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.

05.40.Ca Noise

05.45.Tp Time series analysis

02.50.Tt Inference methods

05.45.Xt Synchronization; coupled oscillators

02.50.Ga Markov processes

Computational physics

Statistical physics and nonlinear systems

Issue 5 (May 2009)

Received 17 February 2009

Published 27 May 2009