Multiple testing of local maxima for detection of peaks on the (celestial) sphere. (English) Zbl 1446.62254

Cheng, Dan; Cammarota, Valentina; Fantaye, Yabebal; Marinucci, Domenico; Schwartzman, Armin

Bernoulli 26, No. 1, 31-60 (2020).

The authors extend the STEM procedure of D. Cheng and A. Schwartzman [Ann. Stat. 45, No. 2, 529–556 (2017; Zbl 1369.62144)] to the case of a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. They also develop a statistical procedure to control the probability of observing false detections, the fraction of false detections (FDP), and the expected FDP in a population. The performance of the proposed procedures is explored numerically for simulated cosmic microwave background radiation fields.
The authors study the sequence of signal-plus-noise models \[ y_{N}(x) = \mu_{N}(x)+ z(x),~x \in \mathbb{S}^{2}, ~N=1,2,\dots \] where \(z\) is a Gaussian, zero-mean, isotropic field on the unit sphere \(\mathbb{S}^{2}\) expressed as \[ z(x)=\sum_{l=1}^{\infty}z_{l}(x),~z_{l}(x)=\sum_{m=-l}^{l}a_{l,m}Y_{l,m}(x) \] with \(Y_{l,m}\) standing for the spherical harmonics, and \(\mu_{N}\) is a sequence of deterministic functions defined by \[ \mu_{N}(x) =\sum_{k=1}^{N}a_{k}h(x, t_{k,N}, \xi_{k}) \] with \(a_{k}>0\) \((k=1, 2, \dots),\) \[ h(x, t, \xi)=\sum_{l=1}^{\infty}\frac{2l+1}{4\pi}e^{-l(l+1)t}P_{l}(\langle x,\xi\rangle), \] \[ P_{l}(u) = \frac{(-1)^{l}}{2^{l}}\frac{d^{l}}{du^{l}}(1 -u^{2})^{l},~ l= 1, 2, \dots, \] and \(\langle \cdot,\cdot\rangle\) standing for the scalar product. It is assumed that \(t_{k,N} \rightarrow 0\) as \(N\rightarrow \infty\).
The observable data is \[ \hat y_{N_j}(x) = \hat \mu_{N_j}(x)+ \hat z_{j}(x),~x \in \mathbb{S}^{2}, ~N_j=1,2,\dots \] where \[ \hat z_{j}(x)=\sum_{l=1}^{\infty}b\left(\frac{l}{B^{j}},s\right)z_{l}(x) \] and \[ \hat \mu_{N_{j}}(x) =\sum_{k=1}^{N} \sum_{l=1}^{\infty}a_{k}b\left(\frac{l}{B^{j}},s\right)\frac{2l+1}{4\pi}e^{-l(l+1)t_{k,N}}P_{l}(\langle x,\xi_{k}\rangle) \] with \(b(u, s) = u^{2s}e^{-u^{2}}\) and bandwidth parameter \(B>1\).
The STEM algorithm is as follows:

1.

Normalize \(\hat y_{N_j}\) to obtain \(\tilde y_{N_j} = \frac{\hat y_{N_j}}{\sqrt{\mathbb{E}[\hat z_{j}^{2}]}}\), where the denominator is assumed to be known.

2.

Find the set of local maxima: \[ T_{j} = \{x \in \mathbb{S}^{2} : \nabla\tilde y_{N_j}(x)=0, \nabla^{2}\tilde y_{N_j}(x) \text{ is negative definite}\}. \]

3.

For each \(x \in T_{j}\), compute the \(p\)-value of the hypothesis “\(\mu_{j}(x')=0\) for all \(x'\in B(x,\rho_{j})\)” where \(B(x,\rho_{j})\) is a geodesic ball centered at \(x\) on the sphere and of radius \(\rho_{j} \sim j^{\nu}B^{-j}\) for some \(\nu>0\).

4.

Apply the multiple testing procedure of Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)] on the set of \(p\)-values obtained, and declare significant all local maxima whose \(p\)-values are smaller than the significance threshold.

The authors obtain explicit upper bounds for the FDP and the expected FDP, and show that the proportion of detected peaks tends to 1 asymptotically. Numerical analysis using simulated cosmic microwave background radiation fields is then performed to assess the accuracy of the analytic approximation for the \(p\)-values in Step 3 of STEM, and the tightness of the upper bounds for FDP and expected FDP.

Reviewer: Tamás Mátrai (Edinburgh)

MSC:

62M40	Random fields; image analysis
62J15	Paired and multiple comparisons; multiple testing
60G60	Random fields
78A40	Waves and radiation in optics and electromagnetic theory
62P35	Applications of statistics to physics
85A25	Radiative transfer in astronomy and astrophysics
62R40	Topological data analysis

Keywords:

cosmic microwave background radiation field; signal-plus-noise model; isotropic field on the unit sphere; false discovery rate; Gaussian random fields; height distribution; Mexican needlet transform; overshoot distribution; STEM algorithm; multiple testing; topological multiple testing scheme

Citations:

Zbl 1369.62144; Zbl 0809.62014

Software:

Healpix

References:

[1]	Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2009). Asymptotics for spherical needlets. Ann. Statist. 37 1150-1171. · Zbl 1160.62087 · doi:10.1214/08-AOS601
[2]	Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289-300. · Zbl 0809.62014 · doi:10.1111/j.2517-6161.1995.tb02031.x
[3]	Cammarota, V., Marinucci, D. and Wigman, I. (2016). On the distribution of the critical values of random spherical harmonics. J. Geom. Anal. 26 3252-3324. · Zbl 1353.60020 · doi:10.1007/s12220-015-9668-5
[4]	Cheng, D., Cammarota, V., Fantaye, Y., Marinucci, D. and Schwartzman, A. (2020). Supplement to “Multiple testing of local maxima for detection of peaks on the (celestial) sphere.” https://doi.org/10.3150/18-BEJ1068SUPP. · Zbl 1446.62254
[5]	Cheng, D. and Schwartzman, A. (2015). Distribution of the height of local maxima of Gaussian random fields. Extremes 18 213-240. · Zbl 1319.60106 · doi:10.1007/s10687-014-0211-z
[6]	Cheng, D. and Schwartzman, A. (2017). Multiple testing of local maxima for detection of peaks in random fields. Ann. Statist. 45 529-556. · Zbl 1369.62144 · doi:10.1214/16-AOS1458
[7]	Cheng, D. and Schwartzman, A. (2018). Expected number and height distribution of critical points of smooth isotropic Gaussian random fields. Bernoulli 24 3422-3446. · Zbl 1407.60051 · doi:10.3150/17-BEJ964
[8]	Delabrouille, J. et al. (2013). The pre-launch Planck Sky Model: A model of sky emission at submillimetre to centimetre wavelengths. Astron. Astrophys. 553 A96.
[9]	Durrer, R. (2008). The Cosmic Microwave Background. Cambridge: Cambridge Univ. Press. · Zbl 0897.58062 · doi:10.1007/BF02768937
[10]	Geller, D. and Mayeli, A. (2009). Continuous wavelets on compact manifolds. Math. Z. 262 895-927. Available at arXiv:0811.4440. · Zbl 1193.42123 · doi:10.1007/s00209-008-0405-7
[11]	Geller, D. and Mayeli, A. (2009). Nearly tight frames and space-frequency analysis on compact manifolds. Math. Z. 263 235-264. Available at arXiv:0706.3642. · Zbl 1193.42124 · doi:10.1007/s00209-008-0406-6
[12]	Górski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M. and Bartelmann, M. (2005). HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 699 759-771.
[13]	Lan, X. and Marinucci, D. (2009). On the dependence structure of wavelet coefficients for spherical random fields. Stochastic Process. Appl. 119 3749-3766. · Zbl 1177.60053 · doi:10.1016/j.spa.2009.07.005
[14]	Loh, W.-L. (2005). Fixed-domain asymptotics for a subclass of Matérn-type Gaussian random fields. Ann. Statist. 33 2344-2394. · Zbl 1086.62111 · doi:10.1214/009053605000000516
[15]	Marinucci, D. and Peccati, G. (2011). Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. London Mathematical Society Lecture Note Series 389. Cambridge: Cambridge Univ. Press. · Zbl 1260.60004
[16]	Marinucci, D. and Peccati, G. (2013). Mean-square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 no. 37, 10. · Zbl 1329.60139 · doi:10.1214/ECP.v18-2400
[17]	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. Mon. Not. R. Astron. Soc. 383 539-545.
[18]	Mayeli, A. (2010). Asymptotic uncorrelation for Mexican needlets. J. Math. Anal. Appl. 363 336-344. · Zbl 1185.42038 · doi:10.1016/j.jmaa.2009.07.044
[19]	Narcowich, F.J., Petrushev, P. and Ward, J.D. (2006). Localized tight frames on spheres. SIAM J. Math. Anal. 38 574-594. · Zbl 1143.42034 · doi:10.1137/040614359
[20]	Planck Collaboration (2016). Planck 2015 results. I. Overview of products and scientific results. Astron. Astrophys. 594 A1. Available at arXiv:1507.02704.
[21]	Planck Collaboration (2016). Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of parameters. Astron. Astrophys. 594 A11. Available at arXiv:1507.02704.
[22]	Planck Collaboration (2016). Planck 2015. XXVI. The second Planck catalogue of compact sources. Astron. Astrophys. 594 A26.
[23]	Schwartzman, A., Gavrilov, Y. and Adler, R.J. (2011). Multiple testing of local maxima for detection of peaks in 1D. Ann. Statist. 39 3290-3319. · Zbl 1246.62173 · doi:10.1214/11-AOS943
[24]	Scodeller, S., Hansen, F.K. and Marinucci, D. (2012). Detection of new point sources in WMAP 7 year data using internal templates and needlets. Astrophys. J. 753 27. Available at arXiv:1201.5852.
[25]	Scodeller, S., Rudjord, O., Hansen, F.K., Marinucci, D., Geller, D. and Mayeli, A. (2011). Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets. Astrophys. J. 733 121.