By Gleb Beliakov
Aggregation of knowledge is of fundamental value within the building of information dependent platforms in numerous domain names, starting from drugs, economics, and engineering to decision-making methods, synthetic intelligence, robotics, and computer studying. This booklet offers a wide creation into the subject of aggregation services, and offers a concise account of the houses and the most sessions of such capabilities, together with classical potential, medians, ordered weighted averaging features, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally offers a few cutting-edge concepts, many graphical illustrations and new interpolatory aggregation features. a specific awareness is paid to id and building of aggregation capabilities from software particular necessities 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 development of choice help, professional, recommender, keep an eye on and lots of different clever systems.
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Additional info for Aggregation Functions: A Guide for Practitioners
This can be done by presenting the experts some prototypical cases (either the input vectors, or domain speciﬁc situations before they are translated into the inputs). If there is more than one expert, their outputs could be either averaged, or translated into the range of possible output values, or the experts could be brought together to ﬁnd a consensus. Third, the data could be collected in an experiment, by asking a group of lay people or experts about their input and output values, but without associating these values with some aggregation rule.
T), t ∈ [0, 1] the output is f (t, t . . , t) = t. 12. 7 The aggregation functions minimum and maximum are the only two functions that are at the same time conjunctive (disjunctive) and averaging, and hence idempotent. 13. The arithmetic mean is an averaging (idempotent) aggregation function 1 f (x) = (x1 + x2 + . . + xn ). 14. The geometric mean is also an averaging (idempotent) aggregation function √ f (x) = n x1 x2 . . xn . 7 Proof: Take any x ∈ [0, 1]n , and denote by p = min(x), q = max(x).
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 ), .