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Self-exciting hurdle models for terrorist activity
Michael D. Porter, Gentry White
TL;DR
Terrorism models are needed to understand activity, yet accurate measures of counter-terrorism effectiveness are lacking. This paper combines self-exciting and hurdle models for terrorist activity; its best model indicates increased short-term risk after an attack.
Problem
Despite increased counter-terrorism spending, accurate measures of these efforts’ effectiveness are lacking, and understanding terrorist activity requires correctly specified models.
Source: §1 · p. 1Method
The paper combines a self-exciting hurdle model for attack-day probability with an event-history-dependent extreme-value model for nonzero attack counts.
Source: §1 · p. 3Results
The best model uses a constant baseline with a self-exciting hurdle term and a Riemann zeta count distribution with a shot-noise-driven parameter, indicating increased short-term terrorism risk after an attack.
Source: §5 · p. 17Takeaways & Limitations
Terrorism risk varies greatly over short periods, so policy responses should address these variations as well as longer-term risk trends.
Source: §5 · p. 18The analysis focuses on attack timing on Indonesia’s islands, omitting location, attack type, and responsible group.
Source: §5 · p. 17
Abstract
from arXiv · showhide
A predictive model of terrorist activity is developed by examining the daily number of terrorist attacks in Indonesia from 1994 through 2007. The dynamic model employs a shot noise process to explain the self-exciting nature of the terrorist activities. This estimates the probability of future attacks as a function of the times since the past attacks. In addition, the excess of nonattack days coupled with the presence of multiple coordinated attacks on the same day compelled the use of hurdle models to jointly model the probability of an attack day and corresponding number of attacks. A power law distribution with a shot noise driven parameter best modeled the number of attacks on an attack day. Interpretation of the model parameters is discussed and predictive performance of the models is evaluated.
1. Introduction.
Terrorism models must accommodate many nonattack days, extreme counts on attack days, and clustered event timing. This paper combines a hurdle model with self-excitation to model and predict activity from event history, without relying on additional covariates.
1. Introduction.
- Earlier approaches include time-series, trajectory, and hazard models, but some hazard models require prespecifying how recent event history affects current events.The literature also uses zero-inflated Poisson models to account for excess zeros.Source: §1 · p. 2
- Terrorism data combine many zero-count days and extreme attack-day counts with clustered timing, making standard independent-count models unsuitable.The paper frames these as two modeling issues: excess zeros and extreme values, and temporal dependence.Source: §1 · p. 2
- A hurdle model separates the probability of zero attacks from the distribution of nonzero counts, accommodating sparse days and extreme counts.It treats extra zeros as arising from a separate hurdle process rather than censoring, which the authors consider more reasonable when attacks are genuinely rare.Source: §1 · pp. 2–3
- A self-exciting component makes event probability depend on previous events, with their influence decreasing over time, to represent clustering and dynamics.The paper motivates this with the clustering hypothesis that events influence the probability of subsequent events.Source: §1 · pp. 2–3
- The paper combines these components for Indonesia and Timor-Leste in 1994–2007, modeling attack-day probability and nonzero counts as functions of event history.The authors state that this approach can predict risk without additional covariates, which may be missing or unreliable.Source: §1 · p. 3
2. Data and exploratory analysis.
The analysis uses daily attack counts from the Global Terrorism Database and finds that the observed data include excess zeros, extreme counts, and temporal clustering. These patterns motivate flexible models that can represent both count distributions and event timing.
2. Data and exploratory analysis.
- The dataset comprises daily counts of attacks in Indonesia, including Timor-Leste, from January 1, 1994, through December 31, 2007.The database is described as an open-source collection of over 87,000 terrorist events worldwide from 1970 through 2008.Source: §2 · p. 3, §2.1 · p. 3
2.1. Exploratory data analysis.
- Of 2,557 training-period days from 1994–2000, 250 attacks occurred on 158 event days, with long quiet stretches followed by periods of increased activity.Figure 1 presents daily and semi-yearly counts; most days had no attacks (93.8%), while some had multiple attacks.Source: §2.1 · pp. 3–4, Figure 1 · p. 4
- The Poisson fit misses tail behavior, while the negative binomial misfits counts of one, two, and three attacks; its goodness-of-fit test gives p = 0.00016.The observed tail includes days with 36, 11, 10, and 6 attacks, so excess zeros and extreme values call for more complex models.Source: §2.1 · pp. 4–5, Table 1 · p. 4
- After adjusting for baseline attack-day probabilities, observed Ripley’s K values exceed the 95% pointwise confidence intervals, indicating unaccounted-for clustering.The intervals use a parametric bootstrap with 1,000 simulations.Source: §2.1 · p. 5
- The observed combination of temporal clustering, excess nonattack days, and possible coordinated attacks motivates flexible models for terrorist activity.The paper develops self-exciting hurdle models in the next section to represent these patterns.Source: §2.1 · p. 5
3. Hurdle models and the self-exciting process.
The hurdle model separates whether an attack day occurs from the number of attacks on that day. A shot-noise process adds history-dependent excitation to event-day probabilities and supports predictions of time until the next event day.
3.1. Hurdle models.
- The hurdle model separates the probability of an event day from the distribution of positive attack counts, and its likelihood can be evaluated separately when the components share no parameters.The count component has no probability mass at zero.Source: §3.1 · pp. 5–7
3.2. Self-exciting process.
- The shot-noise process sums past event-day contributions, with event magnitudes controlling their strength and a normalized decay function governing how their influence fades over time.The decay function has positive-integer support and sums to one, leaving each event’s magnitude parameter responsible for its total contribution.Source: §3.2 · pp. 7–8
3.3. Self-exciting Bernoulli hurdle process.
- The event-day probability depends on a baseline process plus self-excitation from prior events, while the baseline can incorporate exogenous variables but excludes internal event history.The transformation ensures a valid probability; under a rare-event Poisson approximation, an event day contributes an expected α_i additional event days, with timing governed by the decay function.Source: §3.3 · pp. 8, 10
3.4. Survival functions.
- The survival function gives the probability that the next event day is more than a specified number of days away, supporting predictions conditional on the current event history.It is calculated assuming no new events have occurred since the starting time.Source: §3.4 · p. 10
4. Results.
The results compare self-exciting and baseline-only models for event-day probabilities, then assess self-exciting count models. In the 2001–2007 testing period, self-exciting models performed better than baseline-only models for event days, and the self-exciting zeta model was preferred for counts.
4.1. Event day modeling.
- The event-day model SE1 had the lowest AIC among evaluated models, beating the best baseline-only model BL5, which included periodic terms.With the shot-noise term at zero, SE1’s baseline hurdle probability was about 0.012; its generated event day had a 37.54-day expected wait and a 16-day median wait.Source: §4.1 · p. 11, Table 2 · p. 12
4.2. Count modeling.
- For event-day counts, the constant-parameter zeta model fit adequately, while the self-exciting count model placed more mass on multi-attack days when excitation was high.The self-exciting count effect decayed rapidly and lasted only a few days; the training-period zeta estimate was s = 2.86, with a bootstrap Kolmogorov–Smirnov p-value of 0.69.Source: §4.2 · pp. 12–14, Table 3 · p. 14
4.3. Testing model predictions.
- Testing-period prediction was evaluated by repeatedly forecasting the next day from past history, updating parameters with the new observation, and scoring relative log-probability gain.The hurdle-component testing results were similar to training-period AIC rankings; the count results favored the self-exciting model more strongly than training AIC suggested.Source: §4.3 · pp. 15–16
- All self-exciting models outperformed baseline-only models for event-day prediction, and the self-exciting zeta model was preferred for count prediction.The count comparison used the constant-parameter zeta model as its reference.Source: §4.3 · p. 16, Table 5 · p. 16
5. Conclusion.
The paper presents a self-exciting hurdle model for predicting terrorist activity and finds that attack risk can vary substantially over short periods. The analysis is limited to attack timing, leaving other attack characteristics unmodeled.
5. Conclusion.
- Resource allocation depends on understanding future terrorist risk, while the paper notes that policy makers face financial, material, and personnel constraints.The airport-security example frames the trade-off between maintaining added security and investing in longer-term intelligence capabilities.Source: §5 · p. 17
- The best model combines a self-exciting hurdle component with a Riemann zeta count distribution whose parameter is driven by shot noise.The self-exciting term supports contagion theory and suggests a significant short-term increase in risk after an attack.Source: §5 · p. 17
- The model provides good fit and predictive capabilities without exogenous variables, while offering an interpretable structure for studying terrorist activity dynamics.The authors identify covariate effects, including counter-terrorism activities and geographic or political factors, as directions for further analysis.Source: §5 · p. 17
- The analysis focuses on attack timing in Indonesia and does not consider location, attack type, or responsible group.The authors suggest spatial proximity terms or multiple group-specific self-exciting terms as ways to model more of this variation.Source: §5 · pp. 17–18
- Risk varies greatly over short periods, so resource allocation and security responses should reflect both short-term variation and longer-term trends.The paper connects short-term risk to airport security deployment and longer-term trends to strategic counter-terrorism resources.Source: §5 · p. 18