An inherent difficulty in the aggregation of multidimensional data

Marek Gągolewski , Raul Perez-Fernandez , B 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 behavior with regard to orthogonal transformations.

Author Marek Gągolewski (FMIS / DIE)
Marek Gągolewski,,
- Department of Integral Equations
, Raul Perez-Fernandez - [Universiteit Gent]
Raul Perez-Fernandez,,
, B De Baets
B De Baets,,
Journal seriesIEEE Transactions on Fuzzy Systems, ISSN 1063-6706, e-ISSN 1941-0034
Issue year2020
Publication size in sheets0.5
Keywords in Polishagregacja danych, centroid, monotoniczność, niezmienniczość zw. na przekształcenia ortogonalne
Keywords in Englishmultidimensional data aggregation, monotonicity, orthogonal equivariance, centroid
ASJC Classification1702 Artificial Intelligence; 1703 Computational Theory and Mathematics; 2207 Control and Systems Engineering; 2604 Applied Mathematics
Abstract in PolishTeoria agregacji danych zbudowana jest wokół założenia, że wyprowadzane przez nią metody powinny zachowywać pewną relację porządku, zwłaszcza w przypadku podsumowywania ciągów złożonych z liczb rzeczywistych. W niniejszej pracy wykazano, że w przypadku agregacji danych wielowymiarowych podejście klasyczne jest zbyt restrykcyjne. Mianowicie udowodniono, że jedynymi idempotentnymi, monotonicznymi i niezmienniczymi względem przekształceń ortogonalnych funkcjami są ważone centroidy.
Languageen angielski
Score (nominal)200
Score sourcejournalList
ScoreMinisterial score = 200.0, 08-07-2020, ArticleFromJournal
Publication indicators Scopus Citations = 1; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 3.314; WoS Impact Factor: 2018 = 8.759 (2) - 2018=9.438 (5)
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