By Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo

ISBN-10: 3319247514

ISBN-13: 9783319247519

ISBN-10: 3319247530

ISBN-13: 9783319247533

This e-book bargains an easy-to-use and practice-oriented reference advisor to mathematical averages. It offers alternative ways of aggregating enter values given on a numerical scale, and of selecting and/or developing aggregating services for particular functions. construction on a prior monograph via Beliakov et al. released by way of Springer in 2007, it outlines new aggregation tools built for the time being, with a unique specialize in the subject of averaging aggregation services. It examines fresh advances within the box, reminiscent of aggregation on lattices, penalty-based aggregation and weakly monotone averaging, and extends some of the already latest equipment, similar to: ordered weighted averaging (OWA), fuzzy integrals and blend capabilities. a considerable mathematical history isn't referred to as for, as the entire correct mathematical notions are defined the following and said on including a wealth of graphical illustrations of precise households of aggregation capabilities. The authors regularly specialise in useful purposes and provides critical significance to the conciseness of exposition, in addition to the relevance and applicability of the pronounced equipment, supplying a beneficial source for computing device scientists, IT experts, mathematicians, procedure architects, wisdom engineers and programmers, in addition to for somebody dealing with the problem of ways to mix a number of inputs right into a unmarried output value.

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A prototypical example of a linear aggregation function is the arithmetic mean. 48 (Local internality) An aggregation function f is called locally internal if for all x1 , . . , xn ∈ In , f (x1 , . . , xn ) ∈ {x1 , . . , xn }. 12, it is an averaging function), but not vice-versa. Projections and order statistics (see Chap. 2), along with the minimum and maximum, are trivial instances of locally internal aggregation functions. Other functions in this class are the left- and right-continuous idempotent uninorms characterized in [DB99].

It consists in building a hyperplane which fits the data best in the least squares sense. Let the equation of the hyperplane be f (x) = a0 + a1 x1 + a2 x2 + · · · + an xn . 7 Supplementary Material: Some Methods for Approximation and Optimization 39 Then the vector of coefficients a can be determined by solving the least squares problem K n (a0 + minimize k=1 ai xik − yk )2 , i=1 where xik is the i-th component of the vector xk . Linear regression problem can be immediately generalized if we choose n f (x) = a0 + ai Bi (xi ), i=1 where Bi are some given functions of the i-th component of x.

And satisfying f (xk ) ≈ yk , k = 1, . . , K . Of course the approximate equalities may be satisfied exactly, if the properties P1 , P2 , . . allow this. We shall also consider a variation of the selection problem when yk are given as intervals, in which case we require f (xk ) ∈ [y k , y k ], or even approximately satisfy this condition. The satisfaction of approximate equalities f (xk ) ≈ yk is usually translated into the following minimization problem. 16) subject to f satisfies P1 , P2 , .

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A Practical Guide to Averaging Functions by Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo

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