The Annals of Statistics

A deconvolution approach to estimation of a common shape in a shifted curves model

Jérémie Bigot and Sébastien Gadat

Full-text: Open access

Abstract

This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 2422-2464.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861253

Digital Object Identifier
doi:10.1214/10-AOS800

Mathematical Reviews number (MathSciNet)
MR2676894

Zentralblatt MATH identifier
1202.62049

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 42C40: Wavelets and other special systems

Keywords
Mean pattern estimation curve registration inverse problem deconvolution Meyer wavelets adaptive estimation Besov space minimax rate

Citation

Bigot, Jérémie; Gadat, Sébastien. A deconvolution approach to estimation of a common shape in a shifted curves model. Ann. Statist. 38 (2010), no. 4, 2422--2464. doi:10.1214/10-AOS800. https://projecteuclid.org/euclid.aos/1278861253


Export citation

References

  • [1] Allassonière, S., Amit, Y. and Trouvé, A. (2007). Toward a coherent statistical framework for dense deformable template estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 3–29.
  • [2] Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1–29.
  • [3] Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 1225–1259.
  • [4] Bigot, J. (2006). Landmark-based registration of curves via the continuous wavelet transform. J. Comput. Graph. Statist. 15 542–564.
  • [5] Bigot, J., Gadat, S. and Loubes, J. M. (2009). Statistical M-estimation and consistency in large deformable models for image warping. J. Math. Imaging Vision 34 270–290.
  • [6] Bigot, J., Gamboa, F. and Vimond, M. (2009). Estimation of translation, rotation and scaling between noisy images using the Fourier Mellin transform. SIAM J. Imaging Sci. 2 614–645.
  • [7] Buckheit, J. B., Chen, S., Donoho, D. L. and Johnstone, I. (1995). Wavelab reference manual. Dept. Statistics, Stanford Univ. Availablte at http://www-stat.stanford.edu/software/wavelab.
  • [8] Castillo, I. and Loubes, J. M. (2009). Estimation of the distribution of random shifts deformation. Math. Methods Statist. 18 21–42.
  • [9] Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843–874.
  • [10] Cavalier, L. and Raimondo, M. (2007). Wavelet deconvolution with noisy eigenvalues. IEEE Trans. Signal Process. 55 2414–2424.
  • [11] Cirelson, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian Sample Functions. Lecture Notes in Math. 550 20–41. Springer, Berlin.
  • [12] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101–126.
  • [13] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57 301–369.
  • [14] Efromovich, S. and Koltchinskii, V. (2001). On inverse problems with unknown operators. IEEE Trans. Inform. Theory 47 2876–2894.
  • [15] Frechet, M. (1948). Les éléments aleatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 215–310.
  • [16] Gasser, T. and Kneip, A. (1995). Searching for structure in curve samples. J. Amer. Statist. Assoc. 90 1179–1188.
  • [17] Gasser, T. and Kneip, A. (1992). Statistical tools to analyze data representing a sample of curves. Ann. Statist. 20 1266–1305.
  • [18] Glasbey, C. A. and Mardia, K. V. (2001). A penalized likelihood approach to image warping (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 63 465–514.
  • [19] Gervini, D. and Gasser, T. (2004). Self-modeling warping functions. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 959–971.
  • [20] Gill, R. D. and Levit, Y. (1995). Applications of the Van Trees inequality: A Bayesian Cramér–Rao bound. Bernoulli 1 59–79.
  • [21] Grenander, U. (1993). General Pattern Theory: A Mathematical Study of Regular Structures. Oxford Univ. Press, New York.
  • [22] Hardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation and Statistical Applications. Springer, New York.
  • [23] Hoffman, M. and Reiss, M. (2008). Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 310–336.
  • [24] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
  • [25] Isserles, U., Ritov, Y. and Trigano, T. (2008). Semiparametric curve alignment and shift density estimation for biological data. Available at http://arxiv.org/abs/0807.1271.
  • [26] Johnstone, I., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547–573.
  • [27] Kneip, A. and Gasser, T. (1988). Convergence and consistency results for self-modelling regression. Ann. Statist. 16 82–112.
  • [28] Kolaczyk, E. D. (1994). Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data. Ph.D. thesis, Dept. Statistics, Stanford Univ.
  • [29] Korostelëv, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, Berlin.
  • [30] Liu, X. and Müller, H. G. (2004). Functional convex averaging and synchronization for time-warped random curves. J. Amer. Statist. Assoc. 99 687–699.
  • [31] Loubes, J. M., Maza, E. and Gamboa, F. (2007). Semi-parametric estimation of shifts. Electron. J. Stat. 1 616–640.
  • [32] Ma, J., Miller, M. I., Trouvé, A. and Younes, L. (2008). Bayesian template estimation in computational anatomy. NeuroImage 42 252–261.
  • [33] Massart, P. (2006). Concentration Inequalities and Model Selection: Ecole d’été de Probabilités de Saint-Flour XXXIII—2003. Springer, Berlin.
  • [34] Meyer, Y. (1992). Wavelets and Operators. Cambridge Univ. Press, Cambridge.
  • [35] Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 307–330.
  • [36] Pensky, M. and Sapatinas, T. (2009). Functional deconvolution in a periodic setting: Uniform case. Ann. Statist. 37 73–104.
  • [37] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • [38] Raimondo, M. and Stewart, M. (2007). The waveD transform in R: Performs fast translation-invariant wavelet deconvolution. J. Softw. 21 1–27.
  • [39] Ramsay, J. O. and Li, X. (2001). Curve registration. J. Roy. Statist. Soc. Ser. B 63 243–259.
  • [40] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • [41] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis. Springer, New York.
  • [42] Ronn, B. (1998). Nonparametric maximum likelihood estimation for shifted curves. J. Roy. Statist. Soc. Ser. B 60 351–363.
  • [43] Rosenthal, H. P. (1972). On the span in Lp of sequences of independent random variables. II. In Proc. Sixth Berkeley Sympos. Math. Statist. Probab. 2 149–167. Univ. California Press, Berkeley, CA.
  • [44] Vimond, M. (2010). Efficient estimation for a subclass of shape invariant models. Ann. Statist. 38 1885–1912.
  • [45] Wang, K. and Gasser, T. (1997). Alignment of curves by dynamic time warping. Ann. Statist. 25 1251–1276.
  • [46] Willer, T. (2005). Deconvolution in white noise with a random blurring function. LPMA, Unpublished technical report.