Aggregation functions.

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  • Author(s): Grabisch, Michel (F-PARIS1) Grabisch, Michel (F-PARIS1) AMS Author Profile; Marichal, Jean-Luc (LUX-CUL) Marichal, Jean-Luc (LUX-CUL) AMS Author Profile; Mesiar, Radko (SK-STU) Mesiar, Radko (SK-STU) AMS Author Profile; Pap, Endre (SE-NOVI) Pap, Endre (SE-NOVI) AMS Author Profile
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    Encyclopedia of Mathematics and its Applications, 127.
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    Cambridge University Press, Cambridge
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    • Authors' Affiliation:
      (F-PARIS1) Département de Mathématiques, Université de Paris I (Panthéon-Sorbonne)
      (LUX-CUL) Séminaire de Mathématiques, Centre Universitaire de Luxembourg
      (SK-STU) Department of Mathematics, Slovak Technical University (SVŠT)
      (SE-NOVI) Department of Mathematics, University of Novi Sad
    • Abstract:
      This book is intended to be a guide for beginners in the field of aggregation operators. An aggregation operator is a way to merge some data into new data. Thus defined, aggregation operators apply to many different fields, and aggregation operators may adopt many different forms. \par The book treats the main families of aggregation operators, each of them in an independent way; even more interesting, it also studies the relationship among them and treats all types of aggregation operators as a whole. \par For each family of aggregation operators, possible extensions of their properties are given, introducing the reader to the different problems that are being studied nowadays. \par The definition of aggregation operator and some typical examples are given in Chapter 1. \par Chapter 2 is devoted to the different properties that are usually required for aggregation operators. Of course, as aggregation operators can be applied to many different domains, the properties may vary. Moreover, they could be opposite; for example, t-norms always attain a value smaller than the minimum (they are conjunctive), t-conorms always attain a value greater than the maximum (they are disjunctive) and means attain a value between the minimum and the maximum (they are internal). \par The rest of the book can be divided into several parts. Chapters 3, 4 and 5 provide a description of the main groups of aggregation operators and the basis for getting started with aggregation operators. Chapters 6, 7, 8, 9, 10 and Appendix A focus on more technical aspects of the theory. Chapter 11 and Appendix B cover the practical use of aggregation operators. \par Chapter 3 studies conjunctive and disjunctive aggregation operators. Usually, these operators are dual, in the sense that given a conjunctive operator, there is another disjunctive operator related to it. The most typical example of conjunctive (resp. disjunctive) operator is the t-norm (resp. t-conorm). These operators are treated in detail in the chapter. From these operators, although appearing in an independent way, we have copulas and their extension, quasi-copulas. The chapter also treats other interesting operators related to t-norms and t-conorms, namely nullnorms and uninorms. \par Chapter 4 is devoted to means, the other big family of aggregation operators. In this chapter, the results related to means (extensions, constructions, and so on) are stated. \par Chapter 5 treats a special case of mean operators, the so-called fuzzy integrals, that are based on non-additive measures and are probably the most important case of mean operators. For this reason, these operators have been studied in depth by many authors. Special attention is paid to Choquet and Sugeno integrals. \par Chapter 6 treats the different ways to build aggregation operators from existing ones. In particular, this chapter treats the problem of building an aggregation operator from another one but with better properties. \par Chapters 7, 8 and 9 treat the problem of which aggregation operators can be used when particular scales are used. For example, if the values are in an ordinal scale, we cannot use an aggregation operator requiring sums or products. Chapter 7 treats this problem in a general way. It also treats transformations of scales that are meaningful. Chapter 8 treats the case of ordinal scales, a typical situation in practice; in particular, it treats lattice polynomials. Chapter 9 treats bipolar scales, another case appearing in many practical problems; bipolar scales focus on the fact that it could be the case that there are differences when treating small inputs and big inputs and the ways to merge these two ways of acting in the aggregation operator. \par Chapter 10 can be divided into two parts: first, it treats different measures of aggregation operators---for example, ways to measure the extent to which an aggregation operator behaves similarly to the minimum. Next, it covers the different indices to measure some important aspects of the aggregation operator---for example, how to study whether two coordinates interact. \par Chapter 11 treats the problem of identification of aggregation operators. As aggregation operators appear in many different fields, the information that is provided in each domain may be different, too. Consequently, there are many different techniques to identify the corresponding aggregation operator. Besides, for a fixed domain, different families may apply and thus the problem may become more or less easy to solve. All these aspects are treated in the chapter. \par For aggregation operators, it is assumed that the cardinal of inputs to aggregate is fixed. Throughout the book, the authors treat the notion of extended aggregation function when this cardinality is not fixed. In Appendix A the authors treat the problem of aggregating infinitely many arguments. In order to solve the problem, extended aggregation operators play a prominent role. \par Finally, Appendix B gives a list of domains in which aggregation operators can be applied, with a concrete application. \par Although the book seems to focus on the mathematical properties of aggregation operators, it also provides a very complete bibliography where applications and more details can be found. \par Briefly speaking, I find the book a complete and interesting guide for those who are interested in aggregation operators and those who want to learn more about a specific family of aggregation operators or need a guide to understand the main aspects of different operators.
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  • Citations
    • ABNT:
      GRABISCH, M. et al. Aggregation functions. [s.l.] : Cambridge University Press, Cambridge, 2009. Disponível em: . Acesso em: 15 set. 2019.
    • AMA:
      Grabisch M, Marichal J-L, Mesiar R, Pap E. Aggregation Functions. (Miranda P, ed.). Cambridge University Press, Cambridge; 2009. Accessed September 15, 2019.
    • APA:
      Grabisch, M., Marichal, J.-L., Mesiar, R., & Pap, E. (2009). Aggregation functions. (P. Miranda, Ed.). Cambridge University Press, Cambridge. Retrieved from
    • Chicago/Turabian: Author-Date:
      Grabisch, Michel, Jean-Luc Marichal, Radko Mesiar, and Endre Pap. 2009. Aggregation Functions. Edited by Pedro Miranda. Encyclopedia of Mathematics and Its Applications, 127. Cambridge University Press, Cambridge.
    • Harvard:
      Grabisch, M. et al. (2009) Aggregation functions. Edited by P. Miranda. Cambridge University Press, Cambridge (Encyclopedia of Mathematics and its Applications, 127). Available at: (Accessed: 15 September 2019).
    • Harvard: Australian:
      Grabisch, M, Marichal, J-L, Mesiar, R & Pap, E 2009, Aggregation functions, P Miranda (ed.), Encyclopedia of Mathematics and its Applications, 127, Cambridge University Press, Cambridge, viewed 15 September 2019, .
    • MLA:
      Grabisch, Michel, et al. Aggregation Functions. Edited by Pedro Miranda, Cambridge University Press, Cambridge, 2009. EBSCOhost,
    • Chicago/Turabian: Humanities:
      Grabisch, Michel, Jean-Luc Marichal, Radko Mesiar, and Endre Pap. Aggregation Functions. Edited by Pedro Miranda. Encyclopedia of Mathematics and Its Applications, 127. Cambridge University Press, Cambridge, 2009.
    • Vancouver/ICMJE:
      Grabisch M, Marichal J-L, Mesiar R, Pap E. Aggregation functions [Internet]. Miranda P, editor. Cambridge University Press, Cambridge; 2009 [cited 2019 Sep 15]. (Encyclopedia of Mathematics and its Applications, 127). Available from: