PCA for Point Processes

TL;DR

This paper develops a functional PCA framework for analyzing replicated point processes, enabling population-level variability study through interpretable measures, with theoretical foundations, estimation strategies, and diverse applications.

Contribution

It introduces a novel fPCA method for point processes based on cumulative mass functions, with theoretical guarantees and practical implementation in R.

Findings

The method provides interpretable principal measures.

Theoretical results include Karhunen-Loève expansion and Mercer Theorem.

Validated through simulations and applications in seismology, biology, and neuroscience.

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.