Approximate solutions of a general stochastic velocity-jump model subject to discrete-time noisy observations

TL;DR

This paper develops approximate solutions for a velocity-jump model of single-agent motion with noisy, discrete observations, enabling faster predictions and better experimental design in high-resolution tracking data analysis.

Contribution

It introduces a series of approximations for the intractable distributions of a Markovian velocity-jump model under noisy observations, validated through simulations.

Findings

Approximations are accurate with infrequent state switching.

The methods enable fast predictions and likelihood-based inference.

Validation confirms the effectiveness of the approximations.

Abstract

Advances in experimental techniques allow the collection of high-resolution spatio-temporal data that track individual motile entities over time. These tracking data motivate the use of mathematical models to characterise the motion observed. In this paper, we aim to describe the solutions of velocity-jump models for single-agent motion in one spatial dimension, characterised by successive Markovian transitions within a finite network of n states, each with a specified velocity and a fixed rate of switching to every other state. In particular, we focus on obtaining the solutions of the model subject to noisy, discrete-time, observations, with no direct access to the agent state. The lack of direct observation of the hidden state makes the problem of finding the exact distributions generally intractable. Therefore, we derive a series of approximations for the data distributions. We verify the accuracy of these approximations by comparing them to the empirical distributions generated through simulations of four example model structures. These comparisons confirm that the approximations are accurate given sufficiently infrequent state switching relative to the imaging frequency. The approximate distributions computed can be used to obtain fast forwards predictions, to give guidelines on experimental design, and as likelihoods for inference and model selection.