Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beran, R., Fisher, N.I.: Nonparametric comparison of mean axes. Ann. Statist. 26, 472–493 (1998)
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)
Bhattacharya, R.N., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds-part II. Ann. Stat. 33, 1211–1245 (2005)
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)
Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27(4), 733–767 (2001)
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)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)
Efron, B.: The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38. SIAM, Philadelphia, Pa (1982)
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)
Fréchet, M.: Les élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948)
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, 2014
Helgason, S.: Differential Geometry and Symmetric Spaces. AMS Chelsea Publishing, AMS, Providence, Rhode Island (2001)
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)
Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)
Kent, J.T.: New directions in shape analysis. In: The Art of Statistical Science, A Tribute to G. S. Watson, pp. 115–127 (1992)
Ma, Y., Soatto, A., Košecká, J., Sastry, S.: An Invitation to 3-D Vision: From Images to Geometric Models. Springer (2005)
Mardia, K.V., Patrangenaru, V.: Directions and projective shapes. Ann. Stat. 33, 1666–1699 (2005)
Patrangenaru, V., Ellingson, L.L.: Nonparametric Statistics on Manifolds and Their Applications. Chapman & Hall/CRC Texts in Statistical Science (2015)
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)
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)
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)
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)
Sughatadasa, S.M.: Affine and Projective Shape Analysis with Applications. Ph.D. dissertation, Texas Tech University (2006)
Wang, H., Marron, J.S.: Object oriented data analysis: sets of trees. Ann. Stat. 35, 1849–1873 (2007)
Watson, G.S.: Statistics on Spheres. Lecture Notes in the Mathematical Sciences. Wiley (1983)
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)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Patrangenaru, V., Yao, K.D., Guo, R. (2016). Extrinsic Means and Antimeans. In: Cao, R., González Manteiga, W., Romo, J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 175. Springer, Cham. https://doi.org/10.1007/978-3-319-41582-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-41582-6_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41581-9
Online ISBN: 978-3-319-41582-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)