球面单分形与多重分形随机场的分析
Analysis of Spherical Monofractal and Multifractal Random Fields
摘要：
Rényi函数在多重分形随机场的分析中起着重要的作用。对于球面上的随机场，文献中有三种模型，其中Rényi函数是显式已知的。本文的理论部分提出了多重分形随机场的领域和开发具体的模型，可以明确计算的雷尼函数。然后将这些结果应用于普朗克任务收集的宇宙微波背景辐射数据。在文献中，用于描述这些数据的主要统计模型是各向同性高斯场。我们提出了基于模拟随机场、计算Rényi函数和多重分形谱的多重分形研究方法。
英文摘要：
The Rényi function plays an important role in the analysis of multifractal random fields. For random fields on the sphere, there are three models in the literature where the Rényi function is known explicitly. The theoretical part of the article presents multifractal random fields on the sphere and develops specific models where the Rényi function can be computed explicitly. Then these results are applied to the Cosmic Microwave Background Radiation data collected from the Planck mission. The main statistical model used to describe these data in the literature is isotropic Gaussian fields. We present numerical multifractality studies and methodology based on simulating random fields, computing the Rényi function and the multifractal spectrum for different scenarios and actual CMB data.
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