By Gleb Beliakov

Aggregation of data is of fundamental value within the building of information dependent platforms in a variety of domain names, starting from drugs, economics, and engineering to decision-making techniques, man made intelligence, robotics, and computer studying. This ebook provides a huge creation into the subject of aggregation capabilities, and gives a concise account of the houses and the most periods of such features, together with classical potential, medians, ordered weighted averaging capabilities, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally provides a few cutting-edge innovations, many graphical illustrations and new interpolatory aggregation services. a selected recognition is paid to identity and building of aggregation services from software particular necessities and empirical facts. This e-book offers scientists, IT experts and method architects with a self-contained easy-to-use advisor, in addition to examples of computing device code and a software program package deal. it's going to facilitate development of selection help, professional, recommender, keep an eye on and lots of different clever systems.

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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.

3) Both the measure of orness and the average orness value are 12 for weighted arithmetic means, and later we will see that both quantities coincide for OWA functions. 2. 2 Classical means Means are often treated synonymously with averaging functions. , [40]) excludes certain types of averaging functions, which have been developed quite recently, in particular ordered weighted averaging and various integrals. , some Gini means) lack monotonicity, and therefore are not aggregation functions. Following the tradition, in this section we will concentrate on various classical means, and present other types of averaging, or mean-type functions in separate sections.

1 × 2 × 3 × . . × n possible permutations of (1, 2, . . , n). We will denote a vector whose components are arranged in the order given by a permutation P by xP = (xP (1) , xP (2) , . . , xP (n) ). In our example, xP = (x5 , x3 , x2 , x1 , x4 ). We will frequently use the following special permutations of the components of x. 18. We denote by x the vector obtained from x by arranging its components in non-decreasing order, that is, x = xP where P is the permutation such that xP (1) ≤ xP (2) ≤ .