SIAM News Blog

Fighting Terrorism with Mathematics

By Stephen Tench

In terms of security threats, terrorism is a prominent issue in today’s world. Reports of terrorist attacks in varied locations pervade news coverage on an almost daily basis, and the threat doesn’t appear to be going away anytime soon. Thus, new insights about terrorist activity and its evolution are greatly needed.

One way to gain such knowledge is via mathematical modelling. This involves taking a quantitative approach by studying datasets of terrorist events with different models and interpreting the results qualitatively for new information. Modelling data is an economically cheap option that lets the data speak for itself, extricating the researchers’ emotions from the equation.

Of particular interest in the field of crime and terrorism studies is a specific model known as the Hawkes self-exciting point process. The model’s basic principle involves looking at how events in the past can influence those in the future; it was initially developed in the field of seismology to study the relation between earthquakes and their aftershocks. But recent studies have identified similar patterns of main and follow-up events in datasets describing areas such as gang violence [1] and violent civilian deaths in conflicts [2]. For a given set of event times \(\{t_i\},\) the equation form of the Hawkes process can be expressed as



\[g(t)=\omega e^{-\omega t}.\]

The three parameters \((\mu,k_0,\omega)\) must be computed to specify this model. One can obtain this parameter set via maximum likelihood estimation (MLE), which finds the parameters maximising the equation

\[\log L = \sum_i \log(\lambda(t_i)) - \int_0^T \lambda(t) dt.\]

This maximisation can be calculated computationally by several different programs or found via a maximising routine coded in a programming language.

Finding the parameter values of the Hawkes process is the first step of the modelling procedure, but turning those values into something actionable requires a qualitative interpretation. Consider, for example, improvised explosive device (IED) attacks carried out by a terrorist group over some period of time, and assume that the Hawkes process has been fitted to the timestamps of the events. The first parameter \(\mu\) describes a background rate of events. This term is time independent and explains the rate of new IED attacks not related to past occurrences.

A temporary jump in the rate of events might follow an IED attack. For example, a successful attack may encourage follow-up attacks or prompt some kind of counterterrorism response from security forces, leading to a tit for tat escalation in violence. In these or similar cases, the parameter \(k_0\) captures the jump in the rate of events. Finally, since it is unrealistic for events to remain high indefinitely—the resources to make IEDs will start running out, for example—there must be a term that reduces the influence of past events in the very-distant future. This idea is contained in the response function \(g,\) which is an exponential decay; the third parameter \(\omega\) controls the spread of decrease.

The following transformation can be used to test whether the Hawkes process is a good model for the event times:

\[\tau_i=\int_0^{t_i}\lambda(t) dt.\]

Theory states that if the Hawkes process is a good-fitting model, the residuals \(\{\tau_i\}\) should be a Poisson process with unit rate. The Kolmogorov-Smirnov test can assess this assumption.

The Hawkes process was used to study terrorism in Northern Ireland, particularly the period known as “The Troubles” occurring from 1969-1998 [3]. During this time, a group called the Provisional Irish Republican Army (PIRA) fought against British rule in Northern Ireland. One of the group’s main weapons was IED attacks, and members went to great lengths to develop and enhance their IED arsenal over the course of their campaign. The PIRA evolved both on the tactical weapon side and as an organization, and sociological research describes the group’s change in five phases:

  • Phase 1 (1969-1976): The PIRA maintained a military structure organised in terms of brigades, battalions, etc.
  • Phase 2 (1977-1980): The Phase 1 structure allowed heavy infiltration by security forces, so the PIRA moved to a cell-based structure consisting of many small units.
  • Phase 3 (1981-1989): The PIRA launched a big wave of IED attacks with the hope of finally securing a victory against the British Armed Forces.
  • Phase 4 (1990-1994): PIRA leadership began secret negotiations with the British Government to end hostilities.
  • Phase 5 (1995-1998): Negotiations for Phase 4 were made public and the conflict ended for many with the ratification of the “Good Friday” agreement.

This description of the phases of the PIRA comes from sociological and historical accounts. The start and end times of each phase were defined qualitatively for convenience. Having a somewhat arbitrary cut-off between phases may be unrealistic, however, as factors influencing IED attacks could extend further back in time. Such a problem lends itself naturally to the Hawkes process, which captures historical dependencies. This is precisely the outcome of [3], which used the aforementioned techniques to find mathematical boundaries for the phases of the PIRA. Figure 1 demonstrates graphically the relationship between IED event times and a fitted Hawkes process intensity function.

Figure 1. A graph comparing the times of IED events during “The Troubles” in Northern Ireland (top) with the intensity function of a fitted Hawkes process (bottom). Image credit: Stephen Tench.

The mathematically-determined change points for the organisation were found to be fairly consistent with the sociological boundaries for phases 2 and 3. However, the mathematical boundaries differed considerably from the sociological ones in phases 4 and 5. The new boundaries found by the Hawkes process raise interesting questions about the potential for novel sociological research to reconcile the qualitative theory of the PIRA with these quantitative results. This last point illustrates the wide applicability of the Hawkes process, which can act as a bridge between the tools and techniques of mathematics and other applied research areas. The following video provides a further discussion of the results obtained in [3].

Video courtesy of Hannah Fry, co-author of [3].

[1] Egesdal, M., Fathauer, C., Louie, K., & Neuman, J. (2010). Statistical and Stochastic Modeling of Gang Rivalries in Los Angeles. SIAM Undergraduate Research Online, 3, 72-94.

[2] Lewis, E., Mohler, G., Brantingham, P.J., & Bertozzi, A.L. (2012). Self-exciting point process models of civilian deaths in Iraq. Security Journal, 25(3), 244-264.

[3] Tench, S., Fry, H., & Gill, P. (2016). Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army. European Journal of Applied Mathematics, 27(Special Issue 03), 377-402.

Stephen Tench is currently a Ph.D. student at University College London (UCL), working at the Jill Dando Institute of Security and Crime Science and the Centre for Advanced Spatial Analysis (CASA).  His research interests lie in spatial, temporal, and spatio-temporal patterns of crime and security issues, with specific focus on terrorism.

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