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Principal manifold estimation via model complexity selection

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  • Kun Meng
  • Ani Eloyan

Abstract

We propose a framework of principal manifolds to model high‐dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve algorithm as special cases. We propose a novel method for model complexity selection to avoid overfitting, eliminate the effects of outliers and improve the computation speed. Additionally, we propose a method for identifying the interiors of circle‐like curves and cylinder/ball‐like surfaces. The proposed approach is compared to existing methods by simulations and applied to estimate tumour surfaces and interiors in a lung cancer study.

Suggested Citation

  • Kun Meng & Ani Eloyan, 2021. "Principal manifold estimation via model complexity selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(2), pages 369-394, April.
    • Handle: RePEc:bla:jorssb:v:83:y:2021:i:2:p:369-394
    DOI: 10.1111/rssb.12416
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    References listed on IDEAS

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    1. Chen Yue & Vadim Zipunnikov & Pierre-Louis Bazin & Dzung Pham & Daniel Reich & Ciprian Crainiceanu & Brian Caffo, 2016. "Parameterization of White Matter Manifold-Like Structures Using Principal Surfaces," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1050-1060, July.
    2. Ani Eloyan & Sujit Ghosh, 2011. "Smooth density estimation with moment constraints using mixture distributions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 513-531.
    3. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
    4. Roger Koenker & Ivan Mizera, 2004. "Penalized triograms: total variation regularization for bivariate smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 145-163, February.
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    Cited by:

    1. Liwen Wang & Yongda Wang & Shifeng Xiong & Jiankui Yang, 2025. "Some aspects of nonlinear dimensionality reduction," Computational Statistics, Springer, vol. 40(2), pages 883-906, February.

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