Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Benjamin Charlier

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 635-649
  • ISSN: 1292-8100

Abstract

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Let ( 𝕊 1 , d 𝕊 1 S 1 , d S 1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μto admit a well defined Fréchet mean on ( 𝕊 1 , d 𝕊 1 S 1 , d S 1 ). We derive a new sufficient condition of existenceP(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

How to cite

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Charlier, Benjamin. "Necessary and sufficient condition for the existence of a Fréchet mean on the circle." ESAIM: Probability and Statistics 17 (2013): 635-649. <http://eudml.org/doc/274376>.

@article{Charlier2013,
abstract = {Let ( $\mathbb \{S\}^1, d_\{\mathbb \{S\}^1\}$ S 1 , d S 1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μto admit a well defined Fréchet mean on ( $\mathbb \{S\}^1,d_\{\mathbb \{S\}^1\}$ S 1 , d S 1 ). We derive a new sufficient condition of existenceP(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.},
author = {Charlier, Benjamin},
journal = {ESAIM: Probability and Statistics},
keywords = {circular data; fréchet mean; uniqueness; Fréchet mean},
language = {eng},
pages = {635-649},
publisher = {EDP-Sciences},
title = {Necessary and sufficient condition for the existence of a Fréchet mean on the circle},
url = {http://eudml.org/doc/274376},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Charlier, Benjamin
TI - Necessary and sufficient condition for the existence of a Fréchet mean on the circle
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 635
EP - 649
AB - Let ( $\mathbb {S}^1, d_{\mathbb {S}^1}$ S 1 , d S 1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μto admit a well defined Fréchet mean on ( $\mathbb {S}^1,d_{\mathbb {S}^1}$ S 1 , d S 1 ). We derive a new sufficient condition of existenceP(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.
LA - eng
KW - circular data; fréchet mean; uniqueness; Fréchet mean
UR - http://eudml.org/doc/274376
ER -

References

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  1. [1] B. Afsari, Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Amer. Math. Soc.139 (2011) 655–673. Zbl1220.53040MR2736346
  2. [2] R. Bhattacharya and V. Patrangenaru, Large sample theory of intrinsic and extrinsic sample means on manifolds, I. Ann. Stat.31 (2003) 1–29. Zbl1020.62026MR1962498
  3. [3] R.S. Buss and J.P. Fillmore, Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph.20 (2001) 95–126. 
  4. [4] J.M. Corcuera and W.S. Kendall, Riemannian barycentres and geodesic convexity. Math. Proc. Cambridge Philos. Soc.127 (1999) 253–269. Zbl0948.53020MR1705458
  5. [5] M. Émery and G. Mokobodzki, Sur le barycentre d’une probabilité dans une variété, in Séminaire de Probabilités, XXV, vol. 1485 of Lect. Notes Math. (1991) 220–233. Zbl0753.60046MR1187782
  6. [6] N.I. Fisher, Statistical analysis of circular data. Cambridge University Press, Cambridge (1993). Zbl0788.62047MR1251957
  7. [7] M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré10 (1948) 215–310. Zbl0035.20802MR27464
  8. [8] H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math.30 (1977) 509–541. Zbl0354.57005MR442975
  9. [9] D. Kaziska and A. Srivastava, The karcher mean of a class of symmetric distributions on the circle. Stat. Probab. Lett. 78 (2008) 1314–1316 (2008). Zbl1144.62323MR2444322
  10. [10] D.G. Kendall, D. Barden, T.K. Carne and H. Le, Shape and shape theory. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester (1999). Zbl0940.60006MR1891212
  11. [11] H. Le, On the consistency of procrustean mean shapes. Adv. Appl. Probab.30 (1998) 53–63. Zbl0906.60007MR1618880
  12. [12] H. Le, Locating Fréchet means with application to shape spaces. Adv. Appl. Probab.33 (2001) 324–338. Zbl0990.60008MR1842295
  13. [13] H. Le, Estimation of Riemannian barycentres. LMS J. Comput. Math.7 (2004) 193–200. Zbl1054.60011MR2085875
  14. [14] K.V. Mardia and P.E. Jupp, Directional statistics. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester (2000). Revised reprint of ıt Statistics of directional data by Mardia [ MR0336854 (49 #1627)]. Zbl0935.62065MR1828667
  15. [15] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab.18 (1990) 1269–1283. Zbl0713.62021MR1062069
  16. [16] J.M. Oller and J.M. Corcuera, Intrinsic analysis of statistical estimation. Ann. Statist.23 (1995) 1562–1581. Zbl0843.62027MR1370296
  17. [17] X. Pennec, Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision25 (2006) 127–154. MR2254442
  18. [18] H. Ziezold, On expected figures and a strong law of large numbers for random elements in quasi-metric spaces, in Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, Prague, 1974). Reidel, Dordrecht (1977) 591–602. Zbl0413.60024MR501230

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