By Gleb Beliakov
A vast advent to the subject of aggregation features is to be present in this publication. It additionally offers a concise account of the houses and the most sessions of such features. a few cutting-edge recommendations are provided, besides many graphical illustrations and new interpolatory aggregation features. specific recognition is paid to id and development of aggregation features from program particular standards and empirical facts.
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Extra resources for Aggregation Functions: A Guide for Practitioners
We discuss them in Chapter 2. 65 (Arithmetic mean). The arithmetic mean is the function M (x) = 1 1 (x1 + x2 + . . + xn ) = n n n xi . 66 (Weighting vector). A vector w = (w1 , . . , wn ) is called a weighting vector if wi ∈ [0, 1] and n wi = 1. 67 (Weighted arithmetic mean). Given a weighting vector w, the weighted arithmetic mean is the function n Mw (x) = w1 x1 + w2 x2 + . . + wn xn = wi xi . 68 (Geometric mean). The geometric mean is the function √ G(x) = n x1 x2 . . xn = 1/n n xi . 69 (Harmonic mean).
1 Definitions Quasi-arithmetic means generalize power means. Consider a univariate continuous strictly monotone function g : [0, 1] → [−∞, ∞], which we call a generating function. , its range may be Ran(g) ⊂ [−∞, ∞]). 17 (Quasi-arithmetic mean). For a given generating function g, the quasi-arithmetic mean is the function Mg (x) = g −1 1 n n g(xi ) . 18 (Weighted quasi-arithmetic mean). For a given generating function g, and a weighting vector w, the weighted quasi-arithmetic mean is the function n Mw,g (x) = g −1 wi g(xi ) .
56. An aggregation function f is stronger than another aggregation function of the same number of arguments g, if for all x ∈ [0, 1]n : g(x) ≤ f (x). It is expressed as g ≤ f . When f is stronger that g, it is equivalently said that g is weaker than f . Not all aggregation functions are comparable. It may happen that f is stronger than g only on some part of the domain, and the opposite is true on the rest of the domain. In this case we say that f and g are incomparable. 57. 9). Any disjunctive aggregation function is stronger than an averaging function, and any averaging function is stronger than a conjunctive one.