Elsevier

Fuzzy Sets and Systems

Volume 346, 1 September 2018, Pages 127-137
Fuzzy Sets and Systems

k-maxitive aggregation functions

https://doi.org/10.1016/j.fss.2017.12.016Get rights and content

Abstract

Inspired by the idea of k-maxitive measures, we introduce and study k-maxitive aggregation functions. In particular, 1-maxitive aggregation functions are shown to be just maxitive aggregation functions, i.e., maxima of distorted inputs. We introduce, among others, a representation of symmetric k-maxitive aggregation functions. Moreover, we show that for any k-maxitive capacity and the smallest universal integral based on an arbitrary fixed semicopula, including the Sugeno and Shilkret integrals, the resulting aggregation function is k-maxitive.

Introduction

Aggregation functions were originally introduced to characterize finite n-tuples of input values from some real scale by means of a unique output value coming from the same scale [1], [2], [13], [17], [19]. The basic constraints of aggregation functions — boundary conditions and monotonicity — can be introduced on more general scales, formally on bounded posets, which also cover linguistic, interval, fuzzy and other scales. One of the typical properties introduced on real scales, coming from applications in natural sciences, is additivity (recall at least the Riemann and Lebesgue integrals or expected value of random variables) which was further generalized into comonotone additivity characterizing the Choquet integrals [8], including OWA operators [31], k-additivity [12], [21], [25], etc. These properties are linked to the summation of reals, and thus, except for some isomorphic transformations, they still act on reals, and cannot be considered on other types of scales. On the other hand, maxitivity can be introduced on any (bounded) lattice scale and thus it can be seen as a universal property. This fact inspired us to take a closer look at the k-maxitivity of aggregation functions, a new property generalizing the standard maxitivity of aggregation functions.

Recall that an aggregation function (say on [0,1]) is additive if and only if it is a weighted arithmetic mean. Note that one can also define related additive capacities which coincide with (discrete) probabilities. Almost 20 years ago, the additivity of capacities was generalized into k-additivity, kN, see [12], [25], and recently, in [21], [22] the k-additivity of aggregation functions has been introduced and studied, including a description of the link between k-additive aggregation functions and k-additive capacities, where a crucial role is played by the Owen extension [28] of the considered capacity. The notion of k-maxitivity of capacities was introduced almost simultaneously with introducing their k-additivity [25], and then further studied, e.g., in [5], [7], [18], [27].

In the framework of aggregation functions, k-maxitive capacities-based Sugeno integrals were studied, e.g., in [5], [7], [27] and as particular aggregation functions deeply discussed in [3], [4], [15], [24]. Recall that in [3] Brabant and Couceiro have shown that k-maxitive capacities-based Sugeno integrals are exactly the idempotent lattice polynomials whose degree is at most k. This result has a striking similarity with the representation of k-additive aggregation functions in the form of polynomials with degree at most k shown in [21], [22]. Further, in [4], Brabant and Couceiro have presented an axiomatization of the Sugeno integral with respect to k-maxitive capacities. However, maxitive aggregation functions need not be representable by means of the Sugeno integrals thought each Sugeno integral with respect to a maxitive capacity is a maxitive aggregation function. As we will show later, this relationship is also preserved in the case of our proposal of k-maxitive aggregation functions.

Observe that maxitive (i.e., 1-maxitive) capacities coincide with (discrete) possibility measures [10], [32], and that maxitive aggregation functions on [0,1] are just the maxima of distorted inputs. We give more details in the next section, where we also introduce the notion of k-maxitive aggregation functions. In Section 3 we study and characterize k-maxitive aggregation functions, putting stress on 2-maxitive aggregation functions. Section 4 is devoted to some examples, especially to integral forms of k-maxitive aggregation functions in connection with k-maxitive capacities. The paper ends by several concluding remarks.

Section snippets

k-maxitive aggregation functions

Let nN. Throughout the paper, the points in [0,1]n will be denoted by bold letters, x=(x1,,xn), in particular, 0=(0,,0) and 1=(1,,1). Given x,y[0,1]n, we write xy if xiyi for each i{1,,n}.

Definition 2.1

A mapping A:[0,1]n[0,1] is called an (n-ary) aggregation function if it satisfies the boundary conditions A(0)=0, A(1)=1, and the monotonicity condition A(x)A(y) whenever x,y[0,1]n, xy.

We say that A is maxitive if A(xy)=A(x)A(y) for all x,y[0,1]n.

It is easy to see that each x[0,1]n can be

Properties and representations of k-maxitive aggregation functions

For any aggregation function A:[0,1]n[0,1] and any non-empty subset I{1,,n} one can define a mapping AI:[0,1]card(I)[0,1] byAI(z)=A(zI)where zI[0,1]n,ziI={zjiifiI,0else, with ji=card({rI|ri}), iI. Then AI satisfies the boundary condition AI(0)=0 and it is also monotone. Similarly, for any x[0,1]n, let xI=(xj1,,xjcard(I)).

As a generalization of the representation (1) of maxitive aggregation functions, we have the following result.

Theorem 3.1

A mapping A:[0,1]n[0,1] is a k-maxitive aggregation

k-maxitive capacities and k-maxitive aggregation functions

As already mentioned in Introduction, 1-maxitive capacities, i.e., possibility measures, generate via integrals 1-maxitive aggregation functions. In this section we show that a similar result also holds in general for k-maxitive capacities.

We first recall that a binary aggregation function :[0,1]2[0,1] is called a semicopula [11] whenever e=1 is its neutral element, i.e., x1=1x=x for each x[0,1].

For any fixed semicopula ⊗ and capacity μ:2{1,,n}[0,1], the mapping I,μ:[0,1]n[0,1] given by

Concluding remarks

We have introduced and discussed k-maxitive aggregation functions which can be seen as monotone extensions of k-maxitive capacities. We have discussed aggregation functions and capacities related to the standard real unit interval [0,1] scale. Our results can also be applied in more general situations, e.g., if discrete or linguistic scales, or distributive bounded lattices are considered. In the latter case, one can consider the Sugeno integral [9] as an appropriate tool for construction of k

Acknowledgements

Both authors kindly acknowledge the support of the project of Science and Technology Assistance Agency under the contract No. APVV-14-0013. A. Kolesárová also acknowledges the support of the project VEGA 1/0614/18. The authors are grateful to Dr. Marek Gagolewski for his motivating idea and to all referees for their comments and proposals leading to improving the original version of the paper.

References (32)

  • N. Shilkret

    Maxitive measure and integration

    Indag. Math.

    (1971)
  • L.A. Zadeh

    Fuzzy sets as a basis for a theory of possibility

    Fuzzy Sets Syst.

    (1978)
  • G. Beliakov et al.

    Aggregation Functions: A Guide for Practitioners

    (2007)
  • G. Beliakov et al.

    A Practical Guide to Averaging Functions

    (2016)
  • Q. Brabant et al.

    Axiomatisation des integrales de Sugeno k-maxitives

  • T. Calvo et al.

    Aggregation operators defined by k-order additive/maxitive fuzzy measures

    Int. J. Uncertain. Fuzziness Knowl.-Based Syst.

    (1998)
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