# Aggregation Functions by Grabisch M., Marichal J.-L., Mesiar R., Pap E.

By Grabisch M., Marichal J.-L., Mesiar R., Pap E.

Aggregation is the method of mixing a number of numerical values right into a unmarried consultant price, and an aggregation functionality plays this operation. those services come up anywhere aggregating details is necessary: utilized and natural arithmetic (probability, records, determination concept, useful equations), operations learn, machine technological know-how, and plenty of utilized fields (economics and finance, development attractiveness and photograph processing, information fusion, etc.). this can be a accomplished, rigorous and self-contained exposition of aggregation capabilities. periods of aggregation capabilities coated comprise triangular norms and conorms, copulas, capacity and averages, and people in accordance with nonadditive integrals. The houses of every approach, in addition to their interpretation and research, are studied extensive, including building tools and sensible identity tools. distinct awareness is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it's an excellent advent for graduate scholars and a distinct source for researchers

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For example, AM, GM, OWAw are symmetric functions. A prominent example of nonsymmetric aggregation functions is the weighted arithmetic mean WAMw . The following result, well known in group theory, shows that the symmetry property can be checked with only two equalities; see for instance Rotman [367, Ex. 9, p. 24]. 33. F : In → R is a symmetric function if and only if, for all x ∈ In , we have (i) F(x2 , x1 , x3 , . . , xn ) = F(x1 , x2 , x3 , . . , xn ), (ii) F(x2 , x3 , . . , xn , x1 ) = F(x1 , x2 , x3 , .

1 Every unary absolutely continuous function is uniformly continuous. Every n-ary uniformly continuous function is continuous. Every n-ary continuous function has the intermediate value property. 10. Let · : Rn → [0, ∞[ be a norm and let D ⊆ In . A function F : In → R is uniformly continuous in D (with respect to · ) if for every ε > 0 there exists δ > 0 such that |F(x) − F(y)| < ε whenever x − y < δ and x, y ∈ D. It is immediate and well known that any uniformly continuous function is continuous while the converse is not true (for instance F(x) = x2 in R).

The case of the function A∗ can be treated in an analogous way. ✷ For infinite real intervals I, each Lipschitzian function (with respect to any norm · ) is continuous. The converse is false in general. 25. The geometric mean GM on the interval [0, ∞[n is a continuous function which is not Lipschitzian. As already mentioned for the extended aggregation function A : ∪n∈N In → R, continuity of A means continuity of each A(n) . However, the Lipschitz condition for A is restricted by the existence of a constant c ∈]0, ∞[ such that each A(n) is c-Lipschitzian.