IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v64y2012i6p1227-1259.html
   My bibliography  Save this article

On the meaning of mean shape: manifold stability, locus and the two sample test

Author

Listed:
  • Stephan Huckemann

Abstract

Various concepts of mean shape previously unrelated in the literature are brought into relation. In particular, for non-manifolds, such as Kendall’s 3D shape space, this paper answers the question, for which means one may apply a two-sample test. The answer is positive if intrinsic or Ziezold means are used. The underlying general result of manifold stability of a mean on a shape space, the quotient due to an proper and isometric action of a Lie group on a Riemannian manifold, blends the slice theorem from differential geometry with the statistics of shape. For 3D Procrustes means, however, a counterexample is given. To further elucidate on subtleties of means, for spheres and Kendall’s shape spaces, a first-order relationship between intrinsic, residual/Procrustean and extrinsic/Ziezold means is derived stating that for high concentration the latter approximately divides the (generalized) geodesic segment between the former two by the ratio 1:3. This fact, consequences of coordinate choices for the power of tests and other details, e.g. that extrinsic Schoenberg means may increase dimension are discussed and illustrated by simulations and exemplary datasets. Copyright The Institute of Statistical Mathematics, Tokyo 2012

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:64:y:2012:i:6:p:1227-1259
    DOI: 10.1007/s10463-012-0352-2
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10463-012-0352-2
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10463-012-0352-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hendriks, Harrie & Landsman, Zinoviy & Ruymgaart, Frits, 1996. "Asymptotic Behavior of Sample Mean Direction for Spheres," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 141-152, November.
    2. 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.
    3. J. Gower, 1975. "Generalized procrustes analysis," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 33-51, March.
    4. Stephan Huckemann, 2011. "Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(3), pages 424-446, September.
    5. Hendriks, Harrie & Landsman, Zinoviy, 1996. "Asymptotic behavior of sample mean location for manifolds," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 169-178, February.
    6. Ian L. Dryden & Alfred Kume & Huiling Le & Andrew T. A. Wood, 2008. "A multi-dimensional scaling approach to shape analysis," Biometrika, Biometrika Trust, vol. 95(4), pages 779-798.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rabi Bhattacharya & Rachel Oliver, 2019. "Nonparametric Analysis of Non-Euclidean Data on Shapes and Images," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 1-36, February.
    2. Huckemann, Stephan & Hotz, Thomas, 2009. "Principal component geodesics for planar shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 699-714, April.
    3. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    4. Kanika, & Kumar, Somesh & SenGupta, Ashis, 2015. "A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 160-172.
    5. 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.
    6. Meyners, Michael & Qannari, El Mostafa, 2001. "Relating principal component analysis on merged data sets to a regression approach," Technical Reports 2001,47, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    7. Juliana Martins Ruzante & Valerie J. Davidson & Julie Caswell & Aamir Fazil & John A. L. Cranfield & Spencer J. Henson & Sven M. Anders & Claudia Schmidt & Jeffrey M. Farber, 2010. "A Multifactorial Risk Prioritization Framework for Foodborne Pathogens," Risk Analysis, John Wiley & Sons, vol. 30(5), pages 724-742, May.
    8. Barbara McGillivray & Gard B. Jenset & Khalid Salama & Donna Schut, 2022. "Investigating patterns of change, stability, and interaction among scientific disciplines using embeddings," Palgrave Communications, Palgrave Macmillan, vol. 9(1), pages 1-15, December.
    9. Wei Wang & Stephen J Lycett & Noreen von Cramon-Taubadel & Jennie J H Jin & Christopher J Bae, 2012. "Comparison of Handaxes from Bose Basin (China) and the Western Acheulean Indicates Convergence of Form, Not Cognitive Differences," PLOS ONE, Public Library of Science, vol. 7(4), pages 1-7, April.
    10. Lisa Sakamoto & Hiromi Kajiya-Kanegae & Koji Noshita & Hideki Takanashi & Masaaki Kobayashi & Toru Kudo & Kentaro Yano & Tsuyoshi Tokunaga & Nobuhiro Tsutsumi & Hiroyoshi Iwata, 2019. "Comparison of shape quantification methods for genomic prediction, and genome-wide association study of sorghum seed morphology," PLOS ONE, Public Library of Science, vol. 14(11), pages 1-15, November.
    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. Ibrahim, Muhammad Sohail & Dong, Wei & Yang, Qiang, 2020. "Machine learning driven smart electric power systems: Current trends and new perspectives," Applied Energy, Elsevier, vol. 272(C).
    13. John Gower & Garmt Dijksterhuis, 1994. "Multivariate analysis of coffee images: A study in the simultaneous display of multivariate quantitative and qualitative variables for several assessors," Quality & Quantity: International Journal of Methodology, Springer, vol. 28(2), pages 165-184, May.
    14. 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.
    15. Modroño Herrán, Juan Ignacio & Fernández Aguirre, María Carmen & Landaluce Calvo, M. Isabel, 2003. "Una propuesta para el análisis de tablas múltiples," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    16. Peter Verboon & Willem Heiser, 1992. "Resistant orthogonal procrustes analysis," Journal of Classification, Springer;The Classification Society, vol. 9(2), pages 237-256, December.
    17. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.
    18. Dahl, Tobias & Naes, Tormod, 2006. "A bridge between Tucker-1 and Carroll's generalized canonical analysis," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3086-3098, July.
    19. Young-Jin Kwon & Do-Hyun Kim & Byung-Chang Son & Kyoung-Ho Choi & Sungbok Kwak & Taehong Kim, 2022. "A Work-Related Musculoskeletal Disorders (WMSDs) Risk-Assessment System Using a Single-View Pose Estimation Model," IJERPH, MDPI, vol. 19(16), pages 1-19, August.
    20. V Alex Sotola & Cody A Craig & Peter J Pfaff & Jeremy D Maikoetter & Noland H Martin & Timothy H Bonner, 2019. "Effect of preservation on fish morphology over time: Implications for morphological studies," PLOS ONE, Public Library of Science, vol. 14(3), pages 1-16, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aistmt:v:64:y:2012:i:6:p:1227-1259. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.