Abstract
Parameter estimation of complex nonlinear turbulent dynamical systems using only partially observed time series is a challenging topic. The nonlinearity and partial observations often impede using closed analytic formulae to recover the model parameters. In this paper, an exact path-wise sampling method is developed, which is incorporated into a Bayesian Markov chain Monte Carlo (MCMC) algorithm in light of data augmentation to efficiently estimate the parameters in a rich class of nonlinear and non-Gaussian turbulent systems using partial observations. This path-wise sampling method exploits closed analytic formulae to sample the trajectories of the unobserved variables, which avoid the numerical errors in the general sampling approaches and significantly increase the overall parameter estimation efficiency. The unknown parameters and the missing trajectories are estimated in an alternating fashion in an adaptive MCMC iteration algorithm with rapid convergence. It is shown based on the noisy Lorenz 63 model and a stochastically coupled FitzHugh–Nagumo model that the new algorithm is very skillful in estimating the parameters in highly nonlinear turbulent models. The model with the estimated parameters succeeds in recovering the nonlinear and non-Gaussian features of the truth, including capturing the intermittency and extreme events, in both test examples.
Highlights
Complex nonlinear turbulent dynamical systems are ubiquitous in many areas, including geophysics, engineering, material science, and neural science [1,2,3,4]
The dimension of many complex turbulent systems is quite large [9,10,11,12], which requires the development of efficient parameter estimation algorithms that can overcome the curse of dimensionality [13,14]
One of the key features of many turbulent dynamical systems is the availability of only the partial observations [15,16,17]
Summary
Complex nonlinear turbulent dynamical systems are ubiquitous in many areas, including geophysics, engineering, material science, and neural science [1,2,3,4]. This paper aims at showing that the conditional Gaussian structure can be explored to the development of an efficient path-wise sampling strategy, which is incorporated into an MCMC algorithm to advance the parameter estimation using only partial observations. The advantage of such a path-wise sampling algorithm is that the sampled trajectories can be written using closed analytic formulae in light of the conditional Gaussian statistics, which are not achievable for general nonlinear systems.
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