Exact and approximate EM estimation of mutually exciting hawkes processes

  • Jamie F. Olson
  • Kathleen M. Carley


Motivated by the availability of continuous event sequences that trace the social behavior in a population e.g. email, we believe that mutually exciting Hawkes processes provide a realistic and informative model for these sequences. For complex mutually exciting processes, the numerical optimization used for univariate self exciting processes may not provide stable estimates. Furthermore, convergence can be exceedingly slow, making estimation computationally expensive and multiple random restarts doubly so. We derive an expectation maximization algorithm for maximum likelihood estimation mutually exciting processes that is faster, more robust, and less biased than estimation based on numerical optimization. For an exponentially decaying excitement function, each EM step can be computed in a single \(O(N)\) pass through the data, for \(N\) observations, without requiring the entire dataset to be in memory. More generally, exact inference is \(\Theta (N^{2})\), but we identify some simple \(\Theta (N)\) approximation strategies that seem to provide good estimates while reducing the computational cost.


Self-exciting point processes Estimation Expectation-maximization 



This work was supported in part by the Office of Naval Research (N00014-06-1-0104) for adversarial assessment and (N00014-08-11186) for rapid ethnographic assessment, the Army Research Office and ERDC-TEC (W911NF0710317) and the National Science Foundation (#0943168). Additional support was provided by CASOS—the center for Computational Analysis of Social and Organizational Systems at Carnegie Mellon University.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

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