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An inherent difficulty in the aggregation of multidimensional data

journal contribution
posted on 2020-03-01, 00:00 authored by Marek Gagolewski, Raul Perez-Fernandez, Bernard De Baets
In the field of information fusion, the problem of data aggregation has been formalized as an order-preserving process that builds upon the property of monotonicity. However, fields such as computational statistics, data analysis and geometry, usually emphasize the role of equivariances to various geometrical transformations in aggregation processes. Admittedly, if we consider a unidimensional data fusion task, both requirements are often compatible with each other. Nevertheless, in this paper we show that, in the multidimensional setting, the only idempotent functions that are monotone and orthogonal equivariant are the over-simplistic weighted centroids. Even more, this result still holds after replacing monotonicity and orthogonal equivariance by the weaker property of orthomonotonicity. This implies that the aforementioned approaches to the aggregation of multidimensional data are irreconcilable, and that, if a weighted centroid is to be avoided, we must choose between monotonicity and a desirable behaviour with regard to orthogonal transformations.

History

Journal

IEEE Transactions on Fuzzy Systems

Volume

28

Pagination

602-606

Location

Piscataway, N.J.

ISSN

1063-6706

eISSN

1941-0034

Language

eng

Publication classification

C1.1 Refereed article in a scholarly journal

Issue

3

Publisher

IEEE