Abstract
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.
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Bigot, J.: A scale-space approach with wavelets to singularity estimation. ESAIM Probab. Stat. 9, 143–164 (2005) (electronic)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Bouroche, J.-M., Saporta, G.: L’analyse des données. Que Sais-Je? Presses Universitaires de France, Paris (1980)
Gamboa, F., Loubes, J.-M., Maza, E.: Semi-parametric estimation of shifts. Electron. J. Stat. 1, 616–640 (2007)
Gervini, D., Gasser, T.: Nonparametric maximum likelihood estimation of the structural mean of a sample of curves. Biometrika 92(4), 801–820 (2005)
Kneip, A., Gasser, T.: Statistical tools to analyze data representing a sample of curves. Ann. Stat. 20(3), 1266–1305 (1992)
Kneip, A., Li, X., MacGibbon, K.B., Ramsay, J.O.: Curve registration by local regression. Can. J. Stat. 28(1), 19–29 (2000)
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
Liu, X., Müller, H.-G.: Functional convex averaging and synchronization for time-warped random curves. J. Am. Stat. Assoc. 99(467), 687–699 (2004)
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)
Ramsay, J.O., Li, X.: Curve registration. J. R. Stat. Soc. Ser. B Stat. Methodol. 60(2), 351–363 (1998)
Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis. Springer Series in Statistics. Springer, New York (2002). Methods and case studies
Rønn, B.B.: Nonparametric maximum likelihood estimation for shifted curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 63(2), 243–259 (2001)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Sakoe, H., Chiba, S.: Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. ASSP-26 1, 43–49 (1978)
van der Vaart, A.W.: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998)
van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996)
Wang, K., Gasser, T.: Synchronizing sample curves nonparametrically. Ann. Stat. 27(2), 439–460 (1999)
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Dupuy, JF., Loubes, JM. & Maza, E. Non parametric estimation of the structural expectation of a stochastic increasing function. Stat Comput 21, 121–136 (2011). https://doi.org/10.1007/s11222-009-9152-9
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DOI: https://doi.org/10.1007/s11222-009-9152-9