A comparative study of maximum likelihood estimators for nonlinear dynamical system models

Abstract

Estimating nonlinear stochastic dynamical system models from discrete observation is discussed. Nonlinear dynamical system models with observation noise as well as system noise is practically useful for describing the time evolution of dynamic phenomena. The models will work only if their parameters are set appropriately. Then, the models must be estimated from real data which are almost always observed at discrete times. Generally nonlinear models in continuous time are not easy to estimate. With linear approximation of a nonlinear dynamical system model, it can be transformed into a discrete state space model. Using the discretized model together with the Kalman filter algorithm, the parameters of the model can be estimated from discrete observation via maximum likelihood technique. What linear approximation is used is critical for performance of estimation. This paper considers two linear approximations; the first order linear approximation used in the extended Kalman filter and a second order linear approximation based on Ito's formula. Applying these linear approximations to Van der Pol's random oscillation and Rayleigh's random oscillation, we make a numerical comparison of the performance of the two maximum likelihood estimators by Monte Carlo experiments. In addition, it is also important for estimating continuous time models from discrete observation to evaluate how much the performance of estimation is dependent on time interval of discrete observation. We examine the influence of time interval on estimation.