×

Subsampling needlet coefficients on the sphere. (English) Zbl 1200.62118

Summary: In a recent paper [J. Multivariate Anal. 99, No. 4, 606–636 (2008; Zbl 1333.62223)], we analyzed the properties of a new kind of spherical wavelets (called needlets) for statistical inference procedures on spherical random fields; the investigation was mainly motivated by applications to cosmological data. In the present work, we exploit the asymptotic uncorrelation of random needlet coefficients at fixed angular distances to construct subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate how such statistics can be used for isotropy tests and for bootstrap estimation of nuisance parameters, even when a single realization of the spherical random field is observed. The asymptotic theory is developed in detail in the high resolution sense.

MSC:

62M40 Random fields; image analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 1333.62223
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Antoine, J.-P. and Vandergheynst, P. (1999). Wavelets on the 2-sphere: A group-theoretical approach., Appl. Comput. Harmon. Anal. 7 262-291. · Zbl 0945.42023 · doi:10.1006/acha.1998.0272
[2] Antoine, J.-P., Demanet, L., Jacques, L. and Vandergheynst, P. (2002). Wavelets on the sphere: Implementation and approximations., Appl. Comput. Harmon. Anal. 13 177-200. · Zbl 1021.42022 · doi:10.1016/S1063-5203(02)00507-9
[3] Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2008). High frequency asymptotics for wavelet-based tests for Gaussianity and Isotropy on the Torus., J. Multivariate Anal. 99 606-636. · Zbl 1333.62223 · doi:10.1016/j.jmva.2007.02.002
[4] Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2006). Asymptotics for spherical needlets., Ann. Statist. 37 1150-1171. · Zbl 1160.62087 · doi:10.1214/08-AOS601
[5] Billingsley, P. (1968)., Convergence of Probability Measures . New York: Wiley. · Zbl 0172.21201
[6] Cabella, P., Hansen, F.K., Marinucci, D., Pagano, D. and Vittorio, N. (2004). Search for non-Gaussianity in pixel, harmonic and wavelets space: Compared and combined., Phys. Rev. D 69 .
[7] DeJong, P. (1990). A central limit theorem for generalized multilinear forms., J. Multivariate Anal. 34 275-289. · Zbl 0709.60019 · doi:10.1016/0047-259X(90)90040-O
[8] Dodelson, S. (2003)., Modern Cosmology . London: Academic Press.
[9] Guilloux, F., Fay, G. and Cardoso, J.-F. (2007). Practical wavelet design on the sphere., Appl. Comput. Harmon. Anal. To appear. Available at · Zbl 1162.42017 · doi:10.1016/j.acha.2008.03.003
[10] Hansen, F.K., Banday, A.J., Eriksen, H.K., Gorski, K.M. and Lilje, P.B. (2006). Foreground subtraction of cosmic microwave background maps using WI-FIT (wavelet based high resolution fitting of internal templates)., Astrophysical J. 648 784-796.
[11] Hansen, F.K., Cabella, P., Marinucci, D. and Vittorio, N. (2004). Asymmetries in the local curvature of WMAP data., Astrophys. J. 607 L67-L70.
[12] Hernandez, E. and Weiss, G. (1996)., A First Course on Wavelets Florida: CRC Press. · Zbl 0885.42018
[13] Jin, J., Starck, J.-L., Donoho, D.L., Aghanim, N. and Forni, O. (2005). Cosmological non-Gaussian signature detection: Comparing performance of different statistical tests., EURASIP J. Appl. Signal Process. 15 2470-2485. · Zbl 1127.94335 · doi:10.1155/ASP.2005.2470
[14] Mardia, K.V. and Jupp, P.E. (2000)., Directional Statistics . Chichester: Wiley. · Zbl 0935.62065
[15] Marinucci, D. (2006). High resolution asymptotics for the angular bispectrum of spherical random fields., Ann. Statist. 34 1-41. · Zbl 1104.60020 · doi:10.1214/009053605000000903
[16] McEwen, J.D., Vielva, P., Hobson, M.P., Martinez-Gonzalez, E. and Lasenby, A.N. (2007). Detection of the integrated Sachs-Wolfe effect and corresponding dark energy constraints made with directional spherical wavelets., Monthly Notices Roy. Astronom. Soc. 376 1211-1226.
[17] McEwen, J.D., Hobson, M.P., Lasenby, A.N. and Mortlock, D.J. (2006). A high-significance detection of non-Gaussianity in the WMAP 3-year data using directional spherical wavelets., Monthly Notices Roy. Astronom. Soc. 371 L50-L54.
[18] Moudden, Y., Cardoso, J.-F., Starck, J.-L. and Delabrouille, J. (2005). Blind component separation in wavelet space. Application to CMB analysis., EURASIP J. Appl. Signal Process. 15 2437-2454. · Zbl 1141.85301 · doi:10.1155/ASP.2005.2437
[19] Narcowich, F.J., Petrushev, P. and Ward, J.D. (2006). Localized tight frames on the sphere., SIAM J. Math. Anal. Appl. 38 574-594. · Zbl 1143.42034 · doi:10.1137/040614359
[20] Narcowich, F.J., Petrushev, P. and Ward, J.D. (2006). Decomposition of Besov and Triebel-Lizorkin spaces on the sphere., J. Funct. Anal. 238 530-564. · Zbl 1114.46026 · doi:10.1016/j.jfa.2006.02.011
[21] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals., Ann. Probab. 33 177-193. · Zbl 1097.60007 · doi:10.1214/009117904000000621
[22] Pietrobon, D., Balbi, A. and Marinucci, D. (2006) Integrated Sachs-Wolfe effect from the cross correlation of WMAP 3 year and the NRAO VLA sky survey data: New results and constraints on dark energy., Phys. Rev. D 74 043524.
[23] Politis, D.N., Romano, J.P. and Wolf, M. (1999)., Subsampling . New York: Springer. · Zbl 0931.62035
[24] Pycke, J.-R. (2007). A decomposition for invariant tests of uniformity on the sphere., Proc. Amer. Math. Soc. 135 2983-2993. (electronic). · Zbl 1112.62051 · doi:10.1090/S0002-9939-07-08804-1
[25] Sanz, J.L., Herranz, D., Lopez-Caniego, M. and Argueso, F. (2006). Wavelets on the sphere. Application to the detection problem. In, Proceedings of the 14th European Signal Processing Conference ( EUSIPCO 2006 ) (F. Gini and E.E. Kuruoglu, eds.).
[26] Starck, J.-L., Moudden, Y., Abrial, P. and Nguyen, M. (2006). Wavelets, ridgelets and curvelets on the sphere., Astronom Astrophys 446 1191-1204.
[27] Surgailis, D. (2003). CLTs for polynomials of linear sequences: diagram formula with illustrations. In, Theory and Applications of Long-range Dependence 111-127. Boston: Birkhäuser. · Zbl 1032.60017
[28] Varshalovich, D.A., Moskalev, A.N. and Khersonskii, V.K. (1988)., Quantum Theory of Angular Momentum . New Jersey: World Scientific. · Zbl 0725.00003
[29] Vielva, P., Martinez-Gonzalez, E., Barreiro, B., Sanz, J. and Cayon, L. (2004). Detection of non-Gaussianity in the WMAP first year data using spherical wavelets., Astrophysical J. 609 22-34.
[30] Vielva, P., Wiaux, I., Martinez-Gonzalez, E. and Vandergheynst, P. (2007). Alignment and signed-intensity anomalies in WMAP data., Mon. Not. R. Astron. Soc. 381 932-942.
[31] Wiaux, I., Jacques, L. and Vandergheynst, P. (2005). Correspondence principle between spherical and Euclidean wavelets., Astrophys. J. 632 15-28.
[32] Wiaux, I., McEwen, J.D., Vandergheynst, P. and Blanc, O. (2008). Exact reconstruction with directional wavelets on the sphere., Mon. Not. R. Astron. Soc. 388 770-788.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.