The Annals of Applied Statistics

Modeling and estimation of multi-source clustering in crime and security data

George Mohler

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While the presence of clustering in crime and security event data is well established, the mechanism(s) by which clustering arises is not fully understood. Both contagion models and history independent correlation models are applied, but not simultaneously. In an attempt to disentangle contagion from other types of correlation, we consider a Hawkes process with background rate driven by a log Gaussian Cox process. Our inference methodology is an efficient Metropolis adjusted Langevin algorithm for filtering of the intensity and estimation of the model parameters. We apply the methodology to property and violent crime data from Chicago, terrorist attack data from Northern Ireland and Israel, and civilian casualty data from Iraq. For each data set we quantify the uncertainty in the levels of contagion vs. history independent correlation.

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Ann. Appl. Stat. Volume 7, Number 3 (2013), 1525-1539.

First available in Project Euclid: 3 October 2013

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Markov Chain Monte Carlo Hawkes process Cox process crime terrorism


Mohler, George. Modeling and estimation of multi-source clustering in crime and security data. Ann. Appl. Stat. 7 (2013), no. 3, 1525--1539. doi:10.1214/13-AOAS647.

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  • Bowers, K. J., Johnson, S. D. and Pease, K. (2004). Prospective hot-spotting the future of crime mapping? British Journal of Criminology 44 641–658.
  • Brix, A. and Diggle, P. J. (2001). Spatiotemporal prediction for log-Gaussian Cox processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 823–841.
  • Cseke, B. and Heskes, T. (2011). Approximate marginals in latent Gaussian models. J. Mach. Learn. Res. 12 417–454.
  • Egesdal, M., Fathauer, C., Louie, K. and Neuman, J. (2010). Statistical and Stochastic Modeling of Gang Rivalries in Los Angeles. SIAM Undergraduate Research Online 3 72–94.
  • Giesecke, K. and Schwenkler, G. (2011). Filtered likelihood for point processes. Available at SSRN 1898344.
  • Girolami, M. and Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 123–214.
  • Hegemann, R., Lewis, E. and Bertozzi, A. (2012). An “Estimate & Score Algorithm” for simultaneous parameter estimation and reconstruction of missing data on social networks. Unpublished manuscript.
  • Jones, P. A., Brantingham, P. J. and Chayes, L. (2010). Statistical models of criminal behavior: The effects of law enforcement actions. Mathematical Models and Methods in Applied Sciences 20 1397–1423.
  • Lewis, E. and Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Unpublished manuscript.
  • Lewis, E., Mohler, G., Brantingham, P. J. and Bertozzi, A. L. (2012). Self-exciting point process models of civilian deaths in Iraq. Security Journal 25 244–264.
  • Mangion, A. Z., Yuan, K., Kadirkamanathan, V., Niranjan, M. and Sanguinetti, G. (2011). Online variational inference for state-space models with point-process observations. Neural Comput. 23 1967–1999.
  • Marsan, D. and Lengliné, O. (2008). Extending earthquakes’ reach through cascading. Science 319 1076–1079.
  • Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P. and Tita, G. E. (2011). Self-exciting point process modeling of crime. J. Amer. Statist. Assoc. 106 100–108.
  • Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes 100. Chapman & Hall/CRC, Boca Raton, FL.
  • Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83 9–27.
  • Porter, M. D. and White, G. (2012). Self-exciting hurdle models for terrorist activity. Ann. Appl. Stat. 6 106–124.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319–392.
  • Shaman, P. (1969). On the inverse of the covariance matrix of a first order moving average. Biometrika 56 595–600.
  • Short, M. B., D’Orsogna, M. R., Brantingham, P. J. and Tita, G. E. (2009). Measuring and modeling repeat and near-repeat burglary effects. Journal of Quantitative Criminology 25 325–339.
  • Short, M., Mohler, G., Brantingham, P. and Tita, G. (2010). Gang rivalry dynamics via coupled point process networks. Unpublished manuscript.
  • Smith, A. C. and Brown, E. N. (2003). Estimating a state-space model from point process observations. Neural Comput. 15 965–991.
  • Sornette, D. and Utkin, S. (2009). Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks. Phys. Rev. E (3) 79 061110, 15.
  • Stomakhin, A., Short, M. B. and Bertozzi, A. L. (2011). Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems 27 115013, 15.
  • Taddy, M. A. (2010). Autoregressive mixture models for dynamic spatial Poisson processes: Application to tracking intensity of violent crime. J. Amer. Statist. Assoc. 105 1403–1417.
  • Townsley, M., Johnson, S. D. and Ratcliffe, J. H. (2008). Space time dynamics of insurgent activity in Iraq. Security Journal 21 139–146.
  • Veen, A. and Schoenberg, F. P. (2008). Estimation of space–time branching process models in seismology using an EM-type algorithm. J. Amer. Statist. Assoc. 103 614–624.
  • Zammit-Mangion, A., Dewar, M., Kadirkamanathan, V. and Sanguinetti, G. (2012). Point process modelling of the Afghan War Diary. Proc. Natl. Acad. Sci. USA 109 12414–12419.