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Aggregation Functions

Book Description

Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating information is important: applied and pure mathematics (probability, statistics, decision theory, functional equations), operations research, computer science, and many applied fields (economics and finance, pattern recognition and image processing, data fusion, etc.). This is a comprehensive, rigorous and self-contained exposition of aggregation functions. Classes of aggregation functions covered include triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals. The properties of each method, as well as their interpretation and analysis, are studied in depth, together with construction methods and practical identification methods. Special attention is given to the nature of scales on which values to be aggregated are defined (ordinal, interval, ratio, bipolar). It is an ideal introduction for graduate students and a unique resource for researchers.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. List of figures
  8. List of tables
  9. Preface
  10. 1. Introduction
    1. 1.1 Main motivations and scope
    2. 1.2 Basic definitions and examples
    3. 1.3 Conventional notation
  11. 2. Properties for aggregation
    1. 2.1 Introduction
    2. 2.2 Elementary mathematical properties
    3. 2.3 Grouping-based properties
    4. 2.4 Invariance properties
    5. 2.5 Further properties
  12. 3. Conjunctive and disjunctive aggregation functions
    1. 3.1 Preliminaries and general notes
    2. 3.2 Generated conjunctive aggregation functions
    3. 3.3 Triangular norms and related conjunctive aggregation functions
    4. 3.4 Copulas and quasi-copulas
    5. 3.5 Disjunctive aggregation functions
    6. 3.6 Uninorms
    7. 3.7 Nullnorms
    8. 3.8 More aggregation functions related to t-norms
    9. 3.9 Restricted distributivity
  13. 4. Means and averages
    1. 4.1 Introduction and definitions
    2. 4.2 Quasi-arithmetic means
    3. 4.3 Generalizations of quasi-arithmetic means
    4. 4.4 Associative means
    5. 4.5 Means constructed from a mean value property
    6. 4.6 Constructing means
    7. 4.7 Further extended means
  14. 5. Aggregation functions based on nonadditive integrals
    1. 5.1 Introduction
    2. 5.2 Set functions, capacities, and games
    3. 5.3 Some linear transformations of set functions
    4. 5.4 The Choquet integral
    5. 5.5 The Sugeno integral
    6. 5.6 Other integrals
  15. 6. Construction methods
    1. 6.1 Introduction
    2. 6.2 Transformed aggregation functions
    3. 6.3 Composed aggregation
    4. 6.4 Weighted aggregation functions
    5. 6.5 Some other aggregation-based construction methods
    6. 6.6 Aggregation functions based on minimal dissimilarity
    7. 6.7 Ordinal sums of aggregation functions
    8. 6.8 Extensions to aggregation functions
  16. 7. Aggregation on specific scale types
    1. 7.1 Introduction
    2. 7.2 Ratio scales
    3. 7.3 Difference scales
    4. 7.4 Interval scales
    5. 7.5 Log-ratio scales
  17. 8. Aggregation on ordinal scales
    1. 8.1 Introduction
    2. 8.2 Order invariant subsets
    3. 8.3 Lattice polynomial functions and some of their properties
    4. 8.4 Ordinal scale invariant functions
    5. 8.5 Comparison meaningful functions on a single ordinal scale
    6. 8.6 Comparison meaningful functions on independent ordinal scales
    7. 8.7 Aggregation on finite chains by chain independent functions
  18. 9. Aggregation on bipolar scales
    1. 9.1 Introduction
    2. 9.2 Associative bipolar operators
    3. 9.3 Minimum and maximum on symmetrized linearly ordered sets
    4. 9.4 Separable aggregation functions
    5. 9.5 Integral-based aggregation functions
  19. 10. Behavioral analysis of aggregation functions
    1. 10.1 Introduction
    2. 10.2 Expected values and distribution functions
    3. 10.3 Importance indices
    4. 10.4 Interaction indices
    5. 10.5 Maximum improving index
    6. 10.6 Tolerance indices
    7. 10.7 Measures of arguments contribution and involvement
  20. 11. Identification of aggregation functions
    1. 11.1 Introduction
    2. 11.2 General formulation
    3. 11.3 The case of parametrized families of aggregation functions
    4. 11.4 The case of generated aggregation functions
    5. 11.5 The case of integral-based aggregation functions
    6. 11.6 Available software
  21. AppendixA: Aggregation of infinitely many arguments
    1. A.1 Introduction
    2. A.2 Infinitary aggregation functions on sequences
    3. A.3 General aggregation of infinite number of inputs
  22. Appendix B: Examples and applications
    1. B.1 Main domains of applications
    2. B.2 A specific application: mixture of uncertainty measures
  23. List of symbols
  24. References
  25. Index