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

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  • 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

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  • 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
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    References listed on IDEAS

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    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.
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    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.

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