By Grabisch M., Marichal J.-L., Mesiar R., Pap E.
Aggregation is the method of mixing numerous numerical values right into a unmarried consultant price, and an aggregation functionality plays this operation. those capabilities come up anywhere aggregating details is critical: utilized and natural arithmetic (probability, information, selection thought, sensible equations), operations examine, machine technology, and lots of utilized fields (economics and finance, trend attractiveness and photo processing, facts fusion, etc.). this can be a accomplished, 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 response to nonadditive integrals. The homes of every approach, in addition to their interpretation and research, are studied extensive, including building tools and sensible identity tools. detailed recognition 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 special source for researchers
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Extra info for Aggregation Functions
Xp1 , . . 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.
Clearly, it would be unnatural to give as an overall score a value that is lower than the lowest partial score, or greater than the highest score, so that only “internal’’ aggregation functions are allowed. Another example concerns the aggregation of opinions in voting procedures. If, as usual, the voters are anonymous, the aggregation function must be symmetric. In this chapter we present some properties that are generally considered as relevant for aggregation. Of course, not all these properties are equally important or serve the same purpose.
According to our convention on unary aggregation functions, namely A(x) = x for all x ∈ I, it follows immediately from the previous proposition that any strongly idempotent extended aggregation function is idempotent. Idempotizable functions We now consider a superset of the class of idempotent functions, called idempotizable functions. 45. A function F : In → R is idempotizable if its diagonal section δF is strictly increasing and satisfies ran(δF ) = ran(F). 4), any nondecreasing and idempotent function F : In → R fulfills ran(F) = I and hence is idempotizable.