Download Aggregation Functions: A Guide for Practitioners by Gleb Beliakov PDF

By Gleb Beliakov

Aggregation of data is of basic value within the development of data established structures in quite a few domain names, starting from medication, economics, and engineering to decision-making approaches, synthetic intelligence, robotics, and computer studying. This e-book offers a huge creation into the subject of aggregation features, and gives a concise account of the houses and the most sessions of such capabilities, together with classical capability, medians, ordered weighted averaging services, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally offers a few state of the art suggestions, many graphical illustrations and new interpolatory aggregation services. a selected recognition is paid to id and building of aggregation features from software particular specifications and empirical facts. This booklet presents scientists, IT experts and process architects with a self-contained easy-to-use consultant, in addition to examples of desktop code and a software program package deal. it is going to facilitate building of determination help, specialist, recommender, regulate and lots of different clever systems.

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Any disjunctive aggregation function has absorbing element a = 1. 9. Some averaging functions also have an absorbing element, for example the geometric mean 1/n n f (x) = xi i=1 has the absorbing element a = 0. 32. An aggregation function with an annihilator in ]0, 1[ cannot have a neutral element10 . But it may have a neutral element if a = 0 or a = 1. 33. The concept of an absorbing element has been recently extended to that of absorbing tuples, see [28]. 10 Proof: Suppose a ∈]0, 1[ is the absorbing element and e ∈ [0, 1] is the neutral element.

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 check that g(x1 , 1) = (1+x11 )2 ≤ x1 ). Now take ψ(t) = 2−t .

E, t, e, . . , e) = t, for t in any position. 25. A neutral element, if it exists, is unique 9 . It can be any number from [0, 1]. 26. Observe that if an aggregation function f has neutral element e = 1 (respectively e = 0) then f is necessarily conjunctive (respectively disjunctive). Indeed, if f has neutral element e = 1, then by monotonicity it is f (x1 , . . , xn ) ≤ f (1, . . , 1, xi , 1, . . , 1) = xi for any i ∈ {1, . . , n}, and this implies f ≤ min (the proof for the case e = 0 is analogous).

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