# Stability in Aggregation Operators

Abstract

Aggregation functions have been widely studied in literature. Nevertheless, few efforts have been dedicated to analyze those properties related with the family of operators in a global way. In this work, we analyze the stability in a family of aggregation operators The stability property for a family of aggregation operators tries to force a family to have a stable/continuous definition in the sense that the aggregation of n − 1 items should be similar to the aggregation of n items if the last item is the aggregation of the previous n − 1 items. Following this idea some definitions and results are given.

Description

Many properties have been studied in relation with the aggregation operatorfunctions such as continuity, commutativity, monotonicity, associativity. But in contrast, few efforts have been dedicated to research the relations among the members of a family of aggregation operators. As has been pointed recently, these common properties (as for example continuity) show us some desirable characteristics related with each aggregation function An, but do not give us any information about the consistency of the family of aggregation operators in the sense of the relations that should exist among its members. In the context of aggregation operators, it is usually assumed that the information that has to be aggregated is given in terms of a vector of elements in the unit interval, assigning to each vector another number in the unit interval, which constitutes the aggregated value of the original information. Taking into account that, in practice, most of the time one cannot guarantee that the cardinal of the information is going to be fixed (some information can get lost or deleted due to errors in observation or transmission, or because sometimes one gets some additional information not previously taken into account), we need to be able to solve each aggregation problem without knowing a priori the cardinal of data. Trying to eliminate the classical assumption that considers the aggregations functions as independent pieces of the aggregation process, we will define here the idea of stability in a family of aggregation functions {An} (from now on FAO) breaking this idea of independence that usually is assumed. This is, the operators that compose a FAO have to be somehow related so the aggregation process remains the same throughout the dimension n of the data. Therefore, it seems logical to study properties giving sense to the sequences A(2), A(3), A(4), . . . ,. Otherwise we may have only a bunch of disconnected operators. In our opinion an aggregation family should never be understood just as a family of n-ary operators. Rather, all these aggregation operatorsmust be deeply related following some building procedure throughout the aggregation process. To this aim, we have presented here two properties that follows such an objective. It is clear that we should not define a family of aggregation operators {An} in which A2 is the mean, A3 geometric mean, A4 is the minimum. Thus, in our opinion the aggregation process demands a conceptual unit idea rather than a mathematical formula. The stability notion proposed in this paper makes emphasize in the idea of robustness-stability-continuity of the family in the sense that the operator defined for n data items should not differ too much of the operator defined for n − 1 elements. Another aspect that should be considered in the aggregation process is related with the structure of the data. The notion of consistency in the relation among the aggregation functions is not trivial and could depend on the structure of the data. A possible definition of consistency in the framework of recursive rules is done. For more general situations, we present a mechanism that permits us to build the aggregation function taking into account the structure of the data that has to be aggregated. Nevertheless, the definition proposed here is just a seminal effort and possible modifications coming from a further analysis merit to be carried out.

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