Similarity and Compatibility in Fuzzy Set Theory

Similarity and Compatibility in Fuzzy Set Theory

Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw, Poland E-mail: kacprzyk@ibspan.waw.pl http://www.springer.de/cgi-binlsearch_book.pl?series = 2941 Further volumes of this series can be found at our homepage. Vol. 72. M. Mares Fuzzy Cooperative Games. 2001 ISBN 3-7908-1392-3 Vol. 73. Y. Yoshida (Ed.) Dynamical Aspects in Fuzzy Decision, 2001 ISBN 3-7908-1397-4 Vol. 74. H.-N. Teodorescu, L.C. Jain and A. Kandel (Eds.) Hardware Implementation of Intelligent Systems, 2001 ISBN 3-7908-1399-0 Vol. 75. V. Loia and S. Sessa (Eds.) Soft Computing Agents, 2001 ISBN 3-7908-1404-0 Vol. 76. D. Ruan, J. Kacprzyk and M. Fedrizzi (Eds.) Soft Computing for Risk Evaluation and Management, 2001 ISBN 3-7908-1406-7 Vol. 77. W. Liu Propositional, Probabilistic and Evidential Reasoning, 2001 ISBN 3-7908-1414-8 Vol. 78. U. Seiffert and L. C. Jain (Eds.) Self-Organizing Neural Networks, 2002 ISBN 3-7908-1417-2 Vol. 79. A. Osyczka Evolutionary Algorithms for Single and Multicriteria Design Optimization, 2002 ISBN 3-7908-1418-0 Vol. 80. P. Wong, F. Aminzadeh and M. Nikravesh (Eds.) Soft Computing for Reservoir Characterization and Modeling, 2002 ISBN 3-7908-1421-0 Vol. 81. V. Dimitrov and V. Korotkich (Eds.) Fuzzy Logic, 2002 ISBN 3-7908-1425-3 Vol. 82. Ch. Carlsson and R. Fuller Fuz...ry Reasoning in Decision Making and Optimization, 2002 ISBN 3-7908-1428-8 Vol. 83. S. Barro and R. Marin (Eds.) Fuzzy Logic in Medicine, 2002 ISBN 3-7908-1429-6 Vol. 84. L.c. Jain and J. Kacprzyk (Eds.) New Learning Paradigms in Soft Computing, 2002 ISBN 3-7908-1436-9 Vol. 85. D. Rutkowska Neuro-Fuzzy Architectures and Hybrid Learning, 2002 ISBN 3-7908-1438-5 Vol. 86. M.B. Gorzalczany Computational Intelligence Systems and Applications, 2002 ISBN 3-7908-1439-3 Vol. 87. C. Bertoluzza, M.A. Gil and D.A. Ralescu (Eds.) Statistical Modeling, Analysis and Management of Fuz...ry Data, 2002 ISBN 3-7908-1440-7 Vol. 88. R.P. Srivastava and T.J. Mock (Eds.) Belief Functions in Business Decisions, 2002 ISBN 3-7908-1451-2 Vol. 89. B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena and R.R. Yager (Eds.) Technologies for Constructing Intelligent Systems I ISBN 3-7908-1454-7 Vol. 90. B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena and R.R. Yager (Eds.) Technologies for Constructing Intelligent Systems 2 ISBN 3-7908-1455-5 Vol. 91. J.J. Buckley, E. Eslami and T. Feuring Fuz...ry Mathematics in Economics and Engineering, 2002 ISBN 3-7908-1456-3 Vol. 92. P. P. Angelov Evolving Rule-Based Models, 2002 ISBN 3-7908-1457-1

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

Associate Professor Valerie V. Cross University of South Carolina Department of Computer Science and Engineering Columbia, SC 29208 USA vcross@engr.sc.edu Professor Thomas A. Sudkamp Wright State University Department of Computer Science 3640 Colonel Glenn Hwy Dayton, OH 45435-0001 USA tsudkamp@cs.wright.edu ISSN 1434-9922 ISBN 978-3-7908-2507-7 ISBN 978-3-7908-1793-5 (ebook) DOI 10.1007/978-3-7908-1793-5 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

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.

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

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

Contents 1. Introduction... 1 Part I. Similarity, Compatibility, and Fuzzy Set Theory 2. The Nature of Similarity... 5 2.1 Dissimilarity, an Opposite of Similarity?................... 5 2.2 Is Similarity Symmetric?... 6 2.3 Multidimensional vs. Multi-Attribute... 6 2.4 Is Similarity Relative?................................... 7 3. Historic Assessment of Compatibility..................... 9 3.1 Taxonomy... 9 3.2 Psychology... 11 3.3 Statistical Similarity.................................... 15 4. Foundations of Fuzzy Set Theory......................... 17 4.1 Representation and Properties of Fuzzy Sets............... 17 4.2 Fuzzy Set Operators.................................... 20 4.3 Aggregation Operators.................................. 23 4.4 Fuzzy Set Theory and Infinite-Valued Logic................ 25 4.5 Fuzzy Relations........................................ 26 4.6 Measuring Uncertainty.................................. 29 4.7 Possibility Theory...................................... 32 5. Compatibility in Fuzzy Inference... 35 5.1 Compositional Rule ofinference.......................... 35 5.2 Compatibility-Modification Inference...................... 36 5.3 Interpolative and Analogical Inference..................... 39 6. Compatibility in Approximate Reasoning................. 45 6.1 Fuzzy Expert Systems... 45 6.2 Fuzzy Logic Control.................................. 50 6.3 Information Retrieval................................... 51 6.4 Fuzzy Relational Databases.............................. 56

X Contents 6.4.1 ~otation and History............................. 56 6.4.2 Relational Algebra Extensions... 57 6.5 Ranking Fuzzy Numbers... 61 6.6 Similarity Assessment Experiments....................... 64 Part II. Taxonomy of Compatibility Measures 7. Set-Theoretic Measures................................... 71 7.1 Inclusion Indices... 73 7.1.1 Requirements... 73 7.1.2 Ordering of Inclusion Indices....................... 77 7.1.3 Reflexivity, Transitivity, and Nesting................ 79 7.2 Partial :\Iatching Indices... 81 7.2.1 Requirements... 82 7.2.2 Ordering of Partial Matching Indices................ 84 7.2.3 Ordering Between I U / g and Pn/ g...... 85 7.2.4 Reflexivity, Transitivity, and Nesting................ 85 7.3 Similarity Indices....................................... 87 7.3.1 Symmetric Difference............................. 87 7.3.2 Similarity Measure Generation..................... 89 7.3.3 Reflexivity, Transitivity, and Nesting................ 92 7.3.4 Ordering Within Classes of Similarity Indices........ 93 7.3.5 Ordering Between Classes of Similarity Indices....... 9S 7.4 Ordering Between Set-Theoretic Classes................... 96 8. Proximity-Based Measures... 97 8.1 Notation and Terminology............................... 97 8.2 Minkowski Compatibility Measures..., 98 8.2.1 Metrics from Symmetric Difference... 100 8.2.2 Ordering of Minkowski Measures... 101 8.3 Angular Coefficients as Compatibility... 102 8.4 Interval-Based Compatibility Measures... los 8.4.1 Ordering of Interval-Based Measures... 113 8.4.2 Relative Distances... 122 8.5 Linguistic Approximation Distance Measures... 132 9. Logic-Based Measures... 133 9.1 Fuzzy Truth Values and Compatibility... 133 9.2 Similarity Relations from Co-Implication... 135 9.3 Ordering of Logic-Based Measures... 136 10. Fuzzy-Valued Similarity Measures... 139

Contents XI Part III. Empirical Analysis of Compatibility Measures 11. Generic Classification Domain............................ 147 11.1 Overview... 147 11.2 Domain and Evidential Knowledge Representation... 148 11.3 Testing Methodology.................................... 150 12. Set-Theoretic Comparative Study... 153 12.1 T3 Aggregator... 153 12.2 Tl Aggregator... 159 12.3 T2 Aggregator... 162 12.4 Modified Mean Aggregator... 164 12.5 Summary of Set-Theoretic Aggregator Study... 165 13. Proximity-Based Comparative Study... 167 13.1 T3 Aggregator... 168 13.2 Gl,m Aggregator... 170 13.3 T2 Aggregator.......................................... 173 14. Logic-Based Comparative Study... 177 14.1 T3 Aggregator... 177 14.2 Gl,m Aggregator... 179 14.3 T2 Aggregator... 180 15. Comparison Among the Three Classes... 183 15.1 Correlated Domain Knowledge... 185 Index of Notation... 189 References.................................................... 193 Index... 207