State and parameter estimation in stochastic dynamical models

This paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model. In contrast to previous studies, the change in background distribution due to changes in model parameters is taken into account. An ensemble approach to solving the maximum likelihood estimates is proposed. An exact solution for the ensemble update based on a square root Kalman Filter is derived. This solution involves a two step procedure in which an ensemble is first produced by a standard ensemble Kalman Filter, and then “corrected” to account for parameter estimation, thereby allowing a user to take advantage of an existing ensemble filter. The solution is illustrated with simple, low-dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters.