Orness For Idempotent Aggregation Functions

Leire Legarreta,Inmaculada Lizasoain,Iraide Mardones-Pérez

doi:10.3390/axioms6030025

Leire Legarreta, Inmaculada Lizasoain + Show 1 more

Open Access

https://doi.org/10.3390/axioms6030025

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Abstract

Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the supremum of the given data, which is guaranteed only when the aggregation functions are idempotent. Ordered weighted averaging (OWA) operators are particular cases of this kind of function, with the particularity that the obtained global value depends on neither the source nor the expert that provides each datum, but only on the set of values. They have been classified by means of the orness—a measurement of the proximity of an OWA operator to the OR-operator. In this paper, the concept of orness is extended to the framework of idempotent aggregation functions defined both on the real unit interval and on a complete lattice with a local finiteness condition.

Highlights

Aggregation functions [1] are a family of operators that allow us to fuse data—either quantitative or qualitative—in order to obtain a global value that captures the information of the given data as faithfully as possible.These functions have been widely used in decision making [2,3], where it is necessary to merge the different opinions of several experts, or in image processing tasks [4], where the values of different pixels must be fused in order to obtain a single one.OWA (Ordered weighted averaging) operators, defined by Yager in [5] for real values, are a kind of aggregation function that include the most widely used ones, such as the arithmetic mean or the order statistics
We have extended the definition of orness only to the class of idempotent aggregation functions, which are exactly those lying between the AND and the OR-operators
Consider the discrete Sugeno integral Sμ : L3 → L defined by means of the L-fuzzy measure μ : P3 → L with μ({1}) = μ({3}) = 0 L ; μ({2}) = a; μ({i, j}) = a, 1 ≤ i < j ≤ 3; μ({1, 2, 3}) = 1 L, Even if Sμ is not an OWA operator, we can compute its quantitative orness with the new definition we have just given

Summary

Introduction

Aggregation functions [1] are a family of operators that allow us to fuse data—either quantitative or qualitative—in order to obtain a global value that captures the information of the given data as faithfully as possible. The present paper analyzes the possibility of extending both concepts of orness to the wider set of all the aggregation functions lying between the AND and the OR-operators, for which the measurement of their proximity to the OR-operator has sense. The extension of the concept of quantitative orness to this kind of function is analyzed in Section 5 where they are defined on lattices with a local finiteness condition.

Preliminaries

Orness for Idempotent Aggregation Functions Defined on the Real Unit Interval

Quantitative Orness for Idempotent Aggregation Functions

Conclusions