Bayesian nonparametric estimation of Random Dynamical Systems

In this talk, a Bayesian nonparametric framework for the estimation and prediction, from observed time series data, of discretized random dynamical systems is presented [1]. The size of the observed time series can be small and the additive noise may not be Gaussian distributed. We show that as the dynamical noise departs from normality, simple Markov Chain Monte Carlo method (MCMC) models are inefficient. The proposed models assume an unknown error process in the form of a countable mixture of zero mean normals, where a–priori the number of the countable normal components and their variances is unknown. Our method infers the number of unknown components and their variances, i.e., infers the density of the error process directly from the observed data. An extension for the joint estimation and prediction of multiple discrete time random dynamical systems based on multiple time-series observations contaminated by additive dynamical noise is presented [2]. In this case the model assumes an unknown joint error process with a pairwise dependence in the sense that to each pair of unknown dynamical error processes, we assign a– priori an independent Geometric Stick-Breaking process mixture of normals with zero mean. These mixtures a–posteriori will capture common characteristics, if there are any, among the pairs of noise processes. We show numerically that when the unknown error processes share common characteristics, it is possible under suitable prior specification to induce a borrowing of strength relationship among the dynamical error pairs. Then time-series with an inadequate sample size for an independent Bayesian reconstruction can benefit in terms of model estimation accuracy. Finally, possible directions for future research will be discussed.