A Practical Guide to Averaging Functions

Beliakov, Gleb; Sola, Humberto Bustince; Calvo, Tomasa

doi:10.1007/978-3-319-24753-3

Cited by 346 publications

(227 citation statements)

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“…For further details the interested reader is referred to Fodor and Roubens [12], Calvo et al [6], Beliakov et al [4], Torra and Narukawa [27], Grabisch et al [20] and Beliakov et al [3].…”

Section: Preliminariesmentioning

confidence: 99%

Consensus-Based Agglomerative Hierarchical Clustering

García-Lapresta

Pérez-Román

2017

Fuzzy Sets, Rough Sets, Multisets and Clustering

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In this contribution, we consider that a set of agents assess a set of alternatives through numbers in the unit interval. In this setting, we introduce a measure that assigns a degree of consensus to each subset of agents with respect to every subset of alternatives. This consensus measure is defined as 1 minus the outcome generated by a symmetric aggregation function to the distances between the corresponding individual assessments. We establish some properties of the consensus measure, some of them depending on the used aggregation function. We also introduce an agglomerative hierarchical clustering procedure that is generated by similarity functions based on the previous consensus measures.

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“…For further details the interested reader is referred to Fodor and Roubens [12], Calvo et al [6], Beliakov et al [4], Torra and Narukawa [27], Grabisch et al [20] and Beliakov et al [3].…”

Section: Preliminariesmentioning

confidence: 99%

Consensus-Based Agglomerative Hierarchical Clustering

García-Lapresta

Pérez-Román

2017

Fuzzy Sets, Rough Sets, Multisets and Clustering

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“…Points in this domain are denoted by

x, y \in {double-struckD}^{n}

. Comprehensive reviews of averaging functions can be found in Fodor and Roubens, Marichal, Marichal et al., Calvo et al., Beliakov et al., Torra and Narukawa, Mesiar et al., Grabisch et al., and Beliakov et al …”

Section: Averaging Functions: Wa and Owa Functionsmentioning

confidence: 99%

The binomial decomposition of OWA functions, the 2-additive and 3-additive cases inndimensions

Bortot

Pereira

Nguyen

2017

Int. J. Intell. Syst.

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In the context of the binomial decomposition of ordered weighted averaging (OWA) functions, we investigate the constraints associated with the 2‐additive and 3‐additive cases in n dimensions. The 2‐additive case depends on one coefficient whose feasible region does not depend on the dimension n. On the other hand, the feasible region of the 3‐additive case depends on two coefficients and is explicitly dependent on the dimension n. This feasible region is a convex polygon with n vertices and n edges, which is strictly expanding in the dimension n. The orness of the OWA functions within the feasible region is linear in the two coefficients, and the vertices associated with maximum and minimum orness are identified. Finally, we discuss the 3‐additive binomial decomposition in the asymptotic infinite dimensional limit.

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“…Discrete integrals with respect to nonadditive normed measures, also known as fuzzy measures, are a class of important aggregation functions . Aggregation functions combine several inputs, such as degrees of satisfaction/support/evidence in multicriteria decision‐making problems, into one representative value used to rank the alternatives.…”

Section: Introductionmentioning

confidence: 99%

“… use a generating function applied to the arguments of the Choquet integral and the inverse of the generating function applied to the output. The process here is the same as when the arithmetic mean is transformed into a quasi‐arithmetic mean (QAM) . Power functions and the logarithm are notable special cases leading to power‐based and geometric Choquet integral.…”

Section: Introductionmentioning

confidence: 99%

A new type of fuzzy integrals for decision making based on bivariate symmetric means

Beliakov

2018

Int. J. Intell. Syst.

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We propose a new generalization of the discrete Choquet integral based on an arbitrary bivariate symmetric averaging function (mean). So far only the means with a natural multivariate extension were used for this purpose. In this paper, we use a general method based on a pruned binary tree to extend symmetric means with no obvious multivariate form, such as the logarithmic, identric, Heronian, Lagrangean, and Cauchy means. The generalized Choquet integral is built by computing the extensions of the bivariate means of the ordered inputs, and includes some existing extensions as special cases. Our construction is illustrated with multiple examples.

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A Practical Guide to Averaging Functions

Cited by 346 publications

References 0 publications

Consensus-Based Agglomerative Hierarchical Clustering

Consensus-Based Agglomerative Hierarchical Clustering

The binomial decomposition of OWA functions, the 2-additive and 3-additive cases inndimensions

A new type of fuzzy integrals for decision making based on bivariate symmetric means

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