By Gleb Beliakov
A large creation to the subject of aggregation features is to be present in this e-book. It additionally presents a concise account of the houses and the most sessions of such features. a few state of the art concepts are awarded, besides many graphical illustrations and new interpolatory aggregation features. specific awareness is paid to identity and development of aggregation services from program particular specifications and empirical information.
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Extra info for Aggregation Functions: A Guide for Practitioners
58 (Lipschitz continuity). 6) where d(x, y) is a distance between x and y 15 . The smallest such number M is called the Lipschitz constant of f (in the distance d). Typically the distance is the Euclidean distance between vectors, d(x, y) = (x1 − y1 )2 + (x2 − y2 )2 + . . + (xn − yn )2 , but it can be chosen as any norm d(x, y) = ||x − y|| as a p-norm. A p-norm, p ≥ 1 is a function ||x||p = p, and ||x||∞ = max |xi |. ,n Thus, if the change in the input is δ = ||x−y||, then the output will change by at most M δ.
N ), input x we do not expect the output f (˜ x) to be substantially diﬀerent from f (x). The Lipschitz constant M bounds the factor by which the noise is magniﬁed. 59. 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.
We consider two simple techniques for constructing new aggregation functions. 5 Composition and transformation of aggregation functions 29 inputs and the second is based on iterative application of aggregation functions. Let us consider univariate strictly increasing bijections (hence continuous) ϕ1 , ϕ2 , . . , ϕn and ψ; ϕi , ψ : [0, 1] → [0, 1]. 82. Let ϕ1 , . . , ϕn , ψ : [0, 1] → [0, 1] be strictly increasing bijections. For any aggregation function f , the function g(x) = ψ(f (ϕ1 (x1 ), ϕ2 (x2 ), .