Abstract
A Hamiltonian approach is introduced for the reconstruction of trajectories and models of complex stochastic dynamics from noisy measurements. The method converges even when entire trajectory components are unobservable and the parameters are unknown. It is applied to reconstruct nonlinear models of rodent–predator oscillations in Finnish Lapland and high-Arctic tundra. The projected character of noisy incomplete measurements is revealed and shown to result in a degeneracy of the likelihood function within certain null-spaces. The performance of the method is compared with that of the conventional Markov chain Monte Carlo (MCMC) technique.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Data measured from complex (multidimensional, nonlinear) stochastic dynamical systems are often very hard to interpret because seemingly essential information is missing. For example the data set may be incomplete and/or very little may be known about the underlying dynamical system. For historic time series where measurements cannot be repeated, missing information can be considered 'lost'. Examples arise in climate evolution, predator–prey communities, and coupled matter–radiation systems. The problem is usually compounded by random fluctuations (noise).
Main results. The paper describes a solution of this enduring problem based on a novel Hamiltonian formalism derived from stochastic dynamics using maximum likelihood estimation. The key result is our discovery of the Hamiltonian equations underlying the problem through analogy with the Wentzel–Friedlin theory of large fluctuations. We apply the formalism to analyse rodent–predator oscillations for which the predator population could not be observed. Contrary to earlier belief, noise-corrupted measurements of the prey dynamics alone are sufficient for computing both the likelihood of the predator population and the ecological model parameters. The method has advantages over the conventional Markov-chain Monte-Carlo technique.
Wider implications. The dynamics of our exemplar predator–prey community has much in common with e.g. models of economic cycles, the susceptible-exposed-infectious-removed-susceptible (SEIRS) model in epidemiology, and competing modes in lasers. So the method will be of immediate relevance to a broad interdisciplinary community. There is also the promise of reconstructing long-past historical events, e.g. the 1918 influenza epidemic, or even the Black Death.