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Extrinsic Regression and Anti-Regression on Projective Shape Manifolds

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  • Vic Patrangenaru

    (Florida State University)

  • Yifang Deng

    (Florida State University)

Abstract

Necessary and sufficient conditions for the existence of the extrinsic mean and extrinsic antimean of a random object (r.o.) X on a compact metric space ℳ , $\mathcal M,$ lead to considerations of extrinsic regression and antiregression functions on manifolds. One derives asymptotic distributions of kernel based estimators for antiregression functions with a numerical predictor, and use these in deriving confidence tubes for antiregression functions. In particular one considers VW-regression and VW-antiregression, for 3D projective shapes depending on a number of covariates. As an example, using 3D projective shape data extracted from multiple digital cameras images of a species of clamshells, one estimates the age dependent VW-regression and VW-antiregression for their 3D projective shapes, where the proxy for age is the number of seasonal ridges marks on shells, and the response is the 3D projective shape of landmark configurations of seven corresponding points marked on a shell’s surface.

Suggested Citation

  • Vic Patrangenaru & Yifang Deng, 2021. "Extrinsic Regression and Anti-Regression on Projective Shape Manifolds," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 629-646, June.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:2:d:10.1007_s11009-020-09789-8
    DOI: 10.1007/s11009-020-09789-8
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    References listed on IDEAS

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    1. Vic Patrangenaru & Mingfei Qiu & Marius Buibas, 2014. "Two Sample Tests for Mean 3D Projective Shapes from Digital Camera Images," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 485-506, June.
    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. Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
    4. R. N. Bhattacharya & L. Ellingson & X. Liu & V. Patrangenaru & M. Crane, 2012. "Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machine vision," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 28(3), pages 222-235, May.
    5. Wang, Yunfan & Patrangenaru, Vic & Guo, Ruite, 2020. "A Central Limit Theorem for extrinsic antimeans and estimation of Veronese–Whitney means and antimeans on planar Kendall shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
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