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