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Density estimation on manifolds with boundary

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  • Berry, Tyrus
  • Sauer, Timothy

Abstract

Density estimation is a crucial component of many machine learning methods, and manifold learning in particular, where geometry is to be constructed from data alone. A significant practical limitation of the current density estimation literature is that methods have not been developed for manifolds with boundary, except in simple cases of linear manifolds where the location of the boundary is assumed to be known. This limitation is overcome by developing a density estimation method for manifolds with boundary that does not require any prior knowledge of the location of the boundary. To construct an appropriate estimator, statistics are introduced that provably approximate the distance and direction of the boundary, which allows us to apply a cut-and-normalize boundary correction. Then, multiple cut-and-normalize estimators are used to build a consistent kernel density estimator that has uniform bias, at interior and boundary points, on manifolds with boundary.

Suggested Citation

  • Berry, Tyrus & Sauer, Timothy, 2017. "Density estimation on manifolds with boundary," Computational Statistics & Data Analysis, Elsevier, vol. 107(C), pages 1-17.
    • Handle: RePEc:eee:csdana:v:107:y:2017:i:c:p:1-17
    DOI: 10.1016/j.csda.2016.09.011
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    References listed on IDEAS

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    1. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    2. Karunamuni, R.J. & Zhang, S., 2008. "Some improvements on a boundary corrected kernel density estimator," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 499-507, April.
    3. 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.
    4. Malec, Peter & Schienle, Melanie, 2014. "Nonparametric kernel density estimation near the boundary," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 57-76.
    5. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
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    Cited by:

    1. Paula Saavedra-Nieves & Rosa M. Crujeiras, 2022. "Nonparametric estimation of directional highest density regions," 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. 16(3), pages 761-796, September.
    2. Cholaquidis, Alejandro & Fraiman, Ricardo & Moreno, Leonardo, 2022. "Level set and density estimation on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Federico Ferraccioli & Eleonora Arnone & Livio Finos & James O. Ramsay & Laura M. Sangalli, 2021. "Nonparametric density estimation over complicated domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 346-368, April.
    4. 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).

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