By Gleb Beliakov
A huge creation to the subject of aggregation capabilities is to be present in this publication. It additionally offers a concise account of the houses and the most periods of such services. a few cutting-edge ideas are awarded, in addition to many graphical illustrations and new interpolatory aggregation features. specific cognizance is paid to identity and building of aggregation features from program particular standards and empirical info.
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Additional resources for Aggregation Functions: A Guide for Practitioners
28. 87. Take the geometric mean f (x) = x1 x2 , which is an averaging function with the absorbing element a = 0. Take ψ(t) = t2 . Composition (ψ ◦ f )(x) = x1 x2 , yields the product function, which is conjunctive. 88. Take the harmonic mean f (x) = 2( x11 + x12 )−1 = x2x , which 1 +x2 2 also has the absorbing element a = 0. Take again ψ(t) = t . Composition (2x1 x2 )2 g(x) = (ψ ◦ f )(x) = (x 2 is a conjunctive aggregation function (we can 1 +x2 ) 4x2 t . A simple computacheck that g(x1 , 1) = (1+x11 )2 ≤ x1 ).
Since f (0) = 0 and f (1) = 1, the Lipschitz √ constant of any aggregation function is M ≥ 1/||1||. For p-norms we have ||1|| = p n · 1 ≤ 1, that is M ≥ n−1/p , so in principle M can be smaller than 1. 60 (p-stable aggregation functions). Given p ≥ 1, an aggregation function is called p-stable if its Lipschitz constant in the p-norm || · ||p is 1. An extended aggregation function is p-stable if it can be represented as a family of p-stable aggregation functions. 15 16 17 A distance between objects from a set S is a function deﬁned on S × S, whose values are non-negative real numbers, with the properties: 1) d(x, y) = 0 if and only if x = y, 2) d(x, y) = d(y, x), and 3) d(x, z) ≤ d(x, y) + d(y, z) (triangular inequality).
1, t, 1, . . , 1)), which is not diﬃcult to verify. 16 when constructing asymmetric conjunctive and disjunctive functions. 86. A similar construction also transforms averaging functions with the absorbing element a = 1, if ψ is a bijection, to disjunctive functions (by using duality). , to construct averaging functions from either conjunctive or disjunctive functions. 28. 87. Take the geometric mean f (x) = x1 x2 , which is an averaging function with the absorbing element a = 0. Take ψ(t) = t2 .
Aggregation Functions: A Guide for Practitioners by Gleb Beliakov