An inherent difficulty in the aggregation of multidimensional data
Authors:
- Marek Gągolewski,
- Raul Perez-Fernandez,
- B De Baets
Abstract
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.
- Record ID
- WUT6190add233354aae9f3627139f97b92f
- Author
- Journal series
- IEEE Transactions on Fuzzy Systems, ISSN 1063-6706, e-ISSN 1941-0034
- Issue year
- 2020
- Vol
- 28
- No
- 3
- Pages
- 602-606
- Publication size in sheets
- 0.50
- Keywords in Polish
- agregacja danych, centroid, monotoniczność, niezmienniczość zw. na przekształcenia ortogonalne
- Keywords in English
- multidimensional data aggregation, monotonicity, orthogonal equivariance, centroid
- ASJC Classification
- ; ; ;
- Abstract in Polish
- Teoria 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.
- DOI
- DOI:10.1109/TFUZZ.2019.2908135 Opening in a new tab
- URL
- https://ieeexplore.ieee.org/document/8675953 Opening in a new tab
- Language
- (en) English
- Score (nominal)
- 200
- Score source
- journalList
- Score
- = 200.0, 19-05-2022, ArticleFromJournal
- Publication indicators
- = 10; = 2; : 2018 = 3.314; : 2020 (2 years) = 12.029 - 2020 (5 years) =10.444
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- Uniform Resource Identifier
- https://repo.pw.edu.pl/info/article/WUT6190add233354aae9f3627139f97b92f/
- URN
urn:pw-repo:WUT6190add233354aae9f3627139f97b92f
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or PerishOpening in a new tab system.