We introduce a novel statistical framework for the analysis of replicated point processes that allows for the study of point pattern variability at a population level. By treating point process realizations as random measures, we adopt a functional analysis perspective and propose a form of functional Principal Component Analysis (fPCA) for point processes. The originality of our method is to base our analysis on the cumulative mass functions of the random measures which gives us a direct and interpretable analysis. Key theoretical contributions include establishing a Karhunen-Lo\`{e}ve expansion for the random measures and a Mercer Theorem for covariance measures. We establish convergence in a strong sense, and introduce the concept of principal measures, which can be seen as latent processes governing the dynamics of the observed point patterns. We propose an easy-to-implement estimation strategy of eigenelements for which parametric rates are achieved. We fully characterize the solutions of our approach to Poisson and Hawkes processes and validate our methodology via simulations and diverse applications in seismology, single-cell biology and neurosiences, demonstrating its versatility and effectiveness. Our method is implemented in the pppca R-package.

翻译：我们提出了一种新颖的统计框架，用于分析重复点过程，使得能够在总体层面研究点模式的变异性。通过将点过程实现视为随机测度，我们采用泛函分析视角，并提出了一种适用于点过程的泛函主成分分析（fPCA）方法。本方法的创新点在于将分析基于随机测度的累积质量函数，从而提供直接且可解释的分析结果。关键的理论贡献包括：建立了随机测度的Karhunen-Loève展开以及协方差测度的Mercer定理。我们证明了强收敛性，并引入了主测度的概念，该概念可视为控制观测点模式动态的潜在过程。我们提出了一种易于实现的特征元估计策略，并实现了参数速率。我们完整刻画了该方法对泊松过程和霍克斯过程的解，并通过模拟以及在地震学、单细胞生物学和神经科学中的多样化应用验证了方法的普适性和有效性。本方法已在pppca R包中实现。