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Statistical Modeling of Curves Using Shapes and Related Features

Author

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  • Sebastian Kurtek
  • Anuj Srivastava
  • Eric Klassen
  • Zhaohua Ding

Abstract

Motivated by the problems of analyzing protein backbones, diffusion tensor magnetic resonance imaging (DT-MRI) fiber tracts in the human brain, and other problems involving curves, in this study we present some statistical models of parameterized curves, in , in terms of combinations of features such as shape, location, scale, and orientation. For each combination of interest, we identify a representation manifold, endow it with a Riemannian metric, and outline tools for computing sample statistics on these manifolds. An important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized. The nuisance variables, including parameterization, are removed by forming quotient spaces under appropriate group actions. In the case of shape analysis, the resulting spaces are quotient spaces of Hilbert spheres, and we derive certain wrapped truncated normal densities for capturing variability in observed curves. We demonstrate these models using both artificial data and real data involving DT-MRI fiber tracts from multiple subjects and protein backbones from the Shape Retrieval Contest of Non-rigid 3D Models (SHREC) 2010 database.

Suggested Citation

  • Sebastian Kurtek & Anuj Srivastava & Eric Klassen & Zhaohua Ding, 2012. "Statistical Modeling of Curves Using Shapes and Related Features," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1152-1165, September.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:499:p:1152-1165
    DOI: 10.1080/01621459.2012.699770
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    Citations

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    Cited by:

    1. Karthik Bharath & Sebastian Kurtek & Arvind Rao & Veerabhadran Baladandayuthapani, 2018. "Radiologic image‐based statistical shape analysis of brain tumours," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 67(5), pages 1357-1378, November.
    2. Jorge R. Sosa Donoso & Miguel Flores & Salvador Naya & Javier Tarrío-Saavedra, 2023. "Local Correlation Integral Approach for Anomaly Detection Using Functional Data," Mathematics, MDPI, vol. 11(4), pages 1-18, February.
    3. Irene Epifanio & Vicent Gimeno & Ximo Gual-Arnau & M. Victoria Ibáñez-Gual, 2020. "A New Geometric Metric in the Shape and Size Space of Curves in R n," Mathematics, MDPI, vol. 8(10), pages 1-19, October.
    4. Juhyun Park & Jeongyoun Ahn, 2017. "Clustering multivariate functional data with phase variation," Biometrics, The International Biometric Society, vol. 73(1), pages 324-333, March.
    5. 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.
    6. Derek Tucker, J. & Shand, Lyndsay & Chowdhary, Kenny, 2021. "Multimodal Bayesian registration of noisy functions using Hamiltonian Monte Carlo," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).
    7. Niels Lundtorp Olsen & Bo Markussen & Lars Lau Raket, 2018. "Simultaneous inference for misaligned multivariate functional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 67(5), pages 1147-1176, November.
    8. Justin Strait & Sebastian Kurtek & Emily Bartha & Steven N. MacEachern, 2017. "Landmark-Constrained Elastic Shape Analysis of Planar Curves," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 521-533, April.
    9. Kelvin Gu & Debdeep Pati & David B. Dunson, 2014. "Bayesian Multiscale Modeling of Closed Curves in Point Clouds," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(508), pages 1481-1494, December.

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