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Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis

Author

Listed:
  • Ruite Guo

    (Florida State University)

  • Hwiyoung Lee

    (University of Maryland)

  • Vic Patrangenaru

    (Florida State University)

Abstract

Methods of testing for the equality of two distributions on a manifold are unveiled in this paper. One defines the extrinsic energy distance associated with two probability measures on a complete metric space embedded in a numerical space. One derives the extrinsic energy statistic test for homogeneity of such distributions. This test is validated via a simulation example on the Kendall space of planar k-ads with a Veronese-Whitney (VW) embedding. Imaging data driven examples are also considered here. In one application, central to the paper, one tests for homogeneity the distributions of planar Kendall shapes of midsections of the Corpus Callosum in a clinically normal population vs a population of ADHD diagnosed individuals; these distributions are not significantly different, although they are known to have highly significant VW-means. On the other hand, in 3D, the reflection shapes of configurations of Acrosterigma Magnum shells are not significantly different, and do not have significantly similar different 3D Schoenberg means.

Suggested Citation

  • Ruite Guo & Hwiyoung Lee & Vic Patrangenaru, 2023. "Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1178-1204, August.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:2:d:10.1007_s13171-023-00310-0
    DOI: 10.1007/s13171-023-00310-0
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    References listed on IDEAS

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    1. Ellingson, Leif & Patrangenaru, Vic & Ruymgaart, Frits, 2013. "Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 317-333.
    2. Bandulasiri, Ananda & Bhattacharya, Rabi N. & Patrangenaru, Vic, 2009. "Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1867-1882, October.
    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. Stephan Huckemann, 2012. "On the meaning of mean shape: manifold stability, locus and the two sample test," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1227-1259, December.
    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).
    6. Victor Patrangenaru & Robert Paige & K. David Yao & Mingfei Qiu & David Lester, 2016. "Projective shape analysis of contours and finite 3D configurations from digital camera images," Statistical Papers, Springer, vol. 57(4), pages 1017-1040, December.
    7. Chao Huang & Martin Styner & Hongtu Zhu, 2015. "Clustering High-Dimensional Landmark-Based Two-Dimensional Shape Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 946-961, September.
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