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 anyplace aggregating details is critical: utilized and natural arithmetic (probability, records, selection thought, practical equations), operations study, laptop technological know-how, and plenty of utilized fields (economics and finance, development attractiveness and picture processing, info fusion, etc.). this can be a accomplished, rigorous and self-contained exposition of aggregation features. sessions of aggregation services lined contain triangular norms and conorms, copulas, skill and averages, and people in line with nonadditive integrals. The homes of every procedure, in addition to their interpretation and research, are studied intensive, including building equipment and useful identity tools. specified realization is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it's a great advent for graduate scholars and a special source for researchers
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Additional info for Aggregation Functions
12) 40 Properties for aggregation for all square matrices · · · x1n .. ∈ In×n . x11 .. . · · · xnn xn1 Bisymmetry expresses the condition that aggregation of all the elements of any square matrix can be performed first on the rows, then on the columns, or conversely. However, since only square matrices are involved, this property seems not to have a good interpretation in terms of aggregation. Its usefulness remains theoretical. We then consider it for extended functions as follows; see Marichal et al.
As we will now show, some k-conjunctive functions F are such that F(x) remains unchanged when x(k+1) , . . , x(n) are replaced with any other values greater than or equal to x(k) . Similarly, some k-disjunctive functions F are such that F(x) remains unchanged when x(1) , . . , x(n−k) are replaced with any other values smaller than or equal to x(n−k+1) . More precisely, we have the following definitions. 57. Assume n 2 and let k ∈ [n − 1]. A function F : In → R is independent of the (n − k) highest arguments if, for any i ∈ [n] and any x, x ∈ In , with xi x(k+1) , we have xi x(k) xj = xj ∀ j = i ⇒ F(x ) = F(x).
Proof. 29, we give a proof of this equivalence in the more general cases of left-continuity and right-continuity. (i) ⇒ (iii) It is well known that continuous unary functions have the intermediate value property (independently of whether they are nondecreasing or not). Now, for 14 Properties for aggregation any x, y ∈ In such that x y and x = y (this is the only nontrivial case), we can define a unary function f : [0, 1] → R as f (t) := F((1 − t)x + ty). This function is continuous and thus for each c ∈ [F(x), F(y)] = [f (0), f (1)] there is some t0 ∈ [0, 1] such that f (t0 ) = c.
Aggregation Functions by Grabisch M., Marichal J.-L., Mesiar R., Pap E.