Abstract
The primary focus is a sequential data assimilation method for count data modelled by an inhomogeneous Poisson process. In particular, a quadratic approximation technique similar to the extended Kalman filter is applied to develop a sub-optimal, discrete-time, filtering algorithm, called the extended Poisson-Kalman filter (ExPKF), where only the mean and covariance are sequentially updated using count data via the Poisson likelihood function. The performance of ExPKF is investigated in several synthetic experiments where the true solution is known. In numerical examples, ExPKF provides a good estimate of the “true” posterior mean, which can be well-approximated by the particle filter (PF) algorithm in the very large sample size limit. In addition, the experiments demonstrates that the ExPKF algorithm can be conveniently used to track parameter changes; on the other hand, a non-filtering framework such as a maximum likelihood estimation (MLE) would require a statistical test for change points or implement time-varying parameters. Finally, to demonstrate the model on real-world data, the ExPKF is used to approximate the uncertainty of urban crime intensity and parameters for self-exciting crime models. The Chicago Police Department’s CLEAR (Citizen Law Enforcement Analysis and Reporting) system data is used as a case study for both univariate and multivariate Hawkes models. An improved goodness of fit measured by the Kolomogrov-Smirnov (KS) statistics is achieved by the filtered intensity. The potential of using filtered intensity to improve police patrolling prioritisation is also tested. By comparing with the prioritisation based on MLE-derived intensity and historical frequency, the result suggests an insignificant difference between them. While the filter is developed and tested in the context of urban crime, it has the potential to make a contribution to data assimilation in other application areas.