Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications

Morgane Goibert, Stephan Clemencon, Ekhine Irurozki, Pavlo Mozharovskyi
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:10376-10406, 2022.

Abstract

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,; \ldots,;{n}\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$’s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogues of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-goibert22a, title = { Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications }, author = {Goibert, Morgane and Clemencon, Stephan and Irurozki, Ekhine and Mozharovskyi, Pavlo}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {10376--10406}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/goibert22a/goibert22a.pdf}, url = {https://proceedings.mlr.press/v151/goibert22a.html}, abstract = { The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,; \ldots,;{n}\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$’s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogues of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments. } }
Endnote
%0 Conference Paper %T Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications %A Morgane Goibert %A Stephan Clemencon %A Ekhine Irurozki %A Pavlo Mozharovskyi %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-goibert22a %I PMLR %P 10376--10406 %U https://proceedings.mlr.press/v151/goibert22a.html %V 151 %X The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,; \ldots,;{n}\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$’s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogues of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.
APA
Goibert, M., Clemencon, S., Irurozki, E. & Mozharovskyi, P.. (2022). Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:10376-10406 Available from https://proceedings.mlr.press/v151/goibert22a.html.

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