Aggregation functions: Means

This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).

[1]  Jan Hidders,et al.  On the expressibility of functions in XQuery fragments , 2008, Inf. Syst..

[2]  Jean-Luc Marichal,et al.  On Sugeno integral as an aggregation function , 2000, Fuzzy Sets Syst..

[3]  H. Imaoka.  A proposal of opposite-Sugeno integral and a uniform expression of fuzzy integrals , 1995, Proceedings of 1995 IEEE International Conference on Fuzzy Systems..

[4]  S. Weber.  ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .

[5]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[6]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[7]  R. Mesiar,et al.  Aggregation Functions (Encyclopedia of Mathematics and its Applications) , 2009 .

[8]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[9]  S. Yau.  Mathematics and its applications , 2002 .

[10]  H. Weisberg,et al.  The Distribution of Linear Combinations of Order Statistics from the Uniform Distribution , 1971 .

[11]  Idriss Ibn Amar.  Dominique PICARD (1998), Politesse, savoir-vivre et relations sociales. Paris, Presses Universitaires de France, Coll. « Que sais-je ? » , 2000 .

[12]  Ronald R. Yager,et al.  The power average operator , 2001, IEEE Trans. Syst. Man Cybern. Part A.

[13]  J. Fodor,et al.  Characterization of some aggregation functions arising from MCDM problems , 1994 .

[14]  Ronald R. Yager,et al.  On the dispersion measure of OWA operators , 2009, Inf. Sci..

[15]  L. R. Berrone,et al.  Lagrangian means , 1998 .

[16]  Jet Wimp.  Quadrature with Generalized Means , 1986 .

[17]  Rita Almeida Ribeiro,et al.  Aggregation with generalized mixture operators using weighting functions , 2003, Fuzzy Sets Syst..

[18]  Janusz Matkowski,et al.  On Invariant Generalized Beckenbach-Gini Means , 2002 .

[19]  E. Czogala,et al.  Associative monotonic operations in fuzzy set theory , 1984 .

[20]  Feng Qi,et al.  A note on monotonicity for generalized weighted mean values , 2004 .

[21]  Feng Qi,et al.  New proofs of monotonicities of generalized weighted mean values , 2004 .

[22]  Janusz Matkowski,et al.  Mean value property and associated functional equations , 1999 .

[23]  Radko Mesiar,et al.  Weighted ordinal means , 2007, Inf. Sci..

[24]  Feng Qi,et al.  NEW PROOFS OF WEIGHTED POWER MEAN INEQUALITIES AND MONOTONICITY FOR GENERALIZED WEIGHTED MEAN VALUES , 2000 .

[25]  A. Cauchy.  Cours d'analyse de l'École royale polytechnique , 1821 .

[26]  Radko Mesiar,et al.  Measure-based aggregation operators , 2004, Fuzzy Sets Syst..

[27]  Marek Kuczma.  On the quasiarithmetic mean in a mean value property and the associated functional equation , 1990 .

[28]  Zsolt Páles.  On the characterization of quasiarithmetic means with weight function , 1987 .

[29]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[30]  Jean-Luc Marichal,et al.  Characterization of some aggregation functions stable for positive linear transformations , 1999, Fuzzy Sets Syst..

[31]  G. Choquet.  Theory of capacities , 1954 .

[32]  Janusz Kacprzyk,et al.  The Ordered Weighted Averaging Operators , 1997 .

[33]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[34]  Lucio R. Berrone,et al.  On means generated through the Cauchy mean value theorem , 2000 .

[35]  László Losonczi,et al.  General inequalities for nonsymmetric means , 1972 .

[36]  P. K. Sahoo,et al.  Mean Value Theorems and Functional Equations , 1998 .

[37]  Jean-Luc Marichal,et al.  On the associativity functional equation , 2000, Fuzzy Sets Syst..

[38]  Fabrizio Durante,et al.  On representations of 2-increasing binary aggregation functions , 2008, Inf. Sci..

[39]  Radko Mesiar,et al.  Weighted means and weighting functions , 2006, Kybernetika.

[40]  J. Aczel,et al.  Functional Equations in Several Variables: With Applications to Mathematics, Information Theory and to the Natural and Social Sciences , 1989 .

[41]  P. Bullen.  Handbook of means and their inequalities , 1987 .

[42]  Jean-Luc Marichal,et al.  Weighted lattice polynomials , 2007, Discret. Math..

[43]  R. Mesiar,et al.  Aggregation operators: new trends and applications , 2002 .

[44]  S. Simić,et al.  Means and Their Inequalities , 1988 .

[45]  Jean-Luc Marichal,et al.  A characterization of the ordered weighted averaging functions based on the ordered bisymmetry property , 1999, IEEE Trans. Fuzzy Syst..

[46]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[47]  Gilberto Arenas-Díaz,et al.  Fuzzy measures and fuzzy integrals , 2013 .

[48]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[49]  Feng Qi,et al.  Generalized weighted mean values with two parameters† , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[50]  P. S. Aleksandrov,et al.  An introduction to the theory of groups , 1960 .

[51]  K. S. Fu,et al.  AN AXIOMATIC APPROACH TO RATIONAL DECISION MAKING IN A FUZZY ENVIRONMENT , 1975 .

[52]  J. Kacprzyk,et al.  The Ordered Weighted Averaging Operators: Theory and Applications , 1997 .

[53]  Radko Mesiar,et al.  Aggregation of infinite sequences , 2008, Inf. Sci..

[54]  Herbert A. David,et al.  Order Statistics , 2011, International Encyclopedia of Statistical Science.

[55]  Janusz Matkowski,et al.  An invariance of geometric mean with respect to Lagrangian means , 2007 .

[56]  Zeshui Xu,et al.  Choquet integrals of weighted intuitionistic fuzzy information , 2010, Inf. Sci..

[57]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[58]  Óscar Valero,et al.  Aggregation of asymmetric distances in Computer Science , 2010, Inf. Sci..

[59]  J. Merigó,et al.  The connection between distortion risk measures and ordered weighted averaging operators , 2013 .

[60]  Radko Mesiar,et al.  Quantitative weights and aggregation , 2004, IEEE Transactions on Fuzzy Systems.

[61]  D. Schmeidler.  Integral representation without additivity , 1986 .

[62]  K. Stolarsky,et al.  Generalizations of the Logarithmic Mean , 1975 .

[63]  R. Bhatia.  Positive Definite Matrices , 2007 .

[64]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[65]  E. Hayes.  Mean Values. , 2022, Science.

[66]  Herman Akdag,et al.  A tool for aggregation with words , 2009, Inf. Sci..

[67]  P. J. Cheek.  25. Italian Contributions to the Methodology of Statistics , 1988 .

[68]  Mitio Nagumo.  Über eine Klasse der Mittelwerte , 1930 .

[69]  Vicenç Torra,et al.  Modeling Decisions: Information Fusion and Aggregation Operators (Cognitive Technologies) , 2006 .

[70]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[71]  Vicenç Torra,et al.  Modeling decisions - information fusion and aggregation operators , 2007 .

[72]  M. Kuczma,et al.  On two mean value properties and functional equations associated with them , 1989 .

[73]  Luis M. de Campos,et al.  Characterization and comparison of Sugeno and Choquet integrals , 1992 .

[74]  Radko Mesiar,et al.  Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes , 2011, Inf. Sci..

[75]  Michel GRABISCH,et al.  The Interaction and Möbius Representations of Fuzzy Measures on Finite Spaces, -Additive Measures: A Survey , 2022 .

[76]  Alan Horwitz.  Invariant Means , 2000 .

[77]  R. Mesiar,et al.  Aggregation operators: properties, classes and construction methods , 2002 .

[78]  József Sándor,et al.  On the identric and logarithmic means , 1990 .

[79]  Michel Grabisch,et al.  Equivalent Representations of Set Functions , 2000, Math. Oper. Res..

[80]  Ali Emrouznejad,et al.  Improving minimax disparity model to determine the OWA operator weights , 2010, Inf. Sci..

[81]  R. Lipschitz.  De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, et praecipue earum, quae per variabilis spatium finitum valorum maximourm et minimorum numerum habent infinitum,disquisitio. , .

[82]  P. Heywood.  Trigonometric Series , 1968, Nature.

[83]  Gleb Beliakov,et al.  Aggregation Functions: A Guide for Practitioners , 2007, Studies in Fuzziness and Soft Computing.

[84]  Radko Mesiar,et al.  Weighted aggregation operators based on minimization , 2008, Inf. Sci..

[85]  M. Roubens,et al.  Characterization of some stable aggregation functions , 1993 .

[86]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU) - an outline , 2005, GrC.

[87]  A ZadehLotfi,et al.  Toward a generalized theory of uncertainty (GTU) , 2005 .

[88]  D. S. Mitrinovic,et al.  Means and Their Inequalities , 1989, Int. J. Math. Math. Sci..

[89]  Alan Horwitz,et al.  Means and Divided Differences , 1995 .

[90]  J. Aczel,et al.  Functional Equations in Several Variables , 2008 .

[91]  Bernadette Bouchon-Meunier,et al.  Fuzzy Logic And Soft Computing , 1995 .

[92]  László Losonczi,et al.  Equality of two variable means revisited , 2006 .

[93]  Giulio Bemporad.  Sul Principio Della Media Aritmetica , 1918 .

[94]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[95]  László Losonczi,et al.  Homogeneous Cauchy Mean Values , 2002 .

[96]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[97]  Alfred Witkowski,et al.  Monotonicity of generalized weighted mean values , 2004 .

[98]  János C. Fodor,et al.  An Extension of Fung-Fu's Theorem , 1996, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[99]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[100]  S. Ovchinnikov.  Max-Min Representation of Piecewise Linear Functions , 2000, math/0009026.

[101]  Anna Kolesárová,et al.  Triangular norm-based iterative compensatory operators , 1999, Fuzzy Sets Syst..