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Nonparametric Measure-transportation-based Methods for Directional Data

Author

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  • Marc Hallin
  • H Lui
  • Thomas Verdebout

Abstract

This paper proposes various nonparametric tools for directional data based on measure transportation. We use optimal transports todefine new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closedformformulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They yield fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper) meridians. Based on this, we also propose a test of uniformity and establish its universal consistency; simulations indicate that this test outperforms the “projected” Cram´er-von Mises, Anderson-Darling, and Rothman procedures recently proposed in the literature. Two real-data examples involving the analysis of Venus craters and proteins structures conclude the paper.

Suggested Citation

  • Marc Hallin & H Lui & Thomas Verdebout, 2022. "Nonparametric Measure-transportation-based Methods for Directional Data," Working Papers ECARES 2022-18, ULB -- Universite Libre de Bruxelles.
    • Handle: RePEc:eca:wpaper:2013/344268
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    References listed on IDEAS

    as
    1. Marc Hallin & Gilles Mordant & Johan Segers, 2020. "Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance," Working Papers ECARES 2020-06, ULB -- Universite Libre de Bruxelles.
    2. Kanika, & Kumar, Somesh & SenGupta, Ashis, 2015. "A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 160-172.
    3. Eustasio del Barrio & Alberto González-Sanz & Marc Hallin, 2022. "Nonparametric Multiple-Output Center-Outward Quantile Regression," Working Papers ECARES 2022-10, ULB -- Universite Libre de Bruxelles.
    4. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    5. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    6. García-Portugués, Eduardo & Crujeiras, Rosa M. & González-Manteiga, Wenceslao, 2013. "Kernel density estimation for directional–linear data," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 152-175.
    7. Thomas Hamelryck & John T Kent & Anders Krogh, 2006. "Sampling Realistic Protein Conformations Using Local Structural Bias," PLOS Computational Biology, Public Library of Science, vol. 2(9), pages 1-13, September.
    8. Peter T. Kim & Ja-Yong Koo & Thanh Mai Pham Ngoc, 2016. "Supersmooth testing on the sphere over analytic classes," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 84-115, March.
    9. Christophe Ley & Yvik Swan & Thomas Verdebout, 2017. "Efficient ANOVA for directional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 39-62, February.
    10. Di Marzio, Marco & Fensore, Stefania & Panzera, Agnese & Taylor, Charles C., 2019. "Kernel density classification for spherical data," Statistics & Probability Letters, Elsevier, vol. 144(C), pages 23-29.
    11. Eduardo GarcÍa-Portugués & Ingrid Van Keilegom & Rosa M. Crujeiras and & Wenceslao González-Manteiga, 2016. "Testing parametric models in linear-directional regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1178-1191, December.
    12. Hongjian Shi & Mathias Drton & Fang Han, 2022. "Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(537), pages 395-410, January.
    13. Pham Ngoc, Thanh Mai, 2019. "Adaptive optimal kernel density estimation for directional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 248-267.
    14. Eduardo Garcia-Portugues & Davy Paindaveine & Thomas Verdebout, 2020. "On optimal tests for rotational symmetry against new classes of hyperspherical distributions," Post-Print hal-03169388, HAL.
    15. Garcia Portugues, Eduardo & Van Keilegom, Ingrid & Crujeiras and, Rosa M. & Gonzalez-Manteiga, Wenceslao, 2016. "Testing parametric models in linear-directional regression," LIDAM Reprints ISBA 2016046, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Aboubacar Amiri & Baba Thiam & Thomas Verdebout, 2017. "On the Estimation of the Density of a Directional Data Stream," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 249-267, March.
    17. Eduardo García-Portugués & Davy Paindaveine & Thomas Verdebout, 2020. "On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(532), pages 1873-1887, December.
    18. Christophe Ley & Camille Sabbah & Thomas Verdebout, 2014. "A new concept of quantiles for directional data and the angular Mahalanobis depth," Working Papers ECARES ECARES 2013-23, ULB -- Universite Libre de Bruxelles.
    19. Abe, Toshihiro & Ley, Christophe, 2017. "A tractable, parsimonious and flexible model for cylindrical data, with applications," Econometrics and Statistics, Elsevier, vol. 4(C), pages 91-104.
    20. Graciela Boente & Daniela Rodriguez & Wenceslao González Manteiga, 2014. "Goodness-of-fit Test for Directional Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 259-275, March.
    21. Mastrantonio, Gianluca, 2018. "The joint projected normal and skew-normal: A distribution for poly-cylindrical data," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 14-26.
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    Keywords

    directional statistics; quantile; ranks; optimal transport;
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