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 features come up at any place aggregating details is necessary: utilized and natural arithmetic (probability, facts, selection conception, useful equations), operations learn, desktop technological know-how, and plenty of utilized fields (economics and finance, development acceptance and photo processing, facts fusion, etc.). this can be a entire, rigorous and self-contained exposition of aggregation services. periods of aggregation capabilities lined comprise triangular norms and conorms, copulas, capacity and averages, and people in keeping with nonadditive integrals. The houses of every approach, in addition to their interpretation and research, are studied extensive, including development tools and functional id tools. specific realization is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it's an incredible creation for graduate scholars and a different source for researchers
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76) = F p · (F(x11 , . . , xp1 ), . . , F(x1n , . . , xpn )) = F F(x11 , . . , xp1 ), . . , F(x1n , . . , xpn ) (strong decomposability) (strong idempotency). 4 Invariance properties One of the main concerns when choosing an appropriate aggregation function is to take into account the scale types of the variables being aggregated. On this issue, Luce  observed that the general form of the functional relationship between variables is greatly restricted if we know the scale types of the dependent and independent variables.
Xn )), F∗(n) (x1 , . . , xn ) := F(F∗(n−1) (x1 , . . , xn−1 ), xn ). 66. It is a fact that an extended function F : ∪n∈N In → I is associative if and only if the backward and forward extensions of F(2) coincide with F. 5). So, it seems interesting to know whether there exists a functional equation, similar to associativity, which can be fulfilled by the arithmetic mean, or even by other means such as the geometric mean, the quadratic mean, etc. On this subject, an acceptable equation, called associativity of means  or barycentric associativity [17, Chap.
Proof. We simply have F(k · x(1) , . . , k · x(n) ) = F (kp1 + · · · + kpn ) · F(x(1) , . . , x(n) ) (strong decomposability) = F(x(1) , . . , x(n) ) (idempotency). 75. 74, we can readily see that, in the statement of the lemma, idempotency can be relaxed to the following condition: ran(F(n) ) ⊆ ran(F(kn) ) for all k, n ∈ N. 76. If F : ∪n∈N In → I is strongly decomposable and idempotent then, for any n, p ∈ N and any x(1) , . . , x(n) ∈ Ip , we have F(x(1) , . . , x(n) ) = F F(x(1) ), . .
Aggregation Functions by Grabisch M., Marichal J.-L., Mesiar R., Pap E.