Skip to main content
SpringerLink
Log in

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

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Afsari, B. (2011). Riemannian L p center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139, 655–773.

  • Anderson, T. (2003). An introduction to multivariate statistical analysis (3rd ed.). New York: Wiley.

  • Bandulasiri, A., Patrangenaru, V. (2005). Algorithms for nonparametric inference on shape manifolds. Proceedings of JSM 2005 Minneapolis, MN, 1617–1622.

  • Bhattacharya A. (2008) Statistical analysis on manifolds: A nonparametric approach for inference on shape spaces. Sankhya, Series A, 70(2): 223–266

    MATH  Google Scholar 

  • Bhattacharya R. N., Patrangenaru V. (2003) Large sample theory of intrinsic and extrinsic sample means on manifolds I. The Annals of Statistics, 31(1): 1–29

    Article  MathSciNet  Google Scholar 

  • Bhattacharya R. N., Patrangenaru V. (2005) Large sample theory of intrinsic and extrinsic sample means on manifolds II. The Annals of Statistics, 33(3): 1225–1259

    Article  MathSciNet  Google Scholar 

  • Billera L., Holmes S., Holmes S., Holmes S. (2001) Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4): 733–767

    Article  MathSciNet  MATH  Google Scholar 

  • Bredon, G. E. (1972). Introduction to compact transformation groups. In Pure and applied mathematics (Vol. 46). New York: Academic Press.

  • Choquet G. (1954) Theory of capacities. Annales de l’Institut de Fourier, 5: 131–295

    Article  MathSciNet  Google Scholar 

  • Dryden I. L., Mardia K. V. (1998) Statistical shape analysis. Wiley, Chichester

    Google Scholar 

  • Dryden, I. L., Kume, A., Le., H., Wood, A. T. A. (2008). A multidimensional scaling approach to shape analysis (to appear).

  • Fréchet M. (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. 10(4): 215–310

    Google Scholar 

  • Gower J. C. (1975) Generalized Procrustes analysis. Psychometrika, 40(33–51): 40 33–51

    Google Scholar 

  • Hendriks H., Landsman Z. (1996) Asymptotic behaviour of sample mean location for manifolds. Statistics and Probability Letters, 26: 169–178

    Article  MathSciNet  MATH  Google Scholar 

  • Hendriks H., Landsman Z. (1998) Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. Journal of Multivariate Analysis, 67: 227–243

    Article  MathSciNet  MATH  Google Scholar 

  • Hendriks H., Landsman Z., Ruymgaart F. (1996) Asymptotic behaviour of sample mean direction for spheres. Journal of Multivariate Analysis, 59: 141–152

    Article  MathSciNet  MATH  Google Scholar 

  • Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V., Skwerer, S. (2012). Sticky central limit theorems on open books. arXiv.org, 1202.4267 [math.PR] [math.MG] [math.ST].

  • Huckemann, S. (2010). R-package for intrinsic statistical analysis of shapes. http://www.mathematik.uni-kassel.de/~huckeman/software/ishapes_1.0.tar.gz.

  • Huckemann S. (2011) Inference on 3D Procrustes means: Tree boles growth, rank-deficient diffusion tensors and perturbation models. Scandinavian Journal of Statistics, 38(3): 424–446

    MathSciNet  MATH  Google Scholar 

  • Huckemann S., Ziezold H. (2006) Principal component analysis for Riemannian manifolds with an application to triangular shape spaces. Advances of Applied Probability (SGSA), 38(2): 299–319

    Article  MathSciNet  MATH  Google Scholar 

  • Huckemann S., Hotz T., Munk A. (2010) Intrinsic MANOVA for Riemannian manifolds with an application to Kendall’s space of planar shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(4): 593–603

    Article  Google Scholar 

  • Huckemann S., Hotz T., Munk A. (2010) Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions (with discussion). Statistica Sinica, 20(1): 1–100

    MathSciNet  MATH  Google Scholar 

  • Jupp P. E. (1988) Residuals for directional data. Journal of Applied Statistics, 15(2): 137–147

    Article  Google Scholar 

  • Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 509–541.

  • Kendall, D. (1974). Foundations of a theory of random sets. In Stochastic geometry, tribute memory Rollo Davidson (pp. 322–376). New York: Wiley.

  • Kendall, D. G., Barden, D., Carne, T. K., Le, H. (1999). Shape and shape theory. Chichester: Wiley.

  • Kendall W. S. (1990) Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proceedings of the London Mathematical Society, 61: 371–406

    Article  MathSciNet  Google Scholar 

  • Kent, J., Hotz, T., Huckemann, S., Miller, E. (2011). The topology and geometry of projective shape spaces (in preparation).

  • Klassen, E., Srivastava, A., Mio, W., Joshi, S. (2004, March). Analysis on planar shapes using geodesic paths on shape spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(3), 372–383.

    Google Scholar 

  • Kobayashi, S., Nomizu, K. (1963). Foundations of differential geometry (Vol. I). Chichester: Wiley.

  • Kobayashi, S., Nomizu, K. (1969). Foundations of differential geometry (Vol. II). Chichester: Wiley.

  • Krim, H., Yezzi, A. J. J. E. (2006). Statistics and analysis of shapes. In Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhäuser.

  • Le H. (2001) Locating Fréchet means with an application to shape spaces. Advances of Applied Probability (SGSA), 33(2): 324–338

    Article  MATH  Google Scholar 

  • Le H. (2004) Estimation of Riemannian barycenters. LMS Journal of Computation and Mathematics, 7: 193–200

    MathSciNet  MATH  Google Scholar 

  • Lehmann, E. L. (1997). Testing statistical hypotheses. In Springer texts in statistics. New York: Springer.

  • Mardia, K., Patrangenaru,. V. (2001). On affine and projective shape data analysis. In K. V. Mardia, R. G. Aykroyd (Eds.), Proceedings of the 20th LASR Workshop on functional and spatial data analysis (pp. 39–45).

  • Mardia K., Patrangenaru V. (2005) Directions and projective shapes. The Annals of Statistics, 33: 1666–1699

    Article  MathSciNet  MATH  Google Scholar 

  • Matheron, G. (1975). Random sets and integral geometry. In Wiley series in probability and mathematical statistics. New York: Wiley.

  • Nash J. (1956) The imbedding problem for Riemannian manifolds. The Annals of Mathematics, 63: 20–63

    Article  MathSciNet  MATH  Google Scholar 

  • Palais, R. S. (1961). On the existence of slices for actions of non-compact Lie groups. The Annals of Mathematics 2nd Series, 73(2), 295–323

    Article  Google Scholar 

  • Schmidt, F. R., Clausen, M., Cremers, D. (2006). Shape matching by variational computation of geodesics on a manifold. In Pattern recognition (Proceedings DAGM), LNCS (Vol. 4174, pp. 142–151). Berlin: Springer.

  • Small, C. G. (1996). The statistical theory of shape. New York: Springer.

  • Zahn C., Roskies R. (1972) Fourier descriptors for plane closed curves. IEEE Transactions on Computers, C 21: 269–281

    Article  MathSciNet  MATH  Google Scholar 

  • Ziezold, H. (1977). Expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In Transaction of the 7th Prague conference on information theory, statistical decision function and random processes, A, pp. 591–602.

  • Ziezold, H. (1994). Mean figures and mean shapes applied to biological figure and shape distributions in the plane. Biometrical Journal, 36, 491–510

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematical Stochastics, University of Göttingen, Goldschmidstr. 5–7, Göttingen, Germany

    Stephan F. Huckemann

Authors
  1. Stephan F. Huckemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephan F. Huckemann.

About this article

Cite this article

Huckemann, S.F. On the meaning of mean shape: manifold stability, locus and the two sample test. Ann Inst Stat Math 64, 1227–1259 (2012). https://doi.org/10.1007/s10463-012-0352-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-012-0352-2

Keywords

Search

Navigation