The Annals of Applied Statistics

Estimating the historical and future probabilities of large terrorist events

Abstract

Quantities with right-skewed distributions are ubiquitous in complex social systems, including political conflict, economics and social networks, and these systems sometimes produce extremely large events. For instance, the 9/11 terrorist events produced nearly 3000 fatalities, nearly six times more than the next largest event. But, was this enormous loss of life statistically unlikely given modern terrorism’s historical record? Accurately estimating the probability of such an event is complicated by the large fluctuations in the empirical distribution’s upper tail. We present a generic statistical algorithm for making such estimates, which combines semi-parametric models of tail behavior and a nonparametric bootstrap. Applied to a global database of terrorist events, we estimate the worldwide historical probability of observing at least one 9/11-sized or larger event since 1968 to be 11–35%. These results are robust to conditioning on global variations in economic development, domestic versus international events, the type of weapon used and a truncated history that stops at 1998. We then use this procedure to make a data-driven statistical forecast of at least one similar event over the next decade.

Article information

Source
Ann. Appl. Stat., Volume 7, Number 4 (2013), 1838-1865.

Dates
First available in Project Euclid: 23 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1387823295

Digital Object Identifier
doi:10.1214/12-AOAS614

Mathematical Reviews number (MathSciNet)
MR3161698

Zentralblatt MATH identifier
1283.62105

Citation

Clauset, Aaron; Woodard, Ryan. Estimating the historical and future probabilities of large terrorist events. Ann. Appl. Stat. 7 (2013), no. 4, 1838--1865. doi:10.1214/12-AOAS614. https://projecteuclid.org/euclid.aoas/1387823295

References

• Adler, R. J., Feldman, R. E. and Tuqqu, M. S., eds. (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhäuser, Boston, MA.
• Asal, V. and Rethemeyer, R. K. (2008). The nature of the beast: Organizational structures and the lethality of terrorist attacks. Journal of Politics 70 437–449.
• Beck, N., King, G. and Zeng, L. (2000). Improving quantitative studies of international conflict: A conjecture. American Political Science Review 94 21–35.
• Boas, M. L. (2005). Mathematical Models in the Physical Sciences. Wiley, Hoboken, NJ.
• Breiman, L., Stone, C. J. and Kooperberg, C. (1990). Robust confidence bounds for extreme upper quantiles. J. Stat. Comput. Simul. 37 127–149.
• Brown, D., Dalton, J. and Hoyle, H. (2004). Spatial forecast methods for terrorist events in urban environments. In Intelligence and Security Informatics. Lecture Notes in Computer Science 3073 426–435. Springer, Heidelberg.
• Bueno de Mesquita, B. (2003). Ruminations on challenges to prediction with rational choice models. Rationality and Society 15 136–147.
• Bueno de Mesquita, B. (2011). A new model for predicting policy choices: Preliminary tests. Conflict Management and Peace Science 28 64–84.
• Caires, S. and Ferreira, J. A. (2005). On the non-parametric prediction of conditionally stationary sequences. Stat. Inference Stoch. Process. 8 151–184.
• Cameron, G. (2000). WMD terrorism in the United States: The threat and possible countermeasures. Nonproliferation Review 7 162–179.
• Cesa-Bianchi, N., Conconi, A. and Gentile, C. (2004). On the generalization ability of on-line learning algorithms. IEEE Trans. Inform. Theory 50 2050–2057.
• Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging. Cambridge Univ. Press, Cambridge.
• Clauset, A. and Gleditsch, K. S. (2012). The developmental dynamics of terrorist organizations. PLoS ONE 7 e48633.
• Clauset, A., Shalizi, C. R. and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Rev. 51 661–703.
• Clauset, A., Young, M. and Gleditsch, K. S. (2007). On the frequency of severe terrorist events. Journal of Conflict Resolution 51 58–87.
• Clauset, A., Young, M. and Gleditsch, K. S. (2010). A novel explanation of the power-law form of the frequency of severe terrorist events: Reply to saperstein. Peace Economics, Peace Science and Public Policy 16 article 12.
• Clements, M. P. and Hendry, D. F. (1999). Forecasting Non-Stationary Economic Time Series. MIT Press, Cambridge, MA.
• Danielsson, J., de Haan, L., Peng, L. andde Vries, C. G. (2001). Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivariate Anal. 76 226–248.
• de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
• Dekkers, A. L. M. and de Haan, L. (1993). Optimal choice of sample fraction in extreme-value estimation. J. Multivariate Anal. 47 173–195.
• Desmarais, B. A. and Cranmer, S. J. (2011). Forecasting the locational dynamics of transnational terrorism: A network analytic approach. In Proc. European Intelligence and Security Informatics Conference 171–177. IEEE Computer Society, Washington, DC.
• Drees, H. and Kaufmann, E. (1998). Selecting the optimal sample fraction in univariate extreme value estimation. Stochastic Process. Appl. 75 149–172.
• Drennan, M. P. (2007). Analyzing catastrophic terrorist events with applications to the food industry. In The Economic Costs and Consequences of Terrorism (H. W. Richardson, P. Gordon and J. E. Moore, eds.). Edward Elgar Publishing, Northampton, MA.
• Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57. Chapman & Hall, New York.
• 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 46 145–165.
• Enders, W. E. and Sandler, T. (2006). The Political Economy of Terrorism. Cambridge Univ. Press, Cambridge.
• Enders, W., Sandler, T. and Gaibulloev, K. (2011). Domestic versus transnational terrorism: Data, decomposition, and dynamics. Journal of Peace Research 48 319–337.
• Farmer, J. D. and Lillo, F. (2004). On the origin of power law tails in price fluctuations. Quant. Finance 4 7–11.
• Financial Crisis Inquiry Commission (2011). Financial crisis inquiry report. U.S. Government Printing Office, Washington. Available at http://bit.ly/hEsCwo.
• Gartzke, E. (1999). War is in the error term. International Organization 53 567–587.
• Grünwald, P. D. (2007). The Minimum Length Description Principle. MIT Press, Cambridge, MA.
• Gumbel, E. J. (1941). The return period of flood flows. Ann. Math. Statist. 12 163–190.
• Gutenberg, B. and Richter, C. F. (1944). Frequency of earthquakes in California. Bull. Seismol. Soc. Amer. 34 185–188.
• Hancock, M. S. and Jones, J. H. (2004). Likelihood-based inference for stochastic models of sexual network evolution. Theoretical Population Biology 65 413–422.
• Hjort, N. L. and Claeskens, G. (2003). Frequentist model average estimators. J. Amer. Statist. Assoc. 98 879–899.
• Jackson, B. A., Baker, J. C., Cragin, K., Parachini, J., Trujillo, H. R. and Chalk, P. (2005). Aptitude for Destruction: Organizational Learning in Terrorist Groups and Its Implications for Combating Terrorism 1. RAND Corporation, Arlington, VA.
• Kardes, E. and Hall, R. (2005). Survey of Literature on Strategic Decision Making in the Presence of Adversaries. Center for Risk and Economic Analysis of Terrorism Events, Los Angeles, CA.
• Kass, R. E. and Raftery, A. E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.
• Kilcullen, D. (2010). Counterinsurgency. Oxford Univ. Press, Oxford.
• King, G. and Zeng, L. (2001). Explaining rare events in international relations. International Organization 55 693–715.
• Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line: The Dynamic Approach. Springer, New York.
• Lee, Y.-T., Turcotte, D. L., Holliday, J. R., Sachs, M. K., Rundle, J. B., Chen, C.-C. and Tiampo, K. F. (2011). Results of the Regional Earthquake Likelihood Model (RELM) test of earthquake forecasts in California. Proc. Natl. Acad. Sci. USA 108 16533–16538.
• Li, Q. (2005). Does democracy promote or reduce transnational terrorist incidents? Journal of Conflict Resolution 49 278–297.
• Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmospheric Sci. 20 130–141.
• Major, J. A. (1993). Advanced techniques for modeling terrorism risk. Journal of Risk Finance 4 15–24.
• McMorrow, D. (2009). Rare events. JASON Report JSR-09-108, The MITRE Corporation, McLean, VA.
• MIPT (2008). Terrorism knowledge base. National Memorial Institute for the Prevention of Terrorism, Oklahoma City, OK.
• Pape, R. A. (2003). The strategic logic of suicide terrorism. American Political Science Review 97 343–361.
• Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge Univ. Press, Cambridge.
• Reed, W. J. and McKelvey, K. S. (2002). Power-law behavior and parametric models for the size-distribution of forest fires. Ecological Modelling 150 239–254.
• Reiss, R. D. and Thomas, M. (2007). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd ed. Birkhäuser, Basel.
• Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
• Rundle, J. B., Holliday, J. R., Yoder, M., Sachs, M. K., Donnellan, A., Turcotte, D. L., Tiampo, K. F., Klein, W. and Kellogg, L. H. (2011). Earthquake precursors: Activation or quiescence? Geophysical Journal International 187 225–236.
• Rustad, S. C. A., Buhaug, H., Falch, Å. and Gates, S. (2011). All conflict is local: Modeling sub-national variation in civil conflict risk. Conflict Management and Peace Science 28 14–39.
• Sageman, M. (2004). Understanding Terror Networks. Univ. Pennsylvania Press, Philadelpha.
• Shalizi, C. R., Jacobs, A. Z., Klinkner, K. L. and Clauset, A. (2011). Adapting to non-stationarity with growing expert ensembles. Preprint. Available at http://arxiv.org/abs/1103.0949.
• Sornette, D. (2009). Dragon-kings, black swans and the prediction of crises. International Journal of Terraspace Science and Engineering 2 1–18.
• Sornette, D. and Ouillon, G., eds. (2012). Eur. Phys. J. Special Topics 205 1–373.
• START (2011). Global terrorism database (data file). Available at www.start.umd.edu/gtd/.
• Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. J. R. Stat. Soc. Ser. B Stat. Methodol. 36 111–133.
• Strogatz, S. H. (2001). Nonlinear Dynamics And Chaos. Westview, Boulder, CO.
• Valenzuela, M. L., Feng, C., Reddy, P., Momen, F., Rozenblit, J. W., Eyck, B. T. and Szidarovszky, F. (2010). A non-numerical predictive model for asymmetric analysis. In Proc. 17th IEEE International Conference and Workshops on the Engineering of Computer-Based Systems 311–315. IEEE Computer Society, Washington, DC.
• Vuong, Q. H. (1989). Likelihood ratio tests for model selection and nonnested hypotheses. Econometrica 57 307–333.
• Ward, M. D., Greenhill, B. D. and Bakke, K. M. (2010). The perils of policy by $p$-value: Predicting civil conflicts. Journal of Peace Research 47 363–375.
• Wulf, W. A., Haimes, Y. Y. and Longstaff, T. A. (2003). Strategic alternative responses to risks of terrorism. Risk Anal. 23 429–444.
• Zammit-Mangion, A., Dewar, M., Kadirkamanathan, V. and Sanguinetti, G. (2012). Point process modeling of the Afghan War Diary. Proc. Natl. Acad. Sci. USA 109 12414–12419.