By Gleb Beliakov
Aggregation of knowledge is of fundamental value within the building of data dependent platforms in a variety of domain names, starting from medication, economics, and engineering to decision-making techniques, synthetic intelligence, robotics, and computing device studying. This ebook offers a huge creation into the subject of aggregation capabilities, and gives a concise account of the homes and the most periods of such features, 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 provides a few state of the art ideas, many graphical illustrations and new interpolatory aggregation features. a specific consciousness is paid to id and development of aggregation services from software particular specifications and empirical information. This booklet presents scientists, IT experts and process architects with a self-contained easy-to-use advisor, in addition to examples of desktop code and a software program package deal. it's going to facilitate development of determination aid, professional, recommender, keep an eye on and plenty of different clever systems.
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Extra info for Aggregation Functions: A Guide for Practitioners
2. Recursive and non-recursive calculation of an associative function. A continuous decomposable extended aggregation function is always idempotent. Another useful property, which generalizes both symmetry and associativity, and is applicable to extended aggregation functions, is called bisymmetry. Consider the situation in which m jurymen evaluate an alternative with respect to n criteria. Let xij , i = 1, . . , m, j = 1, . . , n denote the score given by the i-th juryman with respect to the j-th criterion.
Xn ) = N (f (N (x1 ), N (x2 ), . . , N (xn ))) is called the dual of f with respect to N , or, for short, the N -dual of f . When using the standard negation, fd is given by fd (x1 , . . , xn ) = 1 − f (1 − x1 , . . , 1 − xn ) and we will simply say that fd is the dual of f . It is evident that the dual of a conjunctive aggregation function is disjunctive, and vice versa, regardless of what strong negation is used. Some functions are self-dual. 55 (Self-dual aggregation function). Given a strong negation N , an aggregation function f is self-dual with respect to N (for short, N -self-dual or N -invariant), if f (x) = N (f (N (x))), where N (x) = (N (x1 ), .
We pay attention to the rate of change of a function because of the ever present input inaccuracies. If the aggregation function receives an inaccurate ˜ = (x1 + δ1 , . . , xn + δn ), contaminated with some noise (δ1 , . . , δn ), 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.
Aggregation Functions: A Guide for Practitioners by Gleb Beliakov
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