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Contributions of Rabi Bhattacharya to the Central Limit Theory and Normal Approximation Read chapter Read first chapter

2016 | OriginalPaper | Chapter

18. Nonparametric Statistics on Manifolds and Beyond

Authors : Stephan Huckemann, Thomas Hotz

Published in: Rabi N. Bhattacharya

Publisher: Springer International Publishing

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Abstract

We review some aspects of the Bhattacharya-Patrangenaru asymptotic theory for intrinsic and extrinsic means on manifolds, some of the problems involved, many of which are still open, and survey some of its impacts on the community.

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previous chapter Nonparametric Statistical Methods on Manifolds
next chapter Reprints: Part VI
Literature
[1]
B. Afsari. Riemannian L p center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139:655–773, 2011.MathSciNetCrossRefMATH
[2]
Burcu Aydın, Gábor Pataki, Haonan Wang, Elizabeth Bullitt, and JS Marron. A principal component analysis for trees. The Annals of Applied Statistics, 3(4):1597–1615, 2009.MathSciNetCrossRefMATH
[3]
Ananda Bandulasiri, Rabi N. Bhattacharya, and Vic Patrangenaru. Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging. Journal of Multivariate Analysis, 100(9):1867–1882, 2009.MathSciNetCrossRefMATH
[4]
Dennis Barden, Huiling Le, and Megan Owen. Central limit theorems for Fréchet means in the space of phylogenetic trees. Electron. J. Probab, 18(25):1–25, 2013.MathSciNetMATH
[5]
A. Bhattacharya. Statistical analysis on manifolds: A nonparametric approach for inference on shape spaces. Sankhyā, Ser. A, 70(2):223–266, 2008.MathSciNetMATH
[6]
R. N. Bhattacharya and V. Patrangenaru. Large sample theory of intrinsic and extrinsic sample means on manifolds I. The Annals of Statistics, 31(1):1–29, 2003.MathSciNetCrossRefMATH
[7]
R. N. Bhattacharya and V. Patrangenaru. Large sample theory of intrinsic and extrinsic sample means on manifolds II. The Annals of Statistics, 33(3):1225–1259, 2005.MathSciNetCrossRefMATH
[8]
R.N. Bhattacharya, L. Ellingson, X. Liu, V. Patrangenaru, and M. Crane. Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machine vision. Applied Stochastic Models in Business and Industry, 28(3):222–235, 2012.MathSciNetCrossRef
[9]
Jérémie Bigot and Sébastien Gadat. A deconvolution approach to estimation of a common shape in a shifted curves model. The Annals of Statistics, 38(4):2422–2464, 2010.MathSciNetCrossRefMATH
[10]
L.J. Billera, S.P. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4):733–767, 2001.MathSciNetCrossRefMATH
[11]
Fred L. Bookstein. Size and shape spaces for landmark data in two dimensions (with discussion). Statistical Science, 1(2):181–222, 1986.CrossRefMATH
[12]
I. L. Dryden and K. V. Mardia. Statistical Shape Analysis. Wiley, Chichester, 1998.MATH
[13]
Ian L. Dryden, Alfred Kume, Huiling Le, and Andrew T. A. Wood. A multidimensional scaling approach to shape analysis. Biometrika, 95(4):779–798, 2008.
[14]
I.L. Dryden, A. Koloydenko, and D. Zhou. Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3(3):1102–1123, 2009.MathSciNetCrossRefMATH
[15]
Leif Ellingson, Vic Patrangenaru, and Frits Ruymgaart. Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours. Journal of Multivariate Analysis, 122:317–333, 2013.MathSciNetCrossRefMATH
[16]
R. Fisher. Dispersion on a sphere. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 217(1130):295–305, 1953.MathSciNetCrossRefMATH
[17]
Maurice Fréchet. Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’Institut de Henri Poincaré, 10(4):215–310, 1948.MATH
[18]
Jean Gayon. History of the concept of allometry. American Zoologist, 40(5):748–758, 2000.
[19]
[20]
David Groisser. On the convergence of some Procrustean averaging algorithms. Stochastics: Internatl. J. Probab. Stochastic Processes, 77(1):51–60, 2005.MathSciNetMATH
[21]
H. Hendriks. Sur le cut-locus d’une sous-variété de l’espace Euclidean. C. R. Acad. Sci. Paris, Série I, 311:637–639, 1990.MathSciNetMATH
[22]
H. Hendriks. Sur le cut-locus d’une sous-variété de l’espace Euclidean. Négligeabilité. C. R. Acad. Sci. Paris, Série I, 315:1275–1277, 1992.
[23]
H. Hendriks and Z. Landsman. Asymptotic behaviour of sample mean location for manifolds. Statistics & Probability Letters, 26:169–178, 1996.MathSciNetCrossRefMATH
[24]
H. Hendriks and Z. Landsman. Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. Journal of Multivariate Analysis, 67:227–243, 1998.MathSciNetCrossRefMATH
[25]
H. Hendriks, Z. Landsman, and F. Ruymgaart. Asymptotic behaviour of sample mean direction for spheres. Journal of Multivariate Analysis, 59:141–152, 1996.MathSciNetCrossRefMATH
[26]
Thomas Hotz and Stephan Huckemann. Intrinsic means on the circle: Uniqueness, locus and asymptotics. Annals of the Institute of Statistical Mathematics, 67(1):177–193, 2015.MathSciNetCrossRefMATH
[27]
Thomas Hotz, Stephan Huckemann, Huiling Le, J. Stephen Marron, Jonathan Mattingly, Ezra Miller, James Nolen, Megan Owen, Victor Patrangenaru, and Sean Skwerer. Sticky central limit theorems on open books. Annals of Applied Probability, 23(6):2238–2258, 2013.
[28]
S. Huckemann, T. Hotz, and A. Munk. 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, 2010.CrossRef
[29]
S. Huckemann, T. Hotz, and A. Munk. Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions (with discussion). Statistica Sinica, 20(1):1–100, 2010.MathSciNetMATH
[30]
Stephan Huckemann. Inference on 3D Procrustes means: Tree boles growth, rank-deficient diffusion tensors and perturbation models. Scandinavian Journal of Statistics, 38(3):424–446, 2011.MathSciNetMATH
[31]
Stephan Huckemann. Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. The Annals of Statistics, 39(2):1098–1124, 2011.MathSciNetCrossRefMATH
[32]
Stephan Huckemann. On the meaning of mean shape: Manifold stability, locus and the two sample test. Annals of the Institute of Statistical Mathematics, 64(6):1227–1259, 2012.MathSciNetCrossRefMATH
[33]
Julian S. Huxley and Georges Teissier. Terminology of relative growth. Nature, 137(3471):780–781, May 1936.
[34]
Sungkyu Jung, Ian L Dryden, and J. S. Marron. Analysis of principal nested spheres. Biometrika, 99(3):551–568, 2012.
[35]
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, XXX:509–541, 1977.
[36]
D. G. Kendall. The diffusion of shape. Adv. Appl. Prob., 9:428–430, 1977.CrossRef
[37]
[38]
D. G. Kendall, D. Barden, T. K. Carne, and H. Le. Shape and Shape Theory. Wiley, Chichester, 1999.CrossRefMATH
[39]
W. S. Kendall. Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proceedings of the London Mathematical Society, 61:371–406, 1990.MathSciNetCrossRefMATH
[40]
Wilfrid S Kendall and Huiling Le. Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables. Brazilian Journal of Probability and Statistics, 25(3):323–352, 2011.
[41]
John T. Kent and Kanti V. Mardia. Consistency of Procrustes estimators. Journal of the Royal Statistical Society, Series B, 59(1):281–290, 1997.MathSciNetCrossRefMATH
[42]
[43]
H. Le. Locating Fréchet means with an application to shape spaces. Advances of Applied Probability (SGSA), 33(2):324–338, 2001.CrossRefMATH
[44]
H. Le and C.G. Small. Multidimensional scaling of simplex shapes. Pattern Recognition, 32:1601–1613, 1999.CrossRef
[45]
[46]
S. Lele. Euclidean distance matrix analysis (EDMA): estimation of mean form and mean form difference. Math. Geol., 25(5):573–602, July 1993.
[47]
K. V. Mardia and P. E. Jupp. Directional Statistics. Wiley, New York, 2000.MATH
[48]
John Mather. Notes on topological stability. Harvard University Cambridge, 1970.MATH
[49]
James E. Mosimann. Size allometry: Size and shape variables with characterizations of the lognormal and generalized gamma distributions. Journal of the American Statistical Association, 65(330):930–945, 1970.CrossRefMATH
[50]
Megan Owen and J Scott Provan. A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 8(1):2–13, 2011.
[51]
Xavier Pennec. Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis., 25(1):127–154, 2006.MathSciNetCrossRef
[52]
Markus Pflaum. Analytic and Geometric Study of Stratified Spaces: Contributions to Analytic and Geometric Aspects, volume 1768. Springer Verlag, 2001.MATH
[53]
Sean Skwerer, Elizabeth Bullitt, Stephan Huckemann, Ezra Miller, Ipek Oguz, Megan Owen, Vic Patrangenaru, Scott Provan, and J.S. Marron. Tree-oriented analysis of brain artery structure. Journal of Mathematical Imaging and Vision, 50(1–2):98–106, 2014.
[54]
[55]
Theophrastus. De Causis Plantarum, Volume I of III. Number 471 in The Loeb classical library. William Heinemann Ltd. and Harvard University Press, London and Cambridge, MA, 1976. with an English translation by Benedict Einarson and George K. K. Link.
[56]
Hongtu Zhu, Yasheng Chen, Joseph G Ibrahim, Yimei Li, Colin Hall, and Weili Lin. Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging. Journal of the American Statistical Association, 104(487):1203–1212, 2009.
[57]
H. Ziezold. Expected figures and a strong law of large numbers for random elements in quasi-metric spaces. Transaction of the 7th Prague Conference on Information Theory, Statistical Decision Function and Random Processes, A:591–602, 1977.
[58]
H. Ziezold. Mean figures and mean shapes applied to biological figure and shape distributions in the plane. Biometrical Journal, 36(4):491–510, 1994.MathSciNetCrossRefMATH
Metadata
Title
Nonparametric Statistics on Manifolds and Beyond
Authors
Stephan Huckemann
Thomas Hotz
Copyright Year
2016
Book
Rabi N. Bhattacharya

Print ISBN: 978-3-319-30188-4

Electronic ISBN: 978-3-319-30190-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-30190-7_18