By Gleb Beliakov
A vast advent to the subject of aggregation capabilities is to be present in this ebook. It additionally offers a concise account of the homes and the most sessions of such services. a few cutting-edge concepts are awarded, besides many graphical illustrations and new interpolatory aggregation services. specific recognition is paid to identity and development of aggregation capabilities from software particular necessities and empirical facts.
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Additional resources for Aggregation Functions: A Guide for Practitioners
Xn−1 ), xn ), starting with f2 = f22 . Each extended aggregation function built from an associative bivariate aggregation function is recursive by Montero, but the converse is not true. 41. Deﬁne f2n (t1 , t2 ) = (n−1)t1 +t2 . n Then fn (x) = 1 n n xi , the i=1 arithmetic mean (which is not associative). 42 (Decomposable extended aggregation function). An extended aggregation function F is decomposable if for all m, n = 1, 2, . . and for all x ∈ [0, 1]m , y ∈ [0, 1]n : fm+n (x1 , . . , xm , y1 , .
9 (Harmonic mean). The harmonic mean is the function n H(x) = n i=1 1 xi −1 . 10 (Weighted harmonic mean). Given a weighting vector w, the weighted harmonic mean is the function n Hw (x) = i=1 wi xi −1 . 11. , W = n i=1 wi = 1, then one can either normalize it ﬁrst by dividing each component by W , or use the alternative expressions for weighted geometric and harmonic means 1/W n i xw i Gw (x) = , i=1 n Hw (x) = W i=1 wi xi −1 . Fig. 2. 3D plots of weighted geometric means G( 1 , 1 ) and G( 1 , 4 ) .
N = a < 1 with 0 in the i-th position, then necessarily ψ(t) = 0 for t ≤ a. If ψ is a bijection, then f must have a = 0 as absorbing element. 85 is that one can construct conjunctive aggregation functions from many types of averaging functions (discussed in Chapter 2) by a simple transformation, and that its condition involves single variate functions ψ(f (1, . . , 1, t, 1, . . , 1)), which is not diﬃcult to verify. 16 when constructing asymmetric conjunctive and disjunctive functions. 86.
Aggregation Functions: A Guide for Practitioners by Gleb Beliakov