The Annals of Statistics

Functional deconvolution in a periodic setting: Uniform case

Marianna Pensky and Theofanis Sapatinas

Full-text: Open access

Abstract

We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model.

We derive minimax lower bounds for the L2-risk in the proposed functional deconvolution model when f(⋅) is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of f(⋅) that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls.

In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.

Article information

Source
Ann. Statist., Volume 37, Number 1 (2009), 73-104.

Dates
First available in Project Euclid: 16 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1232115928

Digital Object Identifier
doi:10.1214/07-AOS552

Mathematical Reviews number (MathSciNet)
MR2488345

Zentralblatt MATH identifier
1274.62253

Subjects
Primary: 62G05: Estimation
Secondary: 62G08: Nonparametric regression 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 35K05: Heat equation 35L05: Wave equation

Keywords
Adaptivity Besov spaces block thresholding deconvolution Fourier analysis functional data Meyer wavelets minimax estimators multichannel deconvolution partial differential equations wavelet analysis

Citation

Pensky, Marianna; Sapatinas, Theofanis. Functional deconvolution in a periodic setting: Uniform case. Ann. Statist. 37 (2009), no. 1, 73--104. doi:10.1214/07-AOS552. https://projecteuclid.org/euclid.aos/1232115928


Export citation

References

  • Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115–129.
  • Casey, S. D. and Walnut, D. F. (1994). Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms. SIAM Rev. 36 537–577.
  • Chesneau, C. (2008). Wavelet estimation via block thresholding: A minimax study under Lp-risk. Statist. Sinica 18 1007–1024.
  • Cirelson, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norm of Gaussian sample function. In Proceedings of the 3rd Japan–U.S.S.R. Symposium on Probability Theory. Lecture Notes in Math. 550 20–41. Springer, Berlin.
  • De Canditiis, D. and Pensky, M. (2004). Discussion on the meeting on “Statistical approaches to inverse problems.” J. Roy. Statist. Soc. Ser. B 66 638–640.
  • De Canditiis, D. and Pensky, M. (2006). Simultaneous wavelet deconvolution in periodic setting. Scand. J. Statist. 33 293–306.
  • Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Computat. Harmon. Anal. 2 101–126.
  • Donoho, D. L. and Raimondo, M. (2004). Translation invariant deconvolution in a periodic setting. Internat. J. Wavelets, Multiresolution and Information Processing 14 415–432.
  • Dozzi, M. (1989). Stochastic Processes with a Multidimensional Parameter. Longman, New York.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problem. Ann. Statist. 19 1257–1272.
  • Fan, J. and Koo, J. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747.
  • Golubev, G. (2004). The principle of penalized empirical risk in severely ill-posed problems. Probab. Theory Related Filelds 130 18–38.
  • Golubev, G. K. and Khasminskii, R. Z. (1999). A statistical approach to some inverse problems for partial differential equations. Problems Inform. Transmission 35 136–149.
  • Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statist. 129. Springer, New York.
  • Harsdorf, S. and Reuter, R. (2000). Stable deconvolution of noisy lidar signals. In Proceedings of EARSeL-SIG-Workshop LIDAR, Dresden/FRG, June 16–17.
  • Hesse, C. H. (2007). The heat equation with initial data corrupted by measurement error and missing data. Statist. Inference Stochastic Processes 10 75–95.
  • Johnstone, I. M. (2002). Function estimation in Gaussian noise: sequence models. Unpublished Monograph. Available at http://www-stat.stanford.edu/~imj/.
  • Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting (with discussion). J. Roy. Statist. Soc. Ser. B 66 547–573.
  • Johnstone, I. M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 1781–1804.
  • Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 58–109.
  • Kerkyacharian, G., Picard, D. and Raimondo, M. (2007). Adaptive boxcar deconvolution on full Lebesgue measure sets. Statist. Sinica 7 317–340.
  • Kolaczyk, E. D. (1994). Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
  • Lattes, R. and Lions, J. L. (1967). Methode de Quasi-Reversibilite et Applications. Travoux et Recherche Mathematiques 15. Dunod, Paris.
  • Mallat, S. G. (1999). A Wavelet Tour of Signal Processing, 2nd ed. Academic Press, San Diego.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Univ. Press.
  • Neelamani, R., Choi, H. and Baraniuk, R. (2004). Forward: Fourier-wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Signal Processing 52 418–433.
  • Park, Y. J., Dho, S. W. and Kong, H. J. (1997). Deconvolution of long-pulse lidar signals with matrix formulation. Applied Optics 36 5158–5161.
  • Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • Pensky, M. and Zayed, A. I. (2002). Density deconvolution of different conditional densities. Ann. Instit. Statist. Math. 54 701–712.
  • Schmidt, W. (1980). Diophantine Approximation. >Lecture Notes in Math. 785. Springer, Berlin.
  • Strauss, W. A. (1992). Partial Differential Equations: An Introduction. Wiley, New York.
  • Walter, G. and Shen, X. (1999). Deconvolution using Meyer wavelets. J. Integral Equations and Applications 11 515–534.
  • Willer, T. (2005). Deconvolution in white noise with a random blurring function. Preprint. arXiv:math/0505142.