Crime Hotspots: Modelling Risk With Self-Exciting Point Processes

Crime does not scatter randomly

If crime occurred purely at random — a fixed background rate, independently at every location — you would expect burglaries, say, to be spread roughly evenly across a city, with the occasional coincidental cluster arising from chance alone. Real crime data looks nothing like that. Decades of criminology research on near-repeat victimisation have shown that a burglary at one address measurably raises the risk of another burglary nearby — often the same house, more often the immediate street — for a period of days to a few weeks, after which the elevated risk decays back toward baseline. Crime clusters in both space and time simultaneously, and it does so because individual events causally influence the near future, not merely because of stable geography.

The Hawkes process: borrowed from earthquakes

The mathematical tool for exactly this pattern is the Hawkes process, a self-exciting point process introduced by Alan Hawkes in 1971 and later adapted for crime by researchers including George Mohler, drawing directly on Yosihiko Ogata's Epidemic-Type Aftershock Sequence (ETAS) model from seismology — the same family of equations used to model how one earthquake raises the probability of aftershocks nearby. The conditional intensity — the instantaneous rate of new events, given everything that has happened so far — combines a steady background rate with a sum of decaying contributions from every past event:

λ(t, x) = μ(x) + Σ_{t_i < t}  g(t − t_i, x − x_i)

μ(x)       background rate — the "always there" baseline risk at location x
g(Δt, Δx)  triggering kernel — extra risk contributed by a past event,
           decaying with elapsed time Δt and distance Δx
           (commonly an exponential or power-law decay in each)

sum runs over every event i that occurred before time t

Boost, flag, and why the distinction matters

Criminologists offer two, not mutually exclusive, explanations for near-repeat patterns. The boost account says an offender genuinely learns something valuable from a successful crime — which window was unlocked, when residents are away, how to avoid a specific alarm — and returns to exploit that same knowledge again, which is precisely the causal, self-exciting mechanism the Hawkes process captures directly. The flag account is more static: some locations were always more vulnerable, due to lighting, sightlines, or easy escape routes, and the first crime simply reveals — flags — that pre-existing vulnerability rather than causing the next one. Distinguishing the two matters for policy, because boost-driven risk responds well to short-term, location-specific intervention right after an incident, while flag-driven risk calls for permanent environmental changes instead.

Predictive policing and its feedback problem

Software like PredPol built directly on ETAS-style Hawkes models to forecast where the next incident is statistically most likely, and to direct patrol resources there. The approach has a well-documented weakness: if officers are sent preferentially to a flagged hotspot, more crime tends to get recorded there simply because more police are present to observe and log it — especially for offences like drug possession that are detected mainly through active policing rather than victim reports. That extra recorded crime then feeds back into the model as further evidence the area is high-risk, entrenching a loop that can reflect historical patrol patterns and bias as much as it reflects the true underlying rate of offending, a concern raised prominently by researchers including Kristian Lum and William Isaac.

What the simulation lets you explore

The model on this page lets you adjust the excitation strength — how strongly each incident raises nearby risk — and toggle a hotspot patrol response on and off, so you can compare how quickly clusters form and dissolve under different assumptions, and see directly how a patrol response that concentrates on the model's own flagged areas changes the pattern of subsequent recorded incidents.

Frequently asked questions

Try it live

Everything above runs in your browser — open Crime Hotspots and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.