Comparison of Expectation-Maximization based parameter estimation using Particle Filter, Unscented and Extended Kalman Filtering techniques

Abstract

The primary requirement of filtering algorithms such as Particle Filter (PF), Unscented Kalman Filter (UKF) and Extended Kalman Filter (EKF) is the availability of an accurate nonlinear state space model. In the absence of good parameter values, one has to estimate both the hidden states and the unknown parameters in a joint framework using measurements available from the process. This problem of joint state and parameter estimation for nonlinear systems can be solved recursively through the combination of a nonlinear smoother and a maximum likelihood parameter estimation scheme. Expectation Maximization (EM) is an efficient optimization algorithm which can provide the maximum likelihood estimate of the model parameters even in the presence of missing data. The algorithm can generate parameter estimates that maximize the likelihood of all the data including those with missing output measurements. This paper presents an approach which combines the EM algorithm with a suitable nonlinear smoother, such as PF, UKF or EKF based smoother. An application of this method to a simulated Continuous Fermentor process with unknown model parameters is presented. A comparative study of the results, when the different smoothing schemes were used in this approach, is presented. The results show that the UKS based technique was able to generate unbiased parameter estimates. The Particle Smoother based parameter estimates converged in the neighbouhood of their true values, but the technique was found to be computationally intensive compared to the UKS and EKS based techniques.