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An inherent difficulty in the aggregation of multidimensional data

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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.
Keywords
Control and Systems Engineering, Computational Theory and Mathematics, Applied Mathematics, Artificial Intelligence, Aggregates, Data aggregation, Data analysis, Geometry, Data integration, Task analysis, Centroid, monotonicity, multidimensional data aggregation, orthogonal equivariance

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MLA
Gagolewski, Marek, et al. “An Inherent Difficulty in the Aggregation of Multidimensional Data.” IEEE TRANSACTIONS ON FUZZY SYSTEMS, vol. 28, no. 3, 2020, pp. 602–06, doi:10.1109/tfuzz.2019.2908135.
APA
Gagolewski, M., Perez Fernandez, R., & De Baets, B. (2020). An inherent difficulty in the aggregation of multidimensional data. IEEE TRANSACTIONS ON FUZZY SYSTEMS, 28(3), 602–606. https://doi.org/10.1109/tfuzz.2019.2908135
Chicago author-date
Gagolewski, Marek, Raul Perez Fernandez, and Bernard De Baets. 2020. “An Inherent Difficulty in the Aggregation of Multidimensional Data.” IEEE TRANSACTIONS ON FUZZY SYSTEMS 28 (3): 602–6. https://doi.org/10.1109/tfuzz.2019.2908135.
Chicago author-date (all authors)
Gagolewski, Marek, Raul Perez Fernandez, and Bernard De Baets. 2020. “An Inherent Difficulty in the Aggregation of Multidimensional Data.” IEEE TRANSACTIONS ON FUZZY SYSTEMS 28 (3): 602–606. doi:10.1109/tfuzz.2019.2908135.
Vancouver
1.
Gagolewski M, Perez Fernandez R, De Baets B. An inherent difficulty in the aggregation of multidimensional data. IEEE TRANSACTIONS ON FUZZY SYSTEMS. 2020;28(3):602–6.
IEEE
[1]
M. Gagolewski, R. Perez Fernandez, and B. De Baets, “An inherent difficulty in the aggregation of multidimensional data,” IEEE TRANSACTIONS ON FUZZY SYSTEMS, vol. 28, no. 3, pp. 602–606, 2020.
@article{8655824,
  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.}},
  author       = {{Gagolewski, Marek and Perez Fernandez, Raul and De Baets, Bernard}},
  issn         = {{1063-6706}},
  journal      = {{IEEE TRANSACTIONS ON FUZZY SYSTEMS}},
  keywords     = {{Control and Systems Engineering,Computational Theory and Mathematics,Applied Mathematics,Artificial Intelligence,Aggregates,Data aggregation,Data analysis,Geometry,Data integration,Task analysis,Centroid,monotonicity,multidimensional data aggregation,orthogonal equivariance}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{602--606}},
  title        = {{An inherent difficulty in the aggregation of multidimensional data}},
  url          = {{http://dx.doi.org/10.1109/tfuzz.2019.2908135}},
  volume       = {{28}},
  year         = {{2020}},
}

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