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Approximate filtering of intensity process for Poisson count data

Banff International Research Station for Mathematical Innovation and Discovery

We develop a sequential data assimilation algorithm for count data modelled by a doubly stochastic Poisson process. We apply an approximation technique similar to the extended Kalman filter to develop a sub-optimal discrete-time filtering algorithm, called the extended Poisson-Kalman lter (ExPKF), where only the mean and covariance are sequentially updated using count data via the Poisson likelihood function. The ExPKF, however, is inconvenient to use when the calculation of the Hessian is difficult. Thus, we also develop an ensemble-based filter based on the Gamma prior assumption; hence, ensemble Poisson-Gamma filter (EnPGF). The implementation of EnPGF is performed in the same manner as the serial-update version of EnKF. The performances of ExPKF and EnPGF are demonstrated in several synthetic experiments where the true solution is known. For the application to real-world data, we use ExPKF to approximate the uncertainty of urban crime intensity and parameters for Hawkes process and highlight the advantage of filtering scheme (over the non-filtering scheme) to track parameter changes.

Mathematics; Game theory, economics, social and behavioral sciences; Computer science

video/mp4

Author affiliation: University of Surrey

2019-09-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71684

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/