Multiple testing of local maxima for detection of peaks on the (celestial) sphere

TL;DR

This paper introduces a topological multiple testing method for detecting peaks on the sphere in Gaussian noise, applicable to astronomy data like CMB, with proven control of false discoveries and demonstrated in simulations.

Contribution

It develops a new multiple testing approach for spherical data using needlet transforms, controlling FDR in a non-Euclidean setting, which is novel for astronomical applications.

Findings

Provides asymptotic FDR control and power consistency.

Demonstrates effectiveness in CMB point-source detection simulations.

Adapts multiple testing to spherical data with non-standard asymptotics.

Abstract

We present 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. Our setting is different from the standard Euclidean/large same asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy. More precisely, we focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large. This novel multiple testing method is illustrated in a simulation of point-source detection in Cosmic Microwave Background radiation (CMB) data.