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Kernel density estimation on Riemannian manifolds

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  • Pelletier, Bruno

Abstract

The estimation of the underlying probability density of n i.i.d. random objects on a compact Riemannian manifold without boundary is considered. The proposed methodology adapts the technique of kernel density estimation on Euclidean sample spaces to this nonEuclidean setting. Under sufficient regularity assumptions on the underlying density, L2 convergence rates are obtained.

Suggested Citation

  • Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    • Handle: RePEc:eee:stapro:v:73:y:2005:i:3:p:297-304
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    References listed on IDEAS

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    1. Hendriks, Harrie, 2003. "Application of fast spherical Fourier transform to density estimation," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 209-221, February.
    2. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    3. Hendriks, H. & Janssen, J. H. M. & Ruymgaart, F. H., 1993. "Strong uniform convergence of density estimators on compact Euclidean manifolds," Statistics & Probability Letters, Elsevier, vol. 16(4), pages 305-311, March.
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    Cited by:

    1. 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.
    2. Asta, Dena Marie, 2021. "Kernel density estimation on symmetric spaces of non-compact type," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    3. Rabi Bhattacharya & Rachel Oliver, 2019. "Nonparametric Analysis of Non-Euclidean Data on Shapes and Images," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 1-36, February.
    4. Khardani, Salah & Yao, Anne Françoise, 2022. "Nonparametric recursive regression estimation on Riemannian Manifolds," Statistics & Probability Letters, Elsevier, vol. 182(C).
    5. Arnone, Eleonora & Ferraccioli, Federico & Pigolotti, Clara & Sangalli, Laura M., 2022. "A roughness penalty approach to estimate densities over two-dimensional manifolds," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    6. Berry, Tyrus & Sauer, Timothy, 2017. "Density estimation on manifolds with boundary," Computational Statistics & Data Analysis, Elsevier, vol. 107(C), pages 1-17.
    7. Ki, Dohyeong & Park, Byeong U., 2021. "Intrinsic Hölder classes of density functions on Riemannian manifolds and lower bounds to convergence rates," Statistics & Probability Letters, Elsevier, vol. 169(C).
    8. Kim, Yoon Tae & Park, Hyun Suk, 2013. "Geometric structures arising from kernel density estimation on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 112-126.
    9. Yiyi Huo & Yingying Fan & Fang Han, 2023. "On the adaptation of causal forests to manifold data," Papers 2311.16486, arXiv.org, revised Dec 2023.
    10. Loubes Jean-Michel & Pelletier Bruno, 2008. "A kernel-based classifier on a Riemannian manifold," Statistics & Risk Modeling, De Gruyter, vol. 26(1), pages 35-51, March.
    11. Guillermo Henry & Daniela Rodriguez, 2009. "Robust nonparametric regression on Riemannian manifolds," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(5), pages 611-628.
    12. Hall, Peter & Yatchew, Adonis, 2010. "Nonparametric least squares estimation in derivative families," Journal of Econometrics, Elsevier, vol. 157(2), pages 362-374, August.
    13. Lizhen Lin & Brian St. Thomas & Hongtu Zhu & David B. Dunson, 2017. "Extrinsic Local Regression on Manifold-Valued Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1261-1273, July.
    14. Cheongjae Jang & Yung-Kyun Noh & Frank Chongwoo Park, 2021. "A Riemannian geometric framework for manifold learning of non-Euclidean data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 673-699, September.
    15. Hielscher, Ralf, 2013. "Kernel density estimation on the rotation group and its application to crystallographic texture analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 119-143.
    16. Ouimet, Frédéric, 2022. "A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    17. Mu Niu & Pokman Cheung & Lizhen Lin & Zhenwen Dai & Neil Lawrence & David Dunson, 2019. "Intrinsic Gaussian processes on complex constrained domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(3), pages 603-627, July.
    18. Hielscher, Ralf & Lippert, Laura, 2021. "Locally isometric embeddings of quotients of the rotation group modulo finite symmetries," Journal of Multivariate Analysis, Elsevier, vol. 185(C).

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    More about this item

    Keywords

    Nonparametric density estimation Kernel density estimation Riemannian manifolds L2 convergence;

    JEL classification:

    • L2 - Industrial Organization - - Firm Objectives, Organization, and Behavior

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