Similarity and Compatibility in Fuzzy Set Theory

by Cross, Valerie V.; Sudkamp, Thomas A.

• ISBN13: 9783790814583
• ISBN10: 379081458X
• Format: Hardcover
• Copyright: 2002-05-01
• Publisher: Physica Verlag

Summary

Assessing the degree to which two objects, an object and a query, or two concepts are similar or compatible is a fundamental component of human reasoning and consequently is critical in the development of automated diagnosis, classification, information retrieval and decision systems. The assessment of similarity has played an important role in such diverse disciplines such as taxonomy, psychology, and the social sciences. Each discipline has proposed methods for quantifying similarity judgments suitable for its particular applications. This book presents a unified approach to quantifying similarity and compatibility within the framework of fuzzy set theory and examines the primary importance of these concepts in approximate reasoning. Examples of the application of similarity measures in various areas including expert systems, information retrieval, and intelligent database systems are provided.

Table of Contents

Introduction	1	(4)
Part I. Similarity, Compatibility, and Fuzzy Set Theory
The Nature of Similarity	5	(4)
Dissimilarity, an Opposite of Similarity?	5	(1)
Is Similarity Symmetric?	6	(1)
Multidimensional vs. Multi-Attribute	6	(1)
Is Similarity Relative?	7	(2)
Historic Assessment of Compatibility	9	(8)
Taxonomy	9	(2)
Psychology	11	(4)
Statistical Similarity	15	(2)
Foundations of Fuzzy Set Theory	17	(18)
Representation and Properties of Fuzzy Sets	17	(3)
Fuzzy Set Operators	20	(3)
Aggregation Operators	23	(2)
Fuzzy Set Theory and Infinite-Valued Logic	25	(1)
Fuzzy Relations	26	(3)
Measuring Uncertainty	29	(3)
Possibility Theory	32	(3)
Compatibility in Fuzzy Inference	35	(10)
Compositional Rule of Inference	35	(1)
Compatibility-Modification Inference	36	(3)
Interpolative and Analogical Inference	39	(6)
Compatibility in Approximate Reasoning	45	(26)
Fuzzy Expert Systems	45	(5)
Fuzzy Logic Control	50	(1)
Information Retrieval	51	(5)
Fuzzy Relational Databases	56	(5)
Notation and History	56	(1)
Relational Algebra Extensions	57	(4)
Ranking Fuzzy Numbers	61	(3)
Similarity Assessment Experiments	64	(7)
Part II. Taxonomy of Compatibility Measures
Set-Theoretic Measures	71	(26)
Inclusion Indices	73	(8)
Requirements	73	(4)
Ordering of Inclusion Indices	77	(2)
Reflexivity, Transitivity, and Nesting	79	(2)
Partial Matching Indices	81	(6)
Requirements	82	(2)
Ordering of Partial Matching Indices	84	(1)
Ordering Between IU/g and P/g	85	(1)
Reflexivity, Transitivity, and Nesting	85	(2)
Similarity Indices	87	(9)
Symmetric Difference	87	(2)
Similarity Measure Generation	89	(3)
Reflexivity, Transitivity, and Nesting	92	(1)
Ordering Within Classes of Similarity Indices	93	(2)
Ordering Between Classes of Similarity Indices	95	(1)
Ordering Between Set-Theoretic Classes	96	(1)
Proximity-Based Measures	97	(36)
Notation and Terminology	97	(1)
Minkowski Compatibility Measures	98	(4)
Metrics from Symmetric Difference	100	(1)
Ordering of Minkowski Measures	101	(1)
Angular Coefficients as Compatibility	102	(3)
Interval-Based Compatibility Measures	105	(27)
Ordering of Interval-Based Measures	113	(9)
Relative Distances	122	(10)
Linguistic Approximation Distance Measures	132	(1)
Logic-Based Measures	133	(6)
Fuzzy Truth Values and Compatibility	133	(2)
Similarity Relations from Co-Implication	135	(1)
Ordering of Logic-Based Measures	136	(3)
Fuzzy-Valued Similarity Measures	139	(8)
Part III. Empirical Analysis of Compatibility Measures
Generic Classification Domain	147	(6)
Overview	147	(1)
Domain and Evidential Knowledge Representation	148	(2)
Testing Methodology	150	(3)
Set-Theoretic Comparative Study	153	(14)
T3 Aggregator	153	(6)
T1 Aggregator	159	(3)
T2 Aggregator	162	(2)
Modified Mean Aggregator	164	(1)
Summary of Set-Theoretic Aggregator Study	165	(2)
Proximity-Based Comparative Study	167	(10)
T3 Aggregator	168	(2)
G1,m Aggregator	170	(3)
T2 Aggregator	173	(4)
Logic-Based Comparative Study	177	(6)
T3 Aggregator	177	(2)
G1,m Aggregator	179	(1)
T2 Aggregator	180	(3)
Comparison Among the Three Classes	183	(6)
Correlated Domain Knowledge	185	(4)
Index of Notation	189	(4)
References	193	(14)
Index	207