Table Of ContentSimilarity and Compatibility
in Fuzzy Set Theory
Studies in Fuzziness and Soft Computing
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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
[email protected]
Professor Thomas A. Sudkamp
Wright State University
Department of Computer Science
3640 Colonel Glenn Hwy
Dayton, OH 45435-0001
USA
[email protected]
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,
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tions 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 1s t 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 compati
bility 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, in
clusion, proximity, and the degree of matching. The quantitative analysis of
these types of relationships is an integral component of inference in auto
mated classification, diagnosis, and analogical reasoning systems.
We begin with a historical overview of techniques for similarity assess
ment 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 proper
ties 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 ap
proximate 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 the
oretical examination of compatibility and the experimental studies on the
interaction of compatibility analysis and support aggregation have been un
dertaken to identify properties of compatibility measures that will assist re
searchers 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 rec
ognized 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 com
patibility 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 IU/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
