Markov chain Monte Carlo estimation of nonlinear dynamics from time series

Much of nonlinear time series analysis is concerned with inferring unmeasured quantities — e.g., system parameters, the shape of attractors in state space — from a noisy measured time series. From a Bayesian perspective, the time series is a vector sample picked at random from a probability density. The density reflects the system dynamics and our subjective uncertainty about system parameters, the measurement function, dynamical noise and measurement noise. The conditional probability density of the system parameters given the measured data is the basis of a Bayesian estimate of the system parameters. Using illustrative chaotic systems with large-amplitude dynamical and measurement noise, we show here that it is feasible to use the Markov chain Monte Carlo (MCMC) technique to generate the Bayesian conditional probabilities. The resulting parameter estimates are markedly superior to those based on conventional least-squares methods: the MCMC-based estimates are unbiased and allow estimates of dynamical parameters on unmeasured components of the state vector. In addition, the MCMC method enables de-noised attractors to be reconstructed, not just in an embedding based on lags of measured variables but in the state space that includes unmeasured components of the dynamics’ state vector. The general purpose MCMC technique effectively combines techniques of nonlinear noise reduction and nonlinear parameter estimation.