Methodological advances for point processes on complex domains

Recent advances in 3D biological imaging technologies now allow spatial point patterns of protein molecules to be observed directly on a cell's outer membrane, where the underlying geometry is complex and must be respected for principled statistical inference. In prior work, we developed functional summary statistics for point patterns on the surfaces of 3D convex shapes, leading to new insights into E. coli outer membrane assembly. In this work, we extend these methods to the multi-
type setting. We begin by developing functional summary statistics for multi-type homogeneous and inhomogeneous point processes on the sphere, and then generalise them to convex 3D surfaces, assuming a known bijective mapping from the shape to the sphere. To support inference in the inhomogeneous case, we employ a plug-in estimator for the intensity function of a spatial point process on a manifold. We demonstrate how these statistics can be used to test for independence between the component processes, with particular focus on methods for generating samples from the null distribution. We conclude with a discussion on extending the framework to a class of non-convex shapes.
 

The useful information contained in spatio-temporal data is often represented in the form of geometric structures and patterns. The filaments or clusters of galaxies in our Universe are one such example.

There are two situations to consider. First, the pattern of interest is hidden in the data set, so the pattern must be detected. Second, the structure of interest is observed, so a relevant characterisation of it should be performed. Probabilistic modelling and Bayesian statistical inference are approaches that can provide answers to these questions.

In this talk, Gibbs-marked point processes with interactions are presented for detection and characterisation of the patterns of interest. Classical and tailored MCMC samplers are presented to simulate the proposed models. Based on these ingredients, global optimisation and posterior sampling algorithms are built to perform statistical inference to detect and characterise the patterns of interest. Applications of the proposed approaches in astronomy, geology and network sciences are also shown.

 

Network point processes often exhibit latent structure that govern the behaviour of the sub-processes. It is not always reasonable to assume that this latent structure is static, and detecting when and how this driving structure changes is often of interest. In this talk, we introduce a novel online methodology for detecting changes within the latent structure of a network point process. We focus on block-homogeneous Poisson processes, where latent node memberships determine the rates of the edge processes. We propose a scalable variational procedure which can be applied on large networks in an online fashion via a Bayesian forgetting factor applied to sequential variational approximations to the posterior distribution. The proposed framework is tested on simulated and real-world data, and it rapidly and accurately detects changes to the latent edge process rates, and to the latent node group memberships, both in an online manner. In particular, in an application on the Santander Cycles bike-sharing network in central London, we detect changes within the network related to holiday periods and lockdown restrictions between 2019 and 2020.