By Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo
This booklet deals an easy-to-use and practice-oriented reference consultant to mathematical averages. It provides alternative ways of aggregating enter values given on a numerical scale, and of selecting and/or developing aggregating capabilities for particular functions. development on a prior monograph through Beliakov et al. released through Springer in 2007, it outlines new aggregation tools built for the time being, with a different specialise in the subject of averaging aggregation capabilities. It examines fresh advances within the box, similar to aggregation on lattices, penalty-based aggregation and weakly monotone averaging, and extends some of the already present equipment, similar to: ordered weighted averaging (OWA), fuzzy integrals and mix capabilities. a considerable mathematical historical past isn't really referred to as for, as all of the appropriate mathematical notions are defined right here and pronounced on including a wealth of graphical illustrations of targeted households of aggregation features. The authors usually concentrate on functional purposes and provides principal significance to the conciseness of exposition, in addition to the relevance and applicability of the suggested equipment, delivering a useful source for computing device scientists, IT experts, mathematicians, method architects, wisdom engineers and programmers, in addition to for a person dealing with the problem of ways to mix numerous inputs right into a unmarried output value.
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Additional resources for A Practical Guide to Averaging Functions
M. A special case arises when the functions gi , h i are linear (or affine). The feasible domain is then a convex polytope, and when in addition to this the objective function is linear or convex quadratic, then special methods of linear and quadratic programming are applied. We also distinguish convex and non-convex optimization. A convex function f satisfies the following condition, f (αx + (1 − α)y) ≤ α f (x) + (1 − α) f (y), for all α ∈ [0, 1] and all x, y ∈ Dom( f ).
66 If an aggregation function is p-stable for a given p > 1, then it is also q-stable for any 1 ≤ q < p. This is because ||x|| p ≤ ||x||q for all x. 67 Aggregation functions which have Lipschitz constant k in 1-norm are called k-Lipschitz. Some properties and constructions are discussed in [BC09, BCJ10b]. 68 The product, minimum and maximum are p-stable extended aggregation functions for any p. The arithmetic mean is also p-stable for any p. 19 √ √ f (x1 , x2 ) = x1 x2 , which is continuous for x1 , x2 ≥ 0, and let x2 = 1.
Xmn ) we can either evaluate the scores given by the i-th juryman, yi = f n (xi1 , . . , xin ), and then aggregate them as z = f m (y1 , . . , compute y˜ j = f m (x1 j , . . , xm j ), and then aggregate these scores as z˜ = f n ( y˜1 , . . , y˜n ). The third alternative is to aggregate all the scores by an aggregation function f mn (x). 1. We can either aggregate scores in each row, and then aggregate the totals in the last column of this table, or we can aggregate scores in each column, and then aggregate the totals in the last row, or aggregate all scores at once.
A Practical Guide to Averaging Functions by Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo