This review is devoted to recent developments in the statistical analysis of spherical data, strongly motivated by applications in cosmology. We start from a brief discussion of cosmological questions and motivations, arguing that most cosmological observables are spherical random fields. Then, we introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We then focus on some specific issues, including tools and algorithms for map reconstruction (i.e., separating the different physical components that contribute to the observed field), geometric tools for testing the assumptions of Gaussianity and isotropy, and multiple testing methods to detect contamination in the field due to point sources. Although these tools are introduced in the cosmological context, they can be applied to other situations dealing with spherical data. Finally, we discuss more recent and challenging issues, such as the analysis of polarization data, which can be viewed as realizations of random fields taking values in spin fiber bundles.

中文翻译：

---

球体上宇宙学数据的几何方法


这篇综述致力于球面数据统计分析的最新发展，其强烈动机是宇宙学中的应用。我们从对宇宙学问题和动机的简要讨论开始，论证大多数宇宙学可观测物都是球形随机场。然后，我们介绍了球面随机场的一些数学背景，包括光谱表示以及针状和小波框架的构造。然后，我们关注一些具体问题，包括用于地图重建的工具和算法（即分离有助于观察场的不同物理成分）、用于测试高斯分布和各向同性假设的几何工具，以及用于检测现场由于点源造成的污染的多种测试方法。尽管这些工具是在宇宙学背景下引入的，但它们可以应用于处理球面数据的其他情况。最后，我们讨论了最近和具有挑战性的问题，例如偏振数据的分析，这可以被视为随机场在自旋光纤束中取值的实现。