Abstract
Many living and complex systems exhibit second order emergent dynamics. Limited experimental access to the configurational degrees of freedom results in data that appears to be generated by a non-Markovian process. This poses a challenge in the quantitative reconstruction of the model from experimental data, even in the simple case of equilibrium Langevin dynamics of Hamiltonian systems. We develop a novel Bayesian inference approach to learn the parameters of such stochastic effective models from discrete finite length trajectories. We first discuss the failure of naive inference approaches based on the estimation of derivatives through finite differences, regardless of the time resolution and the length of the sampled trajectories. We then derive, adopting higher order discretization schemes, maximum likelihood estimators for the model parameters that provide excellent results even with moderately long trajectories. We apply our method to second order models of collective motion and show that our results also hold in the presence of interactions.
Highlights
Recent experimental findings on a variety of living systems, from cell migration [1], bacterial propulsion [2], and worm dynamics [3] to the larger scale of animal groups on the move [4,5,6,7], indicate that the observed behavior cannot be explained with a first-order dynamical process but requires a higher-order description
We proposed a maximum-likelihood inference strategy to tackle the problem of learning the best continuous inertial stochastic model from time lapse recordings of an observed process
The problems arising in this context are general, as they stem from the combination of the following three ingredients: the second- order nature of the process, when described in terms of the directly measurable degrees of freedom, stochasticity, and the use of discrete sequences of data points
Summary
Recent experimental findings on a variety of living systems, from cell migration [1], bacterial propulsion [2], and worm dynamics [3] to the larger scale of animal groups on the move [4,5,6,7], indicate that the observed behavior cannot be explained with a first-order dynamical process but requires a higher-order description. In the absence of an explicit solution for the stochastic process, the most reasonable thing to do is to transform the stochastic differential equation (SDE) into an approximated difference equation Such discretization must be performed very carefully, since the resulting equation should correctly represent the underlying stochastic process both at the scales of the sampled data (at which inference works) and in the microscopic limit of vanishing increments. These two problems are quite general and do not depend on the presence of interactions in the system but rather on the nature of the dynamics. VII, we summarize all our results, discuss their conceptual relevance, and outline their potential for applications to real data
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