Maximum Likelihood Estimation in a Parametric Stochastic Trajectory Model

Abstract

In this work, we consider maximum likelihood estimation of parameters in a stochastic trajectory model. The velocity paths are generated from an Ornstein-Uhlenbeck process and thus revert to a latent expected value. In addition to this expected velocity, parameters that specify the reversion characteristics and the process noise covariance determine the behaviour of typical trajectories of the model. Estimation of these parameters from trajectory samples facilitates learning of patterns and training of predictive models using trajectory data, e.g., automatic identification system (AIS) messages transmitted by vessels. We propose a six-degrees-of-freedom parameterisation and investigate the identifiability of these parameters using the Cramér-Rao bound matrix which we estimate using Monte Carlo methods. We demonstrate that some parameter configurations of interest are identifiable and their maximum likelihood estimate can be found using iterative optimisation algorithms. We demonstrate the efficacy of this approach on both simulated and real data.