Marco Scutari, Francesca Panero & Manuel Proissl

Silvia Chiappa, Thomas P. S. Gillam

Evgenii Chzhen, Nicolas Schreuder

Sam Corbett-Davies, Johann D. Gaebler, Hamed Nilforoshan, Ravi Shroff, Sharad Goel

Jack is a Senior Lecturer in the Department of Econometrics and Business Statistics at Monash University, Melbourne, Australia. Jack's research focuses on methodological and computational challenges in modern Bayesian Inference. Particular areas of interest include Model Misspecification, Robustness, Inference with Loss Functions, and Variable/Model Selection

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-Loève 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.