The Annals of Applied Statistics

Testing the isotropy of high energy cosmic rays using spherical needlets

Gilles Faÿ, Jacques Delabrouille, Gérard Kerkyacharian, and Dominique Picard

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For many decades, ultrahigh energy charged particles of unknown origin that can be observed from the ground have been a puzzle for particle physicists and astrophysicists. As an attempt to discriminate among several possible production scenarios, astrophysicists try to test the statistical isotropy of the directions of arrival of these cosmic rays. At the highest energies, they are supposed to point toward their sources with good accuracy. However, the observations are so rare that testing the distribution of such samples of directional data on the sphere is nontrivial. In this paper, we choose a nonparametric framework that makes weak hypotheses on the alternative distributions and allows in turn to detect various and possibly unexpected forms of anisotropy. We explore two particular procedures. Both are derived from fitting the empirical distribution with wavelet expansions of densities. We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574–594 (electronic)], the so-called needlets. The expansions are truncated at scale indices no larger than some $J^{\star}$, and the $L^{p}$ distances between those estimates and the null density are computed. One family of tests (called MULTIPLE) is based on the idea of testing the distance from the null for each choice of $J=1,\ldots,J^{\star}$, whereas the so-called PLUGIN approach is based on the single full $J^{\star}$ expansion, but with thresholded wavelet coefficients. We describe the practical implementation of these two procedures and compare them to other methods in the literature. As alternatives to isotropy, we consider both very simple toy models and more realistic nonisotropic models based on Physics-inspired simulations. The Monte Carlo study shows good performance of the MULTIPLE test, even at moderate sample size, for a wide sample of alternative hypotheses and for different choices of the parameter $J^{\star}$. On the 69 most energetic events published by the Pierre Auger Collaboration, the needlet-based procedures suggest statistical evidence for anisotropy. Using several values for the parameters of the methods, our procedures yield $p$-values below 1%, but with uncontrolled multiplicity issues. The flexibility of this method and the possibility to modify it to take into account a large variety of extensions of the problem make it an interesting option for future investigation of the origin of ultrahigh energy cosmic rays.

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Ann. Appl. Stat., Volume 7, Number 2 (2013), 1040-1073.

First available in Project Euclid: 27 June 2013

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Nonparametric test isotropy test multiple tests ultrahigh energy cosmic rays wavelet procedure


Faÿ, Gilles; Delabrouille, Jacques; Kerkyacharian, Gérard; Picard, Dominique. Testing the isotropy of high energy cosmic rays using spherical needlets. Ann. Appl. Stat. 7 (2013), no. 2, 1040--1073. doi:10.1214/12-AOAS619.

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  • Abraham, J. et al. (2008). Observation of the suppression of the flux of cosmic rays above $4{\times}10^{19}$ eV. Phys. Rev. Lett. 101 061101.
  • Abraham, J. et al. (2010a). Measurement of the depth of maximum of extensive air showers above $10^{18}$ eV. Phys. Rev. Lett. 104 091101.
  • Abraham, J. et al. (2010b). Measurement of the energy spectrum of cosmic rays above $10^{18}$ eV using the Pierre Auger observatory. Phys. Lett. B 685 239–246.
  • Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2009a). Adaptive density estimation for directional data using needlets. Ann. Statist. 37 3362–3395.
  • Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2009b). Asymptotics for spherical needlets. Ann. Statist. 37 1150–1171.
  • Beck, R. (2011). Cosmic magnetic fields: Observations and prospects. In American Institute of Physics Conference Series (F. A. Aharonian, W. Hofmann and F. M. Rieger, eds.) 1381 117–136. American Institute of Physics, Melville, NY.
  • Benoît, A. et al. (2004). First detection of polarization of the submillimetre diffuse galactic dust emission by Archeops. Astron. Astrophys. 424 571–582.
  • Bhattacharjee, P. and Sigl, G. (2000). Origin and propagation of extremely high energy cosmic rays. Phys. Rep. 327 109–247.
  • Bickel, P., Kleijn, B. and Rice, J. (2008). Event-weighted tests for detecting periodicity in photon arrival times. Astrophysical Journal 685 384–389.
  • Bickel, P. J., Ritov, Y. and Stoker, T. M. (2006). Tailor-made tests for goodness of fit to semiparametric hypotheses. Ann. Statist. 34 721–741.
  • Butucea, C. and Tribouley, K. (2006). Nonparametric homogeneity tests. J. Statist. Plann. Inference 136 597–639.
  • Cronin, J. W. (2005). The highest-energy cosmic rays. Nuclear Physics B Proceedings Supplements 138 465–491.
  • Crutcher, R. M., Wandelt, B., Heiles, C., Falgarone, E. and Troland, T. H. (2010). Magnetic fields in interstellar clouds from Zeeman observations: Inference of total field strengths by Bayesian analysis. Astrophysical Journal 725 466–479.
  • Delabrouille, J., Cardoso, J. F., Le Jeune, M., Betoule, M., Faÿ, G. and Guilloux, F. (2009). A full sky, low foreground, high resolution CMB map from WMAP. Astronomy and Astrophysics 493 835–857.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • Efron, B. and Petrosian, V. (1995). Testing isotropy versus clustering of Gamma-Ray bursts. Astrophysical Journal 449 216.
  • Faÿ, G. and Guilloux, F. (2011). Spectral estimation on the sphere with needlets: High frequency asymptotics. Stat. Inference Stoch. Process. 14 47–71.
  • Faÿ, G., Delabrouille, J., Kerkyacharian, G. and Picard, D. (2013). Supplement to “Testing the isotropy of high energy cosmic rays using spherical needlets.” DOI:10.1214/12-AOAS619SUPP.
  • Faÿ, G., Guilloux, F., Betoule, M., Cardoso, J. F., Delabrouille, J. and Le Jeune, M. (2008). CMB power spectrum estimation using wavelets. Phys. Rev. D 78 083013.
  • Fosalba, P., Lazarian, A., Prunet, S. and Tauber, J. A. (2002). Statistical properties of galactic starlight polarization. Astrophysical Journal 564 762–772.
  • Fromont, M. and Laurent, B. (2006). Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 680–720.
  • Giné M., E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243–1266.
  • Górski, K., Hivon, E., Banday, A., Wandelt, B., Hansen, F., Reinecke, M. and Bartelmann, M. (2005). HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophysical Journal 622 759–771.
  • Greisen, K. (1966). End to the cosmic-ray spectrum? Phys. Rev. Lett. 16 748–750.
  • Han, J. L., Manchester, R. N., Lyne, A. G., Qiao, G. J. andvan Straten, W. (2006). Pulsar rotation measures and the large-scale structure of the galactic magnetic field. Astrophysical Journal 642 868–881.
  • Harari, D., Mollerach, S. and Roulet, E. (2002). Astrophysical magnetic field reconstruction and spectroscopy with ultra high energy cosmic rays. J. High Energy Phys. 7 6.
  • Heiles, C. (1996). A comprehensive view of the galactic magnetic field, especially near the Sun. In Polarimetry of the Interstellar Medium (W. G. Roberge and D. C. B. Whittet, eds.). Astronomical Society of the Pacific Conference Series 97 457. Astonomical Society of the Pacific, San Francisco, CA.
  • Hillas, A. M. (1984). The origin of ultra-high-energy cosmic rays. Annual Review of Astronomy and Astrophysics 22 425–444.
  • Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85–114.
  • Ingster, Y. I. (2000). Adaptive chi-square tests. J. Math. Sciences 99 1110–1120.
  • Ingster, Y. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. Springer, New York.
  • Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on $\mathbb{R}$. Bernoulli 10 187–220.
  • Kachelriess, M. and Semikoz, D. V. (2006). Clustering of ultra-high energy cosmic ray arrival directions on medium scales. Astroparticle Physics 26 10–15.
  • Kerkyacharian, G., Pham Ngoc, T. M. and Picard, D. (2011). Localized spherical deconvolution. Ann. Statist. 39 1042–1068.
  • Kerkyacharian, G., Petrushev, P., Picard, D. and Willer, T. (2007). Needlet algorithms for estimation in inverse problems. Electron. J. Stat. 1 30–76.
  • Kerkyacharian, G., Kyriazis, G., Le Pennec, E., Petrushev, P. and Picard, D. (2010). Inversion of noisy Radon transform by SVD based needlets. Appl. Comput. Harmon. Anal. 28 24–45.
  • Kotera, K. and Olinto, A. V. (2011). The astrophysics of ultrahigh-energy cosmic rays. Annual Review of Astron and Astrophys 49 119–153.
  • Lacour, C. andPham Ngoc, T. M. (2012). Goodness-of-fit test for noisy directional data. Available at arXiv:1203.2008.
  • Lan, X. and Marinucci, D. (2009). On the dependence structure of wavelet coefficients for spherical random fields. Stochastic Process. Appl. 119 3749–3766.
  • Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
  • Marinucci, D., Pietrobon, D., Balbi, A., Baldi, P., Cabella, P., Kerkyacharian, G., Natoli, P., Picard, D. and Vittorio, N. (2008). Spherical needlets for CMB data analysis. Monthly Notices of the Royal Astronomical Society 383 539–545.
  • Martínez, V. J. and Saar, E. (2002). Statistics of the galaxy distribution. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge Univ. Press, Cambridge. Translated from the 1990 French original by D. H. Salinger.
  • Narayan, R. and Piran, T. (1993). Do gamma-ray burst sources repeat? Monthly Notices of the Royal Astronomical Society 265 L65–L68.
  • Narcowich, F., Petrushev, P. and Ward, J. (2006a). Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J. Funct. Anal. 238 530–564.
  • Narcowich, F. J., Petrushev, P. and Ward, J. D. (2006b). Localized tight frames on spheres. SIAM J. Math. Anal. 38 574–594 (electronic).
  • Page, L. et al. (2007). Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Polarization analysis. Astrophysical Journal Supplement 170 335–376.
  • Petrushev, P. and Xu, Y. (2008). Localized polynomial frames on the ball. Constr. Approx. 27 121–148.
  • Pierre Auger Collaboration (Abraham, J. et al.) (2008). Correlation of the highest-energy cosmic rays with the positions of nearby active galactic nuclei. Astroparticle Physics 29 188–204.
  • Pierre Auger Collaboration (Abraham, J. et al.) (2009). Upper limit on the cosmic-ray photon fraction at EeV energies from the Pierre Auger observatory. Astroparticle Physics 31 399–406.
  • Pierre Auger Collaboration (Abreu, P. et al.) (2010). Update on the correlation of the highest energy cosmic rays with nearby extragalactic matter. Astroparticle Physics 34 314–326.
  • Pietrobon, D., Balbi, A. and Marinucci, D. (2006). Integrated Sachs–Wolfe effect from the cross correlation of WMAP3 year and the NRAO VLA sky survey data: New results and constraints on dark energy. Phys. Rev. D 74 043524.
  • Pietrobon, D., Amblard, A., Balbi, A., Cabella, P., Cooray, A. and Marinucci, D. (2008). Needlet detection of features in the WMAP CMB sky and the impact on anisotropies and hemispherical asymmetries. Phys. Rev. D 78 103504.
  • Quashnock, J. M. and Lamb, D. Q. (1993). Evidence for the Galactic origin of gamma-ray bursts. Monthly Notices of the Royal Astronomical Society 265 L45–L50.
  • Rudjord, Ø., Hansen, F. K., Lan, X., Liguori, M., Marinucci, D. and Matarrese, S. (2009). An estimate of the primordial non-Gaussianity parameter $f_{\mathrm{NL}}$ using the needlet bispectrum from WMAP. Astrophysical Journal 701 369–376.
  • Sommers, P. (2001). Cosmic ray anisotropy analysis with a full-sky observatory. Astroparticle Physics 14 271–286.
  • Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
  • Starck, J. L., Moudden, Y., Abrial, P. and Nguyen, M. (2006). Wavelets, ridgelets and curvelets on the sphere. Astronomy and Astrophysics 446 1191–1204.
  • Torres, D. F. and Anchordoqui, L. A. (2004). Astrophysical origins of ultrahigh energy cosmic rays. Rep. Progr. Phys. 67 1663–1730.
  • Triebel, H. (1992). Theory of Function Spaces. II. Monographs in Mathematics 84. Birkhäuser, Basel.
  • Zatsepin, G. T. and Kuz’min, V. A. (1966). Upper limit of the spectrum of cosmic rays. Soviet Journal of Experimental and Theoretical Physics Letters 4 78.

Supplemental materials

  • Supplementary material: Supplement to “Testing the isotropy of high energy cosmic rays with spherical needlets”. In the supplement, we recall the construction of the needlet decomposition on the sphere, and discuss its practical usage. We also complete the Section 5 of this paper with more results obtained from Monte-Carlo simulations.