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Analysis of principal nested spheres

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  • Sungkyu Jung
  • Ian L. Dryden
  • J. S. Marron

Abstract

A general framework for a novel non-geodesic decomposition of high-dimensional spheres or high-dimensional shape spaces for planar landmarks is discussed. The decomposition, principal nested spheres, leads to a sequence of submanifolds with decreasing intrinsic dimensions, which can be interpreted as an analogue of principal component analysis. In a number of real datasets, an apparent one-dimensional mode of variation curving through more than one geodesic component is captured in the one-dimensional component of principal nested spheres. While analysis of principal nested spheres provides an intuitive and flexible decomposition of the high-dimensional sphere, an interesting special case of the analysis results in finding principal geodesics, similar to those from previous approaches to manifold principal component analysis. An adaptation of our method to Kendall's shape space is discussed, and a computational algorithm for fitting principal nested spheres is proposed. The result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation, as exemplified by several datasets. Copyright 2012, Oxford University Press.

Suggested Citation

  • Sungkyu Jung & Ian L. Dryden & J. S. Marron, 2012. "Analysis of principal nested spheres," Biometrika, Biometrika Trust, vol. 99(3), pages 551-568.
    • Handle: RePEc:oup:biomet:v:99:y:2012:i:3:p:551-568
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    File URL: http://hdl.handle.net/10.1093/biomet/ass022
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    Cited by:

    1. Georgios I. Papayiannis & Stelios Psarakis & Athanasios N. Yannacopoulos, 2023. "Modelling of Functional Profiles and Explainable Shape Shifts Detection: An Approach Combining the Notion of the Fréchet Mean with the Shape-Invariant Model," Mathematics, MDPI, vol. 11(21), pages 1-24, October.
    2. Tucker, J. Derek & Wu, Wei & Srivastava, Anuj, 2013. "Generative models for functional data using phase and amplitude separation," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 50-66.
    3. Stefan Sommer, 2019. "An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 37-62, February.
    4. Ley, Christophe & Verdebout, Thomas, 2017. "Skew-rotationally-symmetric distributions and related efficient inferential procedures," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 67-81.
    5. Arthur Pewsey & Eduardo García-Portugués, 2021. "Rejoinder on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 76-82, March.
    6. Kim, Byungwon & Schulz, Jörn & Jung, Sungkyu, 2020. "Kurtosis test of modality for rotationally symmetric distributions on hyperspheres," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    7. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    8. Ian L. Dryden & Kwang-Rae Kim & Huiling Le, 2019. "Bayesian Linear Size-and-Shape Regression with Applications to Face Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 83-103, February.
    9. Jupp, P.E. & Kume, A., 2020. "Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    10. J. Derek Tucker & Drew Yarger, 2024. "Elastic functional changepoint detection of climate impacts from localized sources," Environmetrics, John Wiley & Sons, Ltd., vol. 35(1), February.
    11. Mardia, Kanti V. & Wiechers, Henrik & Eltzner, Benjamin & Huckemann, Stephan F., 2022. "Principal component analysis and clustering on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    12. Stephan F. Huckemann, 2021. "Comments on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 71-75, March.
    13. Lazar, Drew & Lin, Lizhen, 2017. "Scale and curvature effects in principal geodesic analysis," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 64-82.

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