By Grabisch M., Marichal J.-L., Mesiar R., Pap E.
Aggregation is the method of mixing numerous numerical values right into a unmarried consultant worth, and an aggregation functionality plays this operation. those capabilities come up anywhere aggregating info is necessary: utilized and natural arithmetic (probability, data, choice idea, practical equations), operations examine, desktop technology, and lots of utilized fields (economics and finance, trend popularity and snapshot processing, facts fusion, etc.). it is a entire, rigorous and self-contained exposition of aggregation capabilities. sessions of aggregation capabilities lined comprise triangular norms and conorms, copulas, skill and averages, and people in accordance with nonadditive integrals. The homes of every technique, in addition to their interpretation and research, are studied extensive, including building equipment and useful identity tools. targeted cognizance is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it's an excellent creation for graduate scholars and a different source for researchers
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Additional info for Aggregation Functions
Disjunctive functions combine values as an “or’’ operator, so that the result of combination is high if at least one value is high. Thus, a high value cannot be compensated by a low value. Such functions are, in this sense, dual of conjunctive functions. 5. The following proposition shows that, due to the monotonicity of aggregation functions, the conjunctiveness (respectively, disjunctiveness) of an aggregation function over the domain [a, b]n can be checked just on the upper (respectively, lower) boundary of that domain.
Additional input data are then simply aggregated with the current aggregated output. As examples of associative functions, recall Min, Max, Σ, Π, PF , PL . Functions like AM and GM are not associative. One can see that the conjunction of associativity and idempotency immediately cancels the effect of repeating arguments in the aggregation procedure. Indeed, under associativity and idempotency, we clearly have F(m · x, n · y) = F F(m · x), F(n · y) = F(x, y) for all m, n ∈ N. For instance, we have F(x, y, .
3 Of course, symmetry is more natural in voting procedures than in multicriteria decision making, where criteria usually have different importances. 34. F : In → R is a symmetric function if and only if there exists a function G : In → R such that, for all x ∈ In , F(x1 , . . , xn ) = G(x(1) , . . , x(n) ). 34 may lead to different functions G. For example, when F = OS1 then, for any K ⊆ [n], with K 1, we can consider G = MinK . In situations when judges, criteria, or individual opinions are not equally important, the symmetry property must be omitted.
Aggregation Functions by Grabisch M., Marichal J.-L., Mesiar R., Pap E.