types of comparisons frequently made between objects or concepts. These relationships include similarity, inclusion, proximity, and the degree of matching. The quantitative analysis of these types of relationships is an integral component of inference in automated classification, diagnosis, and analogical reasoning systems. We begin with a historical overview of techniques for similarity assessment developecl and employed in taxonomy and psychology. While the objects considered in these disciplines were described by crisp sets of attributes, the techniques developed provided the foundation for much of the subsequent work in assessing compatibility when fuzzy sets are used to define properties and attribute values. The generalization from crisp to fuzzy descriptions permits the representation of vagueness, ambiguity, and imprecision in the definitions of objects and concepts. One objective of this book is to provide a link between the early research in taxonomic and psychological assessment of compatibility and strategies being developed in fuzzy set theory and approximate reasoning. A framework for classifying measures that assess the compatibility of fuzzy sets is presented. Compatibility measures are divided into three classes, set-theoretic measures, proximity-based measures, and logic-based measures, based on the underlying operations used to produce the measure. Within each class, the methods used for generating measures are introduced and the relationships among the various measures are examined. It is our hope that the reader will benefit from this book by gaining an overall understanding of similarity and compatibility assessment. The theoretical examination of compatibility and the experimental studies on the interaction of compatibility analysis and support aggregation have been undertaken to identify properties of compatibility measures that will assist researchers and system designers in making the selection of a compatibility measure that is appropriate for the unique characteristics of their particular application. Finally, we hope that the presentation of the variety of methods used for assessing relationships such as similarity, inclusion, and partial matching

7 VIn Preface illustrates the complexity and the versatility of these concepts. Zadeh recognized the importance of versatility in approximate reasoning when noting "that by avoiding a commitment to fixed, context-independent definitions, fuzzy set theory and fuzzy logic achieve a pluralism which enhances their flexibility and expressive capabililty." The analysis and measurement of compatibility exemplifies the flexibility available in system design using fuzzy set theory for information representation and analysis. Valerie V. Cross Thomas A. Sudkamp

8 Contents 1. Introduction... 1 Part I. Similarity, Compatibility, and Fuzzy Set Theory 2. The Nature of Similarity Dissimilarity, an Opposite of Similarity? Is Similarity Symmetric? Multidimensional vs. Multi-Attribute Is Similarity Relative? Historic Assessment of Compatibility Taxonomy Psychology Statistical Similarity Foundations of Fuzzy Set Theory Representation and Properties of Fuzzy Sets Fuzzy Set Operators Aggregation Operators Fuzzy Set Theory and Infinite-Valued Logic Fuzzy Relations Measuring Uncertainty Possibility Theory Compatibility in Fuzzy Inference Compositional Rule ofinference Compatibility-Modification Inference Interpolative and Analogical Inference Compatibility in Approximate Reasoning Fuzzy Expert Systems Fuzzy Logic Control Information Retrieval Fuzzy Relational Databases

9 X Contents ~otation and History Relational Algebra Extensions Ranking Fuzzy Numbers Similarity Assessment Experiments Part II. Taxonomy of Compatibility Measures 7. Set-Theoretic Measures Inclusion Indices Requirements Ordering of Inclusion Indices Reflexivity, Transitivity, and Nesting Partial :\Iatching Indices Requirements Ordering of Partial Matching Indices Ordering Between I U / g and Pn/ g Reflexivity, Transitivity, and Nesting Similarity Indices Symmetric Difference Similarity Measure Generation Reflexivity, Transitivity, and Nesting Ordering Within Classes of Similarity Indices Ordering Between Classes of Similarity Indices S 7.4 Ordering Between Set-Theoretic Classes Proximity-Based Measures Notation and Terminology Minkowski Compatibility Measures..., Metrics from Symmetric Difference Ordering of Minkowski Measures Angular Coefficients as Compatibility Interval-Based Compatibility Measures... los Ordering of Interval-Based Measures Relative Distances Linguistic Approximation Distance Measures Logic-Based Measures Fuzzy Truth Values and Compatibility Similarity Relations from Co-Implication Ordering of Logic-Based Measures Fuzzy-Valued Similarity Measures

10 Contents XI Part III. Empirical Analysis of Compatibility Measures 11. Generic Classification Domain Overview Domain and Evidential Knowledge Representation Testing Methodology Set-Theoretic Comparative Study T3 Aggregator Tl Aggregator T2 Aggregator Modified Mean Aggregator Summary of Set-Theoretic Aggregator Study Proximity-Based Comparative Study T3 Aggregator Gl,m Aggregator T2 Aggregator Logic-Based Comparative Study T3 Aggregator Gl,m Aggregator T2 Aggregator Comparison Among the Three Classes Correlated Domain Knowledge Index of Notation References Index