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Bibliografická citace

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0 (hodnocen0 x )
BK
1st pub.
Cambridge : Cambridge University, 2009
xvii, 460 s. : il. ; 24 cm

objednat
ISBN 978-0-521-51926-7 (váz.)
Encyclopedia of mathematics and its applications ; 127
Obsahuje bibliografické odkazy a rejstřík
000186981
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 readable book provides 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. // Michel Grabisch is a Professor of computer Sciences at universitŕ Paris I, Pantheon-Sorbonne. // Jean-Luc Marichal is an Associate Professor in the Mathematics Research unitatthe university of Luxembourg. // Radko Mesiar is chairman of the Department of Mathematics and Descriptive Geometry at the Slovak university of Technology, Bratislava. // Endre Pap is a Professor in
the Department of Mathematics and informatics at the university of Novi Sad, Serbia. // Encyclopedia of Mathematics and its Applications // This series is devoted to significanttopics orthemes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. // Books in the Encyclopedia of Mathematics and its Applications cover theirsubjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects. // For a list of titles in the series see p. ii. // Cambridge // UNIVERSITY PRESS www.cambridge.org // ISBN 978-0-521-51926-7 // List of figures page x // List of tables xii // Preface xiii // 1 Introduction 1 // 1.1 Main motivations and scope 1 // 1.2 Basic definitions and examples 2 // 1.3 Conventional notation 9 // 2 Properties for aggregation 11 // 2.1 Introduction 11 // 2.2 Elementary mathematical properties 12 // 2.3 Grouping-based properties 31 // 2.4 Invariance properties 41 // 2.5 Further properties 49 // 3 Conjunctive and disjunctive aggregation functions 56 // 3.1 Preliminaries and general notes 56 // 3.2 Generated conjunctive aggregation functions 59 // 3.3 Triangular norms and related conjunctive aggregation functions 64 // 3.4 Copulas and quasi-copulas 88 // 3.5 Disjunctive aggregation functions 100 // 3.6 Uninorms 106 // 3.7 Nullnorms 115 // 3.8 More aggregation functions related to t-norms 119 // 3.9 Restricted distributivity 123 // 4 Means and averages 130 // 4.1 Introduction and definitions 130 // 4.2 Quasi-arithmetic means 132 // vii // Contents // viii // 4.3 Generalizations of quasi-arithmetic means 139 //
4.4 Associative means 161 // 4.5 Means constructed from a mean value property 163 // 4.6 Constructing means 166 // 4.7 Further extended means 168 // 5 Aggregation functions based on nonadditive integrals 171 // 5.1 Introduction 171 // 5.2 Set functions, capacities, and games 172 // 5.3 Some linear transformations of set functions 177 // 5.4 The Choquet integral 181 // 5.5 The Sugeno integral 207 // 5.6 Other integrals 227 // 6 Construction methods 234 // 6.1 Introduction 234 // 6.2 Transformed aggregation functions 234 // 6.3 Composed aggregation 242 // 6.4 Weighted aggregation functions 247 // 6.5 Some other aggregation-based construction methods 252 // 6.6 Aggregation functions based on minimal dissimilarity 257 // 6.7 Ordinal sums of aggregation functions 261 // 6.8 Extensions to aggregation functions 266 // 7 Aggregation on specific scale types 272 // 7.1 Introduction 272 // 7.2 Ratio scales 273 // 7.3 Difference scales 280 // 7.4 Interval scales 284 // 7.5 Log-ratio scales 289 // 8 Aggregation on ordinal scales 292 // 8.1 Introduction 292 // 8.2 Order invariant subsets 293 // 8.3 Lattice polynomial functions and some of their properties 296 // 8.4 Ordinal scale invariant functions 300 // 8.5 Comparison meaningful functions on a single ordinal scale 304 // 8.6 Comparison meaningful functions on independent ordinal scales 308 // 8.7 Aggregation on finite chains by chain independent functions 310 // 9 Aggregation on bipolar scales 317 // 9.1 Introduction 317 // 9.2 Associative bipolar operators 319 // Contents // ix // 9.3 Minimum and maximum on symmetrized linearly ordered sets 325 // 9.4 Separable aggregation functions 332 // 9.5 Integral-based aggregation functions 334 // 10 Behavioral analysis of aggregation functions 348 // 10.1 Introduction 348 // 10.2 Expected values and distribution functions 348 // 10.3 Importance indices 361 // 10.4 Interaction indices 367 // 10.5 Maximum improving index 370 //
10.6 Tolerance indices 372 // 10.7 Measures of arguments contribution and involvement 378 // 11 Identification of aggregation functions 382 // 11.1 Introduction 382 // 11.2 General formulation 383 // 11.3 The case of parametrized families of aggregation functions 386 // 11.4 The case of generated aggregation functions 388 // 11.5 The case of integral-based aggregation functions 391 // 11.6 Available software 396 // Appendix A: Aggregation of infinitely many arguments 397 // A.l Introduction 397 // A.2 Infinitary aggregation functions on sequences 397 // A. 3 General aggregation of infinite number of inputs 405 // Appendix B: Examples and applications 410 // B. l Main domains of applications 410 // B.2 A specific application: mixture of uncertainty measures 414 // List of symbols 420 // References 428 // Index 454

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