Nonlinear Filtering and Spatial Asymptotic Consistency for SPDEs Observed via Spatio-Temporal Point Processes

TL;DR

This paper develops a mathematical framework for filtering stochastic PDEs observed through point processes, demonstrating asymptotic consistency and validating results with simulations of biophysical intracellular wave data.

Contribution

It introduces new filtering equations for SPDEs with point process observations and proves their asymptotic consistency, advancing the understanding of such complex systems.

Findings

Derived unnormalized and normalized filtering equations for SPDEs with point process observations.

Proved asymptotic consistency of the filtering methods.

Validated theoretical results with extensive simulations on synthetic and real data.

Abstract

In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.