The Annals of Applied Statistics

Hidden Markov models for the activity profile of terrorist groups

Vasanthan Raghavan, Aram Galstyan, and Alexander G. Tartakovsky

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The main focus of this work is on developing models for the activity profile of a terrorist group, detecting sudden spurts and downfalls in this profile, and, in general, tracking it over a period of time. Toward this goal, a $d$-state hidden Markov model (HMM) that captures the latent states underlying the dynamics of the group and thus its activity profile is developed. The simplest setting of $d=2$ corresponds to the case where the dynamics are coarsely quantized as Active and Inactive, respectively. A state estimation strategy that exploits the underlying HMM structure is then developed for spurt detection and tracking. This strategy is shown to track even nonpersistent changes that last only for a short duration at the cost of learning the underlying model. Case studies with real terrorism data from open-source databases are provided to illustrate the performance of the proposed methodology.

Article information

Ann. Appl. Stat., Volume 7, Number 4 (2013), 2402-2430.

First available in Project Euclid: 23 December 2013

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Hidden Markov model self-exciting hurdle model terrorism terrorist groups Colombia Peru Indonesia point process spurt detection


Raghavan, Vasanthan; Galstyan, Aram; Tartakovsky, Alexander G. Hidden Markov models for the activity profile of terrorist groups. Ann. Appl. Stat. 7 (2013), no. 4, 2402--2430. doi:10.1214/13-AOAS682.

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  • Baddeley, A. J., Møller, J. and Waagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerl. 54 329–350.
  • Cho, Y. S., Galstyan, A., Brantingham, P. J. and Tita, G. (2013). Latent point process models for spatial-temporal networks. Available at arXiv:1302.2671.
  • Clauset, A. and Gleditsch, K. S. (2012). The developmental dynamics of terrorist organizations. PLoS ONE 7 e48633.
  • Clauset, A., Young, M. and Gleditsch, K. S. (2007). On the frequency of severe terrorist events. Journal of Conflict Resolution 51 58–87.
  • Cox, D. R. and Isham, V. (1980). Point Processes. Chapman & Hall, London.
  • Cragin, K. and Daly, S. A. (2004). The Dynamic Terrorist Threat: An Assessment of Group Motivations and Capabilities in a Changing World. RAND Corporation, Santa Monica, CA.
  • Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.
  • Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd ed. Edward Arnold, London.
  • Dixon, P. M. (2002). Ripley’s $K$ function. In Encyclopedia of Environmetrics (A. H. El-Shaarawi and W. W. Piegorsc, eds.) 2 1796–1803. Wiley, Chichester.
  • Dugan, L., LaFree, G. and Piquero, A. (2005). Testing a rational choice model of airline hijackings. Criminology 43 1031–1066.
  • Durbin, J. (1973). Distribution Theory for Tests Based on the Sample Distribution Function. SIAM, Philadelphia, PA.
  • Enders, W. and Sandler, T. (1993). The effectiveness of antiterrorism policies: A vector autoregression-intervention analysis. The American Political Science Review 87 829–844.
  • Enders, W. and Sandler, T. (2000). Is transnational terrorism becoming more threatening? A time-series investigation. Journal of Conflict Resolution 44 307–332.
  • Enders, W. and Sandler, T. (2002). Patterns of transnational terrorism, 1970–1999: Alternative time-series estimates. International Studies Quarterly 2 145–165.
  • Haugaard, L., Isacson, A. and Olson, J. (2005). Erasing the lines: Trends in U.S. military programs with Latin America. Technical report, Center for International Policy, Washington, DC.
  • Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83–90.
  • ITERATE (2004). International terrorism: Attributes of terrorist events. Available at
  • LaFree, G. and Dugan, L. (2007). Introducing the global terrorism database. Terrorism and Political Violence 19 181–204.
  • LaFree, G., Morris, N. A. and Dugan, L. (2010). Cross-national patterns of terrorism, comparing trajectories for total, attributed and fatal attacks, 1970–2006. British Journal of Criminology 50 622–649.
  • Lewis, E., Mohler, G. O., Brantingham, P. J. and Bertozzi, A. (2011). Self-exciting point process models of civilian deaths in Iraq. Security Journal 25 244–264.
  • Lindberg, M. (2010). Factors contributing to the strength and resilience of terrorist groups. Grupo de Estudios Estrategicos (GEES) Publication.
  • Midlarsky, M. I. (1978). Analyzing diffusion and contagion effects: The urban disorders of the 1960s. The American Political Science Review 72 996–1008.
  • Midlarsky, M. I., Crenshaw, M. and Yoshida, F. (1980). Why violence spreads: The contagion of international terrorism. International Studies Quarterly 24 262–298.
  • 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.
  • Mueller, J. and Stewart, M. G. (2011). Terrorism, Security, and Money: Balancing the Risks, Benefits, and Costs of Homeland Security. Oxford Univ. Press, London.
  • Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83 9–27.
  • Ogata, Y. (1998). Space–time point process models for earthquake occurrences. Ann. Inst. Statist. Math. 50 379–402.
  • Porter, M. D. and White, G. (2012). Self-exciting hurdle models for terrorist activity. Ann. Appl. Stat. 6 106–124.
  • Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE 77 257–286.
  • Raghavan, V., Galstyan, A. and Tartakovsky, A. G. (2012). Hidden Markov models for the activity profile of terrorist groups. Available at arXiv:1207.1497.
  • Raghavan, V., Galstyan, A. and Tartakovsky, A. G. (2013a). Supplement to “Hidden Markov models for the activity profile of terrorist groups.” DOI:10.1214/13-AOAS682SUPPA.
  • Raghavan, V., Galstyan, A. and Tartakovsky, A. G. (2013b). Supplement to “Hidden Markov models for the activity profile of terrorist groups.” DOI:10.1214/13-AOAS682SUPPB.
  • RDWTI. RAND database of worldwide terrorism incidents. Available at
  • Santos, D. N. (2011). What constitutes terrorist network resiliency? Small Wars Journal 7.
  • Seshadri, V., Csorgo, M. and Stephens, M. A. (1969). Tests for the exponential distribution using Kolmogorov-type statistics. J. R. Stat. Soc. Ser. B Stat. Methodol. 31 499–509.
  • Teerapabolarn, K. (2012). A pointwise approximation of generalized binomial by Poisson distribution. Appl. Math. Sci. (Ruse) 6 1095–1104.
  • Veen, A. and Schoenberg, F. P. (2006). Assessing spatial point process models using weighted $K$-functions. In Case Studies in Spatial Point Process Modeling (A. Baddeley et al., eds.). Lecture Notes in Statistics 185 293–306. Springer, New York.

Supplemental materials

  • Supplementary material A: Information on models for the number of attacks per day studied in this work. This section derives the ML and Baum–Welch estimate of model parameter(s) under the geometric and hurdle-based geometric assumptions on $\{M_{i}\}$.
  • Supplementary material B: Background information on FARC and shining path. This section motivates the choice of the terrorist groups and the corresponding time periods of interest that are the focus of this work.