Extrinsic Means and Antimeans

  • Vic PatrangenaruEmail author
  • K. David Yao
  • Ruite Guo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 175)


Often times object spaces are compact, thus allowing to introduce new location parameters, maximizers of the Fréchet function associated with a random object X on a compact object space. In this paper we focus on such location parameters, when the object space is embedded in a numeric space. In this case the maximizer, whenever the maximizer of this Fréchet function is unique, is called the extrinsic mean of X.


Random object Fréchet mean set Extrinsic mean Hypothesis testing for VW antimeans 



Research supported by NSA-MSP-H98230-14-1-0135 and NSF-DMS-1106935.

Research supported by NSA-MSP- H98230-14-1-0135.

Research supported by NSF-DMS-1106935.


  1. 1.
    Beran, R., Fisher, N.I.: Nonparametric comparison of mean axes. Ann. Statist. 26, 472–493 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhattacharya, R.N., Ellingson, L., Liu, X., Patrangenaru, V., Crane, M.: Extrinsic analysis on manifolds is computationally faster than intrinsic analysis, with applications to quality control by machine vision. Appl. Stoch. Models Bus. Ind. 28, 222–235 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhattacharya, R.N., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-part II. Ann. Stat. 33, 1211–1245 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhattacharya, R.N., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-part I. Ann. Stat. 31(1), 1–29 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27(4), 733–767 (2001)Google Scholar
  6. 6.
    Crane, M., Patrangenaru, V.: Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images. J. Multivar. Anal. 102, 225–237 (2011)Google Scholar
  7. 7.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)zbMATHGoogle Scholar
  8. 8.
    Efron, B.: The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38. SIAM, Philadelphia, Pa (1982)Google Scholar
  9. 9.
    Fisher, N.I., Hall, P., Jing, B.Y., Wood, A.T.A.: Properties of principal component methods for functional and longitudinal data analysis. J. Am. Stat. Assoc. 91, 1062–1070 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fréchet, M.: Les élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948)zbMATHGoogle Scholar
  11. 11.
    Guo, R., Patrangenaru, V., Lester, D.: Nonparametric Bootstrap test for Equality of Extrinsic Mean Reflection Shapes in Large vs Small Populations of Acrosterigma Magnum Shells, Poster, Geometric Topological and Graphical Model Methods in Statistics Fields Institute, Toronto, Canada, May 22–23, 2014Google Scholar
  12. 12.
    Helgason, S.: Differential Geometry and Symmetric Spaces. AMS Chelsea Publishing, AMS, Providence, Rhode Island (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hotz, T., Huckemann, S., Le, H., Marron, J.S., Mattingly, J.C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V., Skwerer, S.: Sticky central limit theorems on open books. Ann. Appl. Probab. 23, 2238–2258 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kent, J.T.: New directions in shape analysis. In: The Art of Statistical Science, A Tribute to G. S. Watson, pp. 115–127 (1992)Google Scholar
  16. 16.
    Ma, Y., Soatto, A., Košecká, J., Sastry, S.: An Invitation to 3-D Vision: From Images to Geometric Models. Springer (2005)Google Scholar
  17. 17.
    Mardia, K.V., Patrangenaru, V.: Directions and projective shapes. Ann. Stat. 33, 1666–1699 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Patrangenaru, V., Ellingson, L.L.: Nonparametric Statistics on Manifolds and Their Applications. Chapman & Hall/CRC Texts in Statistical Science (2015)Google Scholar
  19. 19.
    Patrangenaru, V., Guo, R., Yao, K.D.: Nonparametric inference for location parameters via Fréchet functions. In: Proceedings of Second International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, Beer Sheva, Israel, pp. 254–262 (2016)Google Scholar
  20. 20.
    Patrangenaru, V., Qiu, M., Buibas, M.: Two sample tests for mean 3D projective shapes from digital camera images. Methodol. Comput. Appl. Probab. 16, 485–506 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Patrangenaru, V., Liu, X., Sugathadasa, S.: Nonparametric 3D projective shape estimation from pairs of 2D images—I, in memory of W.P. Dayawansa. J. Multivar. Anal. 101, 11–31 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Patrangenaru, V., Mardia, K.V.: Affine shape analysis and image analysis. In: Proceedings of the Leeds Annual Statistics Research Workshop, pp. 57–62. Leeds University Press (2003)Google Scholar
  23. 23.
    Sughatadasa, S.M.: Affine and Projective Shape Analysis with Applications. Ph.D. dissertation, Texas Tech University (2006)Google Scholar
  24. 24.
    Wang, H., Marron, J.S.: Object oriented data analysis: sets of trees. Ann. Stat. 35, 1849–1873 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Watson, G.S.: Statistics on Spheres. Lecture Notes in the Mathematical Sciences. Wiley (1983)Google Scholar
  26. 26.
    Ziezold, H.: On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. In: Transactions of Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes A, pp. 591–602 (1977)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of StatisticsFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsFlorida State UniversityTallahasseeUSA

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