PCA for Point Processes

2024-04-30

AI Summary

This paper addresses the challenge of modeling population-level variability in replicated point processes. We propose a novel functional principal component analysis (fPCA) framework grounded in random measures and the cumulative mass function (CMF). Introducing the concept of “principal measures,” we establish the Karhunen–Loève expansion for random measures and derive a Mercer-type theorem for their covariance measures, enabling consistent parameter-rate estimation of eigencomponents. The method integrates fPCA, random measure theory, and nonparametric/semiparametric estimation, yielding closed-form solutions for Poisson and Hawkes processes. Evaluated on seismological, single-cell spatial transcriptomic, and neurophysiological datasets, our approach significantly improves both the accuracy of identifying population-level variation structures in point patterns and their biological interpretability.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

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.

Problem

• Analyzing variability in replicated point processes at population level
• Developing functional PCA for point processes using cumulative mass functions
• Characterizing solutions for Poisson and Hawkes processes with applications

Innovation

• Functional PCA for point processes analysis
• Principal measures as latent dynamic processes
• Easy-to-implement estimation with parametric rates