Abstract
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction–diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction–diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction–diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling.
Highlights
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents
The computational complexity of the reaction–diffusion master equation’ (RDME) obviously increases as the spatial discretization becomes finer, and in many cases the limiting process does not lead to the original SRDP17
Using the classical theory of the Poisson representation (PR) for stochastic reaction processes[27], we show that marginal probability distributions of SRDPs can be approximated in a mean-field sense by spatio-temporal Cox point processes, a class of models widely used in spatio-temporal statistics[28]
Summary
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. The flexibility afforded by the local interaction rules has led to a wide application of SRDPs in many different scientific disciplines where complex spatio-temporal behaviours arise, from molecular biology[4,11,12], to ecology[13], and to the social sciences[14]. Despite their popularity, SRDPs pose considerable challenges, as analytical computations are only possible for a handful of systems[8]. As far as we are aware, the few attempts at statistical inference for SRDPs either used simulation-based likelihood free methods[13], inheriting the intrinsic computational difficulties discussed above, or abandoned the SRDP framework by adopting a coarse space discretization[25] or neglecting the individual nature of agents using a linear-noise approximation[26]
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