The estimation of time-invariant parameters of noisy nonlinear oscillatory systems

The inverse problem of estimating time-invariant (static) parameters of a nonlinear system exhibiting noisy oscillation is considered in this paper. Firstly, a Markov Chain Monte Carlo (MCMC) simulation is used for the time-invariant parameter estimation which exploits a non-Gaussian filter, namely the Ensemble Kalman Filter (EnKF) for state estimation required to compute the likelihood function. Secondly, a recently proposed Particle Filter (PF) (that uses the EnKF for its proposal density for the state estimation) has been adapted for combined state and parameter estimation. Numerical illustrations highlight the strengths and limitations of the MCMC, EnKF and PF algorithms for time-invariant parameter estimation. For low measurement noise and dense measurement data, the performances of the MCMC, EnKF and PF algorithms are comparable. For high measurement noise and sparse observational data, the EnKF fails to provide accurate parameter estimates. Hence the adapted PF algorithm becomes necessary in order to obtain parameter estimates comparable in accuracy to the MCMC simulation with EnKF. It highlights the fact that the augmented state space model for the combined state and parameter estimation contains stronger nonlinearity than the original state space model. Hence the EnKF effectively handles the state estimation of the original state space model, but it fails for the combined state and parameter estimation using the augmented system. The effectiveness of the EnKF for the state estimation is therefore leveraged in the MCMC simulation for the time-invariant parameter estimation. In order to obtain accurate parameter estimates using the augmented system, the adapted PF becomes necessary to match the parameter estimates obtained using the MCMC simulation complemented by EnKF for likelihood function computation.