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Smoothing splines on Riemannian manifolds, with applications to 3D shape space

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  • Kwang‐Rae Kim
  • Ian L. Dryden
  • Huiling Le
  • Katie E. Severn

Abstract

There has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling, unwrapping and wrapping originally proposed by Jupp and Kent for spherical data. In particular, we develop such a fitting procedure for shapes of configurations in general m‐dimensional Euclidean space, extending our previous work for two‐dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first‐order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data.

Suggested Citation

  • Kwang‐Rae Kim & Ian L. Dryden & Huiling Le & Katie E. Severn, 2021. "Smoothing splines on Riemannian manifolds, with applications to 3D shape space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 108-132, February.
    • Handle: RePEc:bla:jorssb:v:83:y:2021:i:1:p:108-132
    DOI: 10.1111/rssb.12402
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    References listed on IDEAS

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    1. Alfred Kume & Ian L. Dryden & Huiling Le, 2007. "Shape-space smoothing splines for planar landmark data," Biometrika, Biometrika Trust, vol. 94(3), pages 513-528.
    2. 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.
    3. Emil Cornea & Hongtu Zhu & Peter Kim & Joseph G. Ibrahim, 2017. "Regression models on Riemannian symmetric spaces," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 463-482, March.
    4. Zhu, Hongtu & Chen, Yasheng & Ibrahim, Joseph G. & Li, Yimei & Hall, Colin & Lin, Weili, 2009. "Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1203-1212.
    5. Peter E. Jupp & John T. Kent, 1987. "Fitting Smooth Paths to Spherical Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(1), pages 34-46, March.
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    Cited by:

    1. Valerio Varano & Stefano Gabriele & Franco Milicchio & Stefan Shlager & Ian Dryden & Paolo Piras, 2022. "Geodesics in the TPS Space," Mathematics, MDPI, vol. 10(9), pages 1-20, May.
    2. García-Morales, Vladimir, 2021. "A constructive theory of shape," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).

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