A Bayesian nonparametric approach to reconstruction and prediction of random dynamical systems

Abstract

We propose a Bayesian nonparametric mixture model for the reconstruction and prediction from observed time series data, of discretized stochastic dynamical systems, based on Markov Chain Monte Carlo methods. Our results can be used by researchers in physical modeling interested in a fast and accurate estimation of low dimensional stochastic models when the size of the observed time series is small and the noise process (perhaps) is non-Gaussian. The inference procedure is demonstrated specifically in the case of polynomial maps of an arbitrary degree and when a Geometric Stick Breaking mixture process prior over the space of densities, is applied to the additive errors. Our method is parsimonious compared to Bayesian nonparametric techniques based on Dirichlet process mixtures, flexible and general. Simulations based on synthetic time series are presented.

A Bayesian nonparametric approach to dynamical noise reduction

“…The novelty of our approach lies on the fact that we make no parametric assumptions for the density of the noise component. Instead, we model the additive error using a highly flexible family of density functions, which are based on a Bayesian nonparametric model, namely the Geometric Stick Breaking process [8], extending previous works regarding reconstruction and prediction of random dynamical systems [13,14,23]. No matter what additive errors are involved, we are confident that our family of densities will be able to capture the right shape and hence statistical inference, for the parameters of interest will be improved and reliable.…”

“…, N n ) (see Ref. [23], and references therein). The d i random variable, denotes the component of the random mixture f in (5), that the observation x i came from.…”

“…We have seen that the reconstruction step, stems from the GSBR-sampler introduced in Ref. [23]. The differences are: the absence of the out-of-sample variables, the more general d-dimensional lag dependence, the application of a conjugate beta prior and the application of an improper prior, on the variables p and (θ, x 1:d ), respectively.…”

Joint reconstruction and prediction\break of random dynamical systems under\break borrowing of strength

“…Finally we randomize the probability-weights by letting λ ∼ Be(α, β); then λ a-posteriori is again beta with its parameters updated by a sufficient statistic of the data. In Merkatas, Kaloudis, and Hatjispyros (2017) it is shown that a G-based Bayesian nonparametric framework for dynamical system estimation is efficient, faster and less complicated when compared to Bayesian nonparametric modeling via the Dirichlet process.…”

A Bayesian nonparametric approach to the approximation of the global stable manifold

“…1. Approximate the quasi-invariant measure, of the underlying random dynamic system, thus implying the existence of a prediction barrier [26].…”