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 capabilities come up at any place aggregating details is critical: utilized and natural arithmetic (probability, information, selection concept, sensible equations), operations study, desktop technological know-how, and lots of utilized fields (economics and finance, development attractiveness and snapshot processing, facts fusion, etc.). it is a accomplished, rigorous and self-contained exposition of aggregation capabilities. periods of aggregation features coated comprise triangular norms and conorms, copulas, skill and averages, and people in line with nonadditive integrals. The houses of every procedure, in addition to their interpretation and research, are studied intensive, including development equipment and useful id tools. certain consciousness is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it really is a terrific advent for graduate scholars and a distinct source for researchers
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N. Proof. We simply have F(x1 , . . , xn ) = F((n − 1) · x1 , . . , (n − 1) · xn ) = F (n − 1) · (xn , . . 74) (strong decomposability) = F(xn , . . , x1 ) (strong idempotency). 78. It is noteworthy that, under idempotency, the definition of strong decomposability can be rewritten by replacing each element xi with a p-tuple x(i) ∈ Ip . 8) can be written with p-tuples x(1) , x(2) , x(3) ∈ Ip . 74). 3 Autodistributivity We now consider the autodistributivity property, also called self-distributivity.
Then, for all x ∈ In , x(k+1) F(x) x(n−k) . 3 Grouping-based properties The properties we will focus on in this section concern the “grouping’’ character of the aggregation functions. That is to say, we assume that it is possible to partition the set of arguments into disjoint subgroups, build the partial aggregation for each subgroup, and then combine these partial results to get the overall value. Such a 32 Properties for aggregation condition may take several forms. A strong one we will first present is associativity.
X(σ(n)) ) 24 Properties for aggregation for all σ ∈ S[n] . Then we have, for all x ∈ Inσ , F(x) = Fσ (x), which describes F on the whole domain since In = ∪σ∈S[n] Inσ . 4 Idempotency In algebra, we say that x is an idempotent element with respect to a binary operation ∗ if x ∗ x = x. This algebraic property can be extended to n-ary functions, thus defining the idempotency property for any n-ary function. Also called unanimity, agreement, or reflexivity, this property means that if all xi are identical, F(x1 , .