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Aggregation functions. (English) Zbl 1196.00002

Encyclopedia of Mathematics and its Applications 127. Cambridge: Cambridge University Press (ISBN 978-0-521-51926-7/hbk). xvii, 460 p. (2009).
This is a carefully written book which is very welcome. The best choice for the reviewer is to cite from the preface of the book: ”The process of combining several numerical values into a single representative one is called aggregation, and the numerical function performing this process is called an aggregation function. This simple definition demonstrates the size of the field of application of aggregation: applied mathematics (e.g. probability, statistics, decision theory), computer sciences (e.g. artificial intelligence, operations research), as well as many applied fields (economics and finance, pattern recognition and image processing, data fusion, multicriteria decision aid, automated reasoning, etc.). ... The rapid growth of the above mentioned application fields, largely due to the arrival of computers, has made necessary the establishment of a sound theoretical basis for aggregation functions. Hence, since the 1980s, aggregation functions have become a genuine research field, rapidly developing, but in a rather scattered way since aggregation functions are rooted in many different fields. ... Yet for the researcher as well as for the practitioner, this abundance of literature, because it is scattered in many domains, is more a handicap than an advantage, and there is a real lack of a unified and complete view of aggregation functions, where one could find the most important concepts and results presented in a clear and rigorous way. This book has been written with the intention of filling this gap: it offers a full, comprehensive, rigorous, and unified treatment of aggregation functions. Our main motivation has been to bring a unified viewpoint of the aggregation problem, and to provide an abstract mathematical presentation and analysis ... and also a unified teminology and notation. ... The book is intended primarily for researchers and graduate students in applied mathematics and computer sciences, secondarily for practitioners ... The reader of the book is assumed to have the basic knowledge of a graduate student in algebra and analysis. ... The main theoretical corpus is given in Chapters 2 to 5. Additional theoretical material is given in Chapters 6 to 9, while Chapters 10 and 11 are more practically oriented. In Chapters 2 to 5, as far as possible, most of the results are given with proof. ...” Contents:
1. Introduction
2. Properties for aggregation
3. Conjunctive and disjunctive aggregation functions (e.g. t-norms, copulas)
4. Means and averages
5. Aggregation functions based on nonadditive integrals (e.g. Choquet integral, Sugeno integral)
6. Construction methods
7. Aggregation on specific scale typs
8. Aggregation on ordinal scales
9. Aggregation on bipolar scales
10. Behavioral analysisi of aggregation functions
11. Identification of aggregation functions

MSC:

00A05 Mathematics in general
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
26-02 Research exposition (monographs, survey articles) pertaining to real functions
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
28E10 Fuzzy measure theory
91B06 Decision theory
91B14 Social choice
91B82 Statistical methods; economic indices and measures
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