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

by Arianna Ceccarelli, Alexander P. Browning, and Ruth E. Baker

We analyse velocity-jump models for single-agent motion in one spatial dimension, in which the agent transitions between n states, each with a set velocity and fixed switching rates to other states. Since the agent’s true state cannot be observed, computing the exact distributions of discrete-time noisy data is generally intractable. Therefore, we derive approximations for the observed data distributions, both for a single measurement and considering the correlation between locations of a tracked agent, and validate them through simulations of four model structures. These approximations enable fast predictions, guide experimental design, and can be used as likelihoods for inference and model selection.

Solutions of an example three-state model. The three-state model (panel B) comprises a forward-movement state, a backward-movement state and a stationary state. A short data track is shown in panel A. A location increment y is defined. as the difference between subsequently measured locations. Panel C compares the empirical distribution, generated with in-silico data, to one of the approximations obtained in the manuscript for single location increments. Panel D compares the empirical distribution for two subsequent location increments to the joint approximation obtained in the manuscript, and to the marginal approximation which does not take into account the correlation between subsequent location increments.