Consistency and stability in aggregation operators: an application to missing data problems
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Fecha
2013
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Aggregation Functions in Theory and in Practise: Proceedings of the 7th International Summer School on Aggregation Operators at the Public University of Navarra, Pamplona, Spain, July 16-20, 2013
Resumen
An aggregation operator is usually defined as a real function A n such that, from n data items x 1, …, x n in [0,1], produces an aggregated value A n (x 1,…,x n ) in [0,1]. This definition can be extended to consider the whole family of operators for any n instead of a single operator for an specific n. This has led to the current standard definition [4, 15] of a family of aggregation operators (FAO) as a set {A n :[0,1]n → [0,1],n ∈ N}, providing instructions on how to aggregate collections of items of any dimension n. This sequence of aggregation functions {A n } n ∈ N is also called extended aggregation functions (EAF) by other authors.
Descripción
In this work, we will deal with two different but related problems for extended aggregation functions or family of aggregation operators. On one hand, let us remark that in practice, it is frequent that some information can get lost, be deleted or added, and each time a cardinality change occurs a new aggregation operator Am has to be used to aggregate the new collection of m elements. However, it is important to remark that a relation between {An} and {Am} does not necessarily exist in a family of aggregation operators as defined in [4]. In this context, it seems natural to incorporate some properties to maintain the logical consistency between operators in a FAO when changes on the cardinality of the data occur, for which we need to be able to build up a definition of family of aggregation operators in terms of its logical consistency, and solve each problem of aggregation without knowing apriori the cardinality of the data. This is, the operators that compose a FAO have to be somehow related, so the aggregation process remains the same throughout the possible changes in the dimension n of the data. Therefore, it seems logical to study properties giving sense to the sequences A(2), A(3), A(4),..., because otherwise we may have only a bunch of disconnected operators.
In this sense, the notion of stability for a family of aggregation operators is inspired in continuity, though our approach focuses in the cardinality of the data rather than in the data itself, so we can assure some robustness in the result of the aggregation process. On the other hand, a problem that has not been received too much attention is how to obtain an aggregation when some of the variables to be aggregated are missing. If the aggregation operator function An present a clear definition for the case in which the dimension is lower, this problem is easily solved, but not always is a trivial task. Following the ideas of stability, in this paper we will deal with the problem of missing data for some well-known families of aggregation operators.
Citación
Gomez, D., Rojas, K., Montero, J., Rodríguez, J.T. (2013). Consistency and Stability in Aggregation Operators: An Application to Missing Data Problems. In: Bustince, H., Fernandez, J., Mesiar, R., Calvo, T. (eds) Aggregation Functions in Theory and in Practise. Advances in Intelligent Systems and Computing, vol 228. Springer, Berlin, Heidelberg.