Statistics and Computing

, Volume 21, Issue 1, pp 121–136 | Cite as

Non parametric estimation of the structural expectation of a stochastic increasing function

  • J.-F. DupuyEmail author
  • J.-M. Loubes
  • E. Maza


This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So, we define the structural expectation of the underlying stochastic function. Then, we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.


Functional data analysis Non parametric warping model Structural expectation Curve registration 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bigot, J.: A scale-space approach with wavelets to singularity estimation. ESAIM Probab. Stat. 9, 143–164 (2005) (electronic) zbMATHCrossRefMathSciNetGoogle Scholar
  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) zbMATHGoogle Scholar
  3. Bouroche, J.-M., Saporta, G.: L’analyse des données. Que Sais-Je? Presses Universitaires de France, Paris (1980) Google Scholar
  4. Gamboa, F., Loubes, J.-M., Maza, E.: Semi-parametric estimation of shifts. Electron. J. Stat. 1, 616–640 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Gervini, D., Gasser, T.: Nonparametric maximum likelihood estimation of the structural mean of a sample of curves. Biometrika 92(4), 801–820 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  6. Kneip, A., Gasser, T.: Statistical tools to analyze data representing a sample of curves. Ann. Stat. 20(3), 1266–1305 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  7. Kneip, A., Li, X., MacGibbon, K.B., Ramsay, J.O.: Curve registration by local regression. Can. J. Stat. 28(1), 19–29 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23. Springer, Berlin (1991). Isoperimetry and processes zbMATHGoogle Scholar
  9. Liu, X., Müller, H.-G.: Functional convex averaging and synchronization for time-warped random curves. J. Am. Stat. Assoc. 99(467), 687–699 (2004) zbMATHCrossRefGoogle Scholar
  10. Loubes, J.-M., Maza, É., Lavielle, M., Rodríguez, L.: Road trafficking description and short term travel time forecasting, with a classification method. Can. J. Stat. 34(3), 475–491 (2006) zbMATHCrossRefGoogle Scholar
  11. Ramsay, J.O., Li, X.: Curve registration. J. R. Stat. Soc. Ser. B Stat. Methodol. 60(2), 351–363 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis. Springer Series in Statistics. Springer, New York (2002). Methods and case studies zbMATHCrossRefGoogle Scholar
  13. Rønn, B.B.: Nonparametric maximum likelihood estimation for shifted curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 63(2), 243–259 (2001) CrossRefMathSciNetGoogle Scholar
  14. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  15. Sakoe, H., Chiba, S.: Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. ASSP-26 1, 43–49 (1978) CrossRefGoogle Scholar
  16. van der Vaart, A.W.: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  17. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996) zbMATHGoogle Scholar
  18. Wang, K., Gasser, T.: Synchronizing sample curves nonparametrically. Ann. Stat. 27(2), 439–460 (1999) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Laboratoire Mathématiques, Image et ApplicationsUniversité de La RochelleLa RochelleFrance
  2. 2.Université Toulouse 3Institut de Mathématiques (UMR 5219)ToulouseFrance
  3. 3.Laboratoire Génomique et Biotechnologie des FruitsINP-ENSAT (UMR 990)Castanet-TolosanFrance

Personalised recommendations