A maximum-likelihood method for estimating parameters, stochastic disturbance intensities and measurement noise variances in nonlinear dynamic models with process disturbances

•We developed a Laplace approximation maximum likelihood estimation (LAMLE) algorithm.•LAMLE can be used to estimate parameters in stochastic differential equation models.•LAMLE was tested using a nonlinear stochastic model for a reactor.•Parameter estimates are compared with two approximate maximum likelihood methods.•LAMLE estimation results are more precise than those obtained from the two methods.

An improved approximate maximum likelihood algorithm is developed for estimating measurement noise variances along with model parameters and disturbance intensities in nonlinear stochastic differential equation (SDE) models. This algorithm uses a Laplace approximation and B-spline basis functions for approximating the likelihood function of the parameters given the measurements. The resulting Laplace approximation maximum likelihood estimation (LAMLE) algorithm is tested using a nonlinear continuous stirred tank reactor (CSTR) model. Estimation results for four model parameters, two process disturbance intensities and two measurement noise variances are obtained using LAMLE and are compared with results from two other maximum-likelihood-based methods, the continuous-time stochastic method (CTSM) of Kristensen and Madsen (2003) and the Fully Laplace Approximation Estimation Method (FLAEM) (Karimi and McAuley, 2014). Parameter estimations using 100 simulated data sets reveal that the LAMLE estimation results tend to be more precise and less biased than corresponding estimates obtained using CTSM and FLAEM.