We address the problems of extracting information generated by one dimensional periodic point processes. These problems arise in numerous situations, from astronomy and biomedical applications to reliability and quality control and signal processing. We divide our analysis into two cases, namely single and then multiple source(s). We wish to extract the fundamental period of the generator(s), and, in the second case, to deinterleave the processes. We present two algorithms, designed to work on all one dimensional periodic processes, but in particular on sparse datasets where other procedures break down. The first algorithm works on data from single period processes, computing an estimate of the underlying period. It is extremely computationally efficient and straightforward, and works on all single period processes, but in particular on sparse datasets where others break down. Its justification, however, rests on some deep mathematics, including a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to analyze data from multiple periodic processes. This second procedure relies on the Riemann zeta function, Weyl's equidistribution theorem, and Wiener's periodogram.

中文翻译：

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周期点过程：理论与应用

我们解决了提取一维周期点过程生成的信息的问题。从天文学和生物医学应用到可靠性和质量控制以及信号处理，这些问题在许多情况下都会出现。我们将分析分为两种情况，即单一来源然后是多个来源。我们希望提取生成器的基本周期，在第二种情况下，希望对过程进行解交织。我们提出了两种算法，设计用于所有一维周期过程，但特别是在其他过程发生故障的稀疏数据集上。第一种算法处理来自单周期过程的数据，计算基础周期的估计。它具有极高的计算效率和直截了当的功能，并且适用于所有单周期流程，但特别是在其他细分的稀疏数据集上。但是，其辩解基于某些深层数学，包括对黎曼zeta函数的概率解释。然后，我们基于此过程来分析来自多个周期性过程的数据。第二个过程依赖Riemann zeta函数，Weyl等分定理和Wiener周期图。