k-maxitive aggregation functions
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 ) 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, , 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 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 . Throughout the paper, the points in will be denoted by bold letters, , in particular, and . Given , we write if for each .
Definition 2.1 A mapping is called an (n-ary) aggregation function if it satisfies the boundary conditions , , and the monotonicity condition whenever , . We say that A is maxitive if for all .
It is easy to see that each can be
Properties and representations of k-maxitive aggregation functions
For any aggregation function and any non-empty subset one can define a mapping by with , . Then satisfies the boundary condition and it is also monotone. Similarly, for any , let .
As a generalization of the representation (1) of maxitive aggregation functions, we have the following result.
Theorem 3.1 A mapping 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 is called a semicopula [11] whenever is its neutral element, i.e., for each .
For any fixed semicopula ⊗ and capacity , the mapping 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 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)
- et al.
k-maxitive Sugeno integrals as aggregation models for ordinal preferences
Fuzzy Sets Syst.
(2018) - et al.
Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices
Fuzzy Sets Syst.
(2010) k-order additive discrete fuzzy measures and their representation
Fuzzy Sets Syst.
(1997)- et al.
The Choquet integral with respect to a level dependent capacity
Fuzzy Sets Syst.
(2011) - et al.
Congruences and the discrete Sugeno integrals on bounded distributive lattices
Inf. Sci.
(2016) - et al.
Triangular norm-based iterative compensatory operators
Fuzzy Sets Syst.
(1999) - et al.
Weighted ordinal means
Inf. Sci.
(2007) - et al.
k-additive aggregation functions and their characterizations
Eur. J. Oper. Res.
(2018) - et al.
OWA operators defined on complete lattices
Fuzzy Sets Syst.
(2013) On Sugeno integral as an aggregation function
Fuzzy Sets Syst.
(2000)
Maxitive measure and integration
Indag. Math.
Fuzzy sets as a basis for a theory of possibility
Fuzzy Sets Syst.
Aggregation Functions: A Guide for Practitioners
A Practical Guide to Averaging Functions
Axiomatisation des integrales de Sugeno k-maxitives
Aggregation operators defined by k-order additive/maxitive fuzzy measures
Int. J. Uncertain. Fuzziness Knowl.-Based Syst.
Cited by (19)
Representation, optimization and generation of fuzzy measures
2024, Information FusionRandom generation of linearly constrained fuzzy measures and domain coverage performance evaluation
2024, Information SciencesOn comonotone k-maxitive aggregation functions
2023, Fuzzy Sets and SystemsCitation Excerpt :It was shown in [11] that the class of all comonotone maxitive aggregation functions coincides with the class of the so-called level-dependent Sugeno integrals. Another generalization of maxitivity was recently studied in [10], where the so-called k-maxitive aggregation functions were introduced. They extend the notion of maxitive aggregation functions in the similar way as k-maxitive fuzzy measures extend the notion of maxitive fuzzy measures (see, e.g. [2,6]).
Aggregation with dependencies: Capacities and fuzzy integrals
2022, Fuzzy Sets and SystemsCitation Excerpt :In general it consists in restricting interactions to subsets of size at most k, and avoiding or pre-setting interactions in larger subsets. The k-additive, k-maxitive/minitive and p-symmetric capacities [37,54,59,79] are the main approaches. For instance the k-additive capacities are based on setting the Möbius coefficients (and consequently the Shapley interaction indices) to zero whenever the cardinality of a subset is larger than k.
Random generation of k-interactive capacities
2022, Fuzzy Sets and SystemsCitation Excerpt :Decision models based on capacities are sophisticated but suffer from exponential (in terms of the number of criteria n) complexity, which stems from the size of power sets. Simplifications to capacities which limit criteria interactions to subsets of small cardinality include k-additive [15], k-maxitive/minitive [24,34], k-tolerant/intolerant [23], p-symmetric [26] and k-interactive capacities [4]. These simplifications keep some criteria interactions but reduce significantly the computational complexity.