Aggregation Functions in Theory and in Practise: Proceedings by Michał Baczyński (auth.), Humberto Bustince, Javier

By Michał Baczyński (auth.), Humberto Bustince, Javier Fernandez, Radko Mesiar, Tomasas Calvo (eds.)

This quantity collects the prolonged abstracts of forty five contributions of members to the 7th overseas summer time college on Aggregation Operators (AGOP 2013), held at Pamplona in July, 16-20, 2013. those contributions disguise a truly huge diversity, from the in simple terms theoretical ones to these with a extra utilized concentration. furthermore, the summaries of the plenary talks and tutorials given on the comparable workshop are included.

Together they supply a very good review of contemporary developments in learn in aggregation services which might be of curiosity to either researchers in Physics or arithmetic engaged on the theoretical foundation of aggregation features, and to engineers who require them for applications.

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Additional info for Aggregation Functions in Theory and in Practise: Proceedings of the 7th International Summer School on Aggregation Operators at the Public University of Navarra, Pamplona, Spain, July 16-20, 2013

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Mesiar, J. Komorník, and M. Komorníková Fig. 1 Formulae for the copula Cq We open a problem of characterizing all diagonal sections δ of bivariate semicopulas such that the formula (7) yields a copula. Note also that if a function Cp given by (7) is a copula, then Cq with q = (M, M, λ , f , f ) is a copula for any λ ∈ [−1, 0]. , consider a function Cp : [0, 1]2 → [0, 1] given by Cp (x, y) = max (0,C(x, y) − C(N(x), N(y))) . , Cp is a semicopula [2, 4]. For arbitrary Frank copula [9] and the standard negation Ns : [0, 1] → [0, 1] given by Ns (x) = 1 − x, we see that Cp = W is a copula.

In the opposite case, we recommend the lecture notes [12]. Recently, several construction methods for bivariate copulas have been proposed. Recall, for example, conic copulas [10], univariate conditioning method proposed in [15], UCS (univariate conditioning stable) copulas [8], a method proposed by Rodríguez–Lallena and Úbeda–Flores [17] and its generalization in [11], another method introduced by Aguilló et al. in [1], quadratic construction introduced in [13], several construction methods based on diagonal or horizontal (vertical) sections discussed in [5, 3, 7], etc.

E. for all x, x ∈ [0, 1], it holds that ω (x ) − ω (x) ≤ x − x . Note that (OD2) implies that ω is absolutely continuous, and hence differentiable almost everywhere. The opposite diagonal section ωC of a copula C is an opposite diagonal function. Conversely, for any opposite diagonal function ω , there exists at least one copula C with opposite diagonal section ωC = ω . For instance, the copula Fω defined by Fω (x, y) = max(x + y − 1, 0) + min{ω (t) | t ∈ [min(x, 1 − y), max(x, 1 − y)]} , is the greatest copula with opposite diagonal section ω [2, 9].

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