# PCA for Point Processes

arXiv: https://arxiv.org/abs/2404.19661 · HTML: https://mlsanity.com/paper/2404.19661/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 64 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 6 of 6)

- [COEVOLVE: A Joint Point Process Model for Information Diffusion and Network Co-evolution](https://mlsanity.com/paper/1507.02293/) (2015)
- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Sparse and low-rank multivariate Hawkes processes](https://mlsanity.com/paper/1501.00725/) (2015)
- [Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity](https://mlsanity.com/paper/2205.04107/) (2022)
- [Neuronal Network Inference and Membrane Potential Model using Multivariate Hawkes Processes](https://mlsanity.com/paper/2108.00758/) (2021)
- [Network Hawkes Process Models for Exploring Latent Hierarchy in Social Animal Interactions](https://mlsanity.com/paper/2012.09598/) (2020)

## Built on by (most cited first, 2 of 2)

- [Multiresolution Analysis and Statistical Thresholding on Dynamic Networks](https://mlsanity.com/paper/2506.01208/) (2025)
- [Non-Negative Matrix Factorization for Event Data](https://mlsanity.com/paper/2606.06205/) (2026)

## Abstract (arXiv)

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`eve 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.