Download A Practical Guide to Averaging Functions by Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo PDF
By Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo
This publication bargains 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 features for particular purposes. development on a prior monograph via Beliakov et al. released by means of Springer in 2007, it outlines new aggregation tools built in the mean time, with a distinct specialise in the subject of averaging aggregation services. It examines fresh advances within the box, corresponding to aggregation on lattices, penalty-based aggregation and weakly monotone averaging, and extends a few of the already current tools, akin to: ordered weighted averaging (OWA), fuzzy integrals and combination features. a considerable mathematical heritage isn't known as for, as all of the suitable mathematical notions are defined right here and suggested on including a wealth of graphical illustrations of targeted households of aggregation features. The authors in general specialize in useful functions and provides imperative value to the conciseness of exposition, in addition to the relevance and applicability of the mentioned tools, delivering a important source for computing device scientists, IT experts, mathematicians, procedure architects, wisdom engineers and programmers, in addition to for a person dealing with the difficulty of ways to mix numerous inputs right into a unmarried output value.
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Extra info for A Practical Guide to Averaging Functions
Example text
The next definition is an even stronger continuity requirement. The reason is that simple, or even uniform continuity is not sufficient to distinguish functions that real function of n arguments is continuous if for any sequences {xi j }, i = 1, . . , n such that lim xi j = yi it holds lim f (x1 j , . . xn j ) = f (y1 , . . , yn ). Because the domain In is a compact 14 A j→∞ j→∞ set, continuity is equivalent to its stronger version, uniform continuity. , we can check continuity by fixing all variables but one, and checking continuity of each univariate function.
16 A distance between objects from a set S is a function defined on S × S, whose values are nonnegative real numbers, with the properties: (1) d(x, y) = 0 if and only if x = y, (2) d(x, y) = d(y, x), and (3) d(x, z) ≤ d(x, y)+ d(y, z) (triangular inequality). Such distance is called a metric. 17 A norm is a function f on a vector space with the properties: (1) f (x) > 0 for all nonzero x and f (0) = 0, (2) f (ax) = |a| f (x), and (3) f (x + y) ≤ f (x) + f (y).. , it is differentiable on its entire domain, except for a subset of measure zero.
The classical expert systems MYCIN and PROSPECTOR [BS84, DHN76] use precisely this type of aggregation (on [−1,1] interval). A different behavior may also be needed: aggregation of both high and low values moves the output towards some intermediate value. Thus certain aggregation functions need to be conjunctive, disjunctive or averaging in different parts of their domain. Uninorms and nullnorms (see [BPC07]) are typical examples of such aggregation functions, but there are many others. 85 The 3 − Π function [YR96] is n xi f (x) = n i=1 i=1 n xi + , (1 − xi ) i=1 with the convention 00 = 0.