Similarity and Compatibility in Fuzzy Set Theory:
Assessment and Applications
Valerie Cross and Thomas Sudkamp
Studies in Fuzziness and Soft Computing,
Number 93,
Physica-Verlag, 2002
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:
Chapter
1. Introduction
Part
I Similarity,
Compatibility, and Fuzzy Set Theory
Chapter
2. The Nature of
Similarity
2.1 Dissimilarity, an opposite of
similarity
2.2
Is similarity symmetric?
2.3
Multidimensional vs. multi-attribute
2.4 Is similarity relative?
Chapter
3. Historic assessment
of compatibility
3.1 Taxonomy
3.2 Psychology
3.3
Statistical similarity
Chapter
4. Foundations of
Fuzzy Set Theory
4.1 Representation and properties of
fuzzy sets
4.2 Fuzzy set operators
4.3 Aggregation
4.4 Fuzzy logic as infinite-valued
logic
4.5 Fuzzy relations
4.6 Measuring uncertainty
Chapter
5. Fuzzy Set Theory
in Approximate Reasoning
5.1 Compositional rule of inference
5.2 Compatibility-modification
inference
5.3 Fuzzy analogical and
interpolation inference
Chapter
6. Applications
of Compatibility Measures
6.1 Fuzzy expert systems
6.2 Fuzzy logic control
6.3 Information retrieval
6.4 Fuzzy relational databases
6.4.1 Notation and history
6.4.2 Relational algebra extensions
6.5 Ranking fuzzy numbers
6.6 Similarity assessment
experiments
Part
II Taxonomy of Compatibility Measures
Chapter
7. Set-Theoretic Measures
7.1 Inclusion indices
7.1.1 Requirements
7.1.2 Ordering of inclusion indices
7.1.3 Reflexivity, transitivity, and nesting
7.2 Partial matching indices
7.2.1 Requirements
7.2.2 Ordering of partial matching indices
7.2.3 Ordering between inclusion and paritial
matching
7.2.4 Reflexivity, transitivity, and nesting
7. Similarity indices
7.3.1 Symmetric difference
7.3.2 Similarity measure generation
7.3.3 Reflexivity, transitivity, and nesting
7.3.4 Ordering within classes of similarity indices
7.3.5 Ordering between classes of similarity indices
7.4 Ordering between set-theoretic
classes
Chapter
8. Proximity
Based Measures
8.1 Notation and terminology
8.2 Minkowski
compatibility measures
8.2.1 Metrics from symmetric difference
8.2.2 Ordering of Minkowski measures
8.3 Angular coefficients as
compatibility
8.4 Interval based compatibility
measures
8.4.1 Ordering of interval measures
8.4.2 Relative distances
8.5 Linguistic approximation
distance measures
Chapter
9. Logic-Based
Measures
9.1 Fuzzy truth values and
compatibility
9.2 Similarity relations from co-implication
9.3 Ordering of logic based measures
Chapter
10. Fuzzy-Valued Similarity Measures
Part
III Empirical Analysis of Compatibility Measures
Chapter
11. Generic
Classification Domain
11.1 Overview
11.2 Domain and evidential knowledge
representation
11.3 Testing methodology
Chapter
12. Set-Theoretic
Comparative Study
12.1 T3 aggregator
12.2 T1 aggregator
12.3 T2 aggregator
12.4 Modified mean aggregator
12.5 Summary of set-theoretic
aggregator study
Chapter
13.
Proximity-Based Comparative Study
13.1 T3 aggregator
13.2 G1,m aggregator
13.3 T2 aggregator
Chapter
14. Logic-Based Comparative
Study
14.1 T3 aggregator
14.2 G1,m aggregator
14.3 T2 aggregator
Chapter
15. Comparison Among the Three Classes
15.1 Correlated domain knowledge
Index of
Notation
References