Statistical Inference for the Duffing Process

The aim of the research concerns inference methods for non-linear dynamical systems. In particular, the focus is on a differential equation called Duffing oscillator. This equation is suitable to model non-linear phenomena like jumps, hysteresis, or subharmonics and it may lead to chaotic behaviour as control parameters vary. Such behaviour have been observed in many different real-world scenarios, as in economics or biology. Inference in the Duffing process is performed with the unscented Kalman filter (UKF) by casting the system in state space form. In the context of ordinary differential equations, the uncertainty of the UKF estimates for chaotic systems is quantified by a simulation study. To overcome the limitations of the UKF when applied to the Duffing process, a new algorithm that matches Bayesian optimization (BO) and approximate Bayesian computation (ABC) within the UKF scheme is proposed. The novelty consists in (i) optimizing the sigma points location by means of maximization of the likelihood of observations with BO, and (ii) initialize the UKF with candidate parameters coming from the ABC scheme. The proposed algorithm can outperform the UKF in complex systems where the likelihood function is highly multi-modal. Concerning stochastic differential equations, a massive simulation study is presented to evaluate the performance of the UKF for parameter estimation. Finally, illustrations of the method with real data and further developments of the research are discussed.