Abstract
This paper deals with the estimation of a function f defined on the sphere \(\mathbb{S}^{d}\) of ℝd+1 from a sample of noisy observation points. We introduce an estimation procedure based on wavelet-like functions on the sphere called needlets and study two estimators f ⊛ and \(f^{\bigstar}\) respectively made adaptive through the use of a stochastic and deterministic needlet-shrinkage method. We show hereafter that these estimators are nearly optimal in the minimax framework, explain why f ⊛ outperforms \(f^{\bigstar}\), and run finite-sample simulations with f ⊛ to demonstrate that our estimation procedure is easy to implement and fares well in practice. We are motivated by applications in geophysical and atmospheric sciences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams R, Fournier J (2003) Sobolev spaces, 2nd edn. Elsevier Science, Oxford
Baldi P, Kerkyacharian G, Marinucci D, Picard D (2009) Adaptive density estimation for directional data using needlets. Ann Stat 37(6):3362–3395
Donoho D, Johnstone I, Kerkyacharian G, Picard D (1995) Wavelet shrinkage: Asymptotia? J R Stat Soc B 57(2):301–369
Faÿ G, Guilloux F, Cardoso JF, Delabrouille J, Le Jeune M (2008) CMB power spectrum estimation using wavelets. Phys Rev D 78(8)
Fengler M (2005) Vector spherical harmonic and vector wavelet based non-linear Garlekin schemes for solving the incompressible Navier-Stokes equation on the sphere. PhD thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group
Freeden W, Michel V (2004) Multiscale potential theory, with applications to geoscience. Birkhäuser, Boston
Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere (with applications to geomathematics). Oxford sciences publication. Clarendon Press, Oxford
Freeden W, Michel D, Michel V (2005) Local multiscale approximations of geostrophic oceanic flow: theoretical background and aspects of scientific computing. Mar Geod 28:313–329
Guilloux F, Faÿ G, Cardoso JF (2009) Practical wavelet design on the sphere. Appl Comput Harmon Anal 26(2):143–160
Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Inter 135:107–124
Kerkyacharian G, Picard D (2004) Regression in random design and warped wavelets. Bernoulli 10(6):1053–1105
Li TH (1999) Multiscale representation and analysis of spherical data by spherical wavelets. SIAM J Sci Comput 21:924–953
Li TH, Oh HS (2004) Estimation of global temperature fields from scattered observations by a spherical-wavelet-based spatially adaptive method. J R Stat Soc B 66:221–238
Maier T (2005) Wavelet-Mie-representations for solenoidal vector fields with applications to ionospheric geomagnetic data. SIAM J Appl Math 65:1888–1912
Marinucci D, Pietrobon D, Balbi A, Baldi P, Cabella P, Kerkyacharian G, Natoli P, Picard D, Vittorio N (2008) Spherical needlets for CMB data analysis. Mon Not R Astron Soc 383(2):539–545
Mayer C (2004) Wavelet modelling of the spherical inverse source problem with application to geomagnetism. Inverse Probl 20:1713–1728
Müller C (1966) Spherical harmonics. Springer, Berlin
Narcowich F, Ward J (1996) Nonstationary wavelets on the m-sphere for scattered data. Appl Comput Harmon Anal 3:324–326
Narcowich F, Petruchev P, Ward J (2006) Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J Funct Anal 238:530–564
Narcowich F, Petruchev P, Ward J (2007) Localized tight frames on spheres. SIAM J Math Anal 38(2):574–594
Panet I, Jamet O, Diament M, Chambodut A (2005) Modelling the Earth’s gravity field using wavelet frames. Springer, Berlin
Petrov VV (1995) Limit theorems of probability theory: sequences of independent random variables. Oxford University Press, Oxford
Schmidt M, Han S, Kusche J, Sánchez L, Shum C (2006) Regional high-resolution spatiotemporal gravity modeling from GRACE data using spherical wavelets. Geophys Res Lett 33(8)
Schmidt M, Fengler M, Mayer-Gürr T, Eicker A, Kusche J, Sánchez L, Han SC (2007) Regional gravity modeling in terms of spherical base functions. J Geod 81:17–38
Stein E, Weiss G (1975) Fourier analysis on Euclidean spaces. Princeton University Press, Princeton
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Monnier, JB. Nonparametric regression on the hyper-sphere with uniform design. TEST 20, 412–446 (2011). https://doi.org/10.1007/s11749-011-0233-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-011-0233-7