Self-exciting point processes explain spatial-temporal patterns in crime

Self-exciting point processes explain spatial-temporal
patterns in crime
G. O. Mohler ∗ M. B. Short † P. J. Brantingham ‡ F. P. Schoenberg §
G. E. Tita
Self-exciting point processes fit to Los Angeles crime data explain spatial-temporal patterns in
crime and, as a consequence, lead to forecasting strategies that significantly outperform widely used
crime hotspot maps. Highly clustered event sequences are observed in crime data due to various
patterns of criminal behavior 1, 2, 3, 4, 5, 6, 7, 8 . We show how these behavioral patterns can be incor-
porated into macroscopic, city-wide models of crime in a quantitative and systematic way. For this
purpose we model crime as a space-time Poisson process of “background” events, each triggering a
sequence of “aftershocks” analogous to those in seismology. Our findings show that this approach
provides a more complete picture of the statistical nature of crime and has important implications for
crime prediction and prevention.
It is increasingly accepted that crime spreads through local environments via a contagion-like process 8 .
Burglars repeatedly attack clusters of nearby targets because local vulnerabilities are well-known to the
offenders 7 . A gang shooting may incite waves of retaliatory violence in the local set space (territory) of the
rival gang 9, 10 . The local, contagious spread of crime leads to the formation of crime clusters in space and
time. Recognizing this, hotspot policing strategies use past crime clusters to estimate where crime is likely
to occur in the future. Here crime hotspot maps are the most widely used tool for the quantification of
future crime risk and are a key element in hotspot policing 11, 12, 13, 14, 15 .
Given space-time crime observations (xk, yk, tk), crime hotspot maps are generated for a time interval
∗ Department of Mathematics and Computer Science, Santa Clara University
† Department of Mathematics, University of California, Los Angeles
‡ Department of Anthropology, University of California, Los Angeles
§ Department of Statistics, University of California, Los Angeles
Department of Criminology, Law, and Society, University of California, Irvine
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[t − T, t] by overlaying a density plot of the function,
λ(x, y, t) =
t−T

function given by (1) with the function,
The key differences between (1) and (2) are:
λ(x, y, t) = µ(x, y) +
g(x − xk, y − yk, t − tk). (2)
tk

i) “Exact-repeat” crimes, multiple crimes occurring at the same city address, dominate the log-likelihood
function and cause the kernel g in (2) to converge to a delta function in space. By replacing the observations
(xk, yk, tk) with (i(k), j(k), tk), where (i(k), j(k)) is the grid cell containing event k, this issue is resolved.
ii) Forecasts are typically generated by dividing a city map into equally sized cells and choosing a percentage
of the cells as likely criminal targets. Thus coarse-graining is useful, either before or after fitting the model,
in order to determine the cells with the highest risk.
In general the background intensity µ in (2) could depend on time 20 , as the incorporation of variables such
as weather and time of day could be useful, but here we assume µ is stationary. The important distinction
between (1) and (2) is that µ is Poisson and thus does not depend on the history of the process.
When the cell widths of the spatial grid are large compared to the distance over which the elevated risk
spreads, in the case of burglary typically a few hundred meters 1, 11 , the coarse-grained version of (2) can be
approximated by the spatially decoupled cell intensities,
λlm(t) = µlm +
{tk

of weeks or months, and for coarse-grained models with cells of a few hundred meters or more we find that
these events can be effectively modeled as background events.
Whereas self-exciting point process models estimate that more than 90% of burglaries and robberies in
the data set are background events, crime hotspot maps over-estimate the contribution of aftershocks to
future crime risk by shifting all weight to the kernel g. Furthermore, the length of the time interval T for
crime hotspot maps is typically determined on the order of several weeks using rules of thumb, but here we
estimate the decay of the excited rate to be much more rapid, on the order of several days. For other data
sets, crime types, and cities, aftershocks may be more prevalent and occur over longer time scales. In such
a circumstance a model of the form (1) would have greater merit. However, we argue that such a model
should be determined through the maximum likelihood estimation of (3) and that µ will likely play some
non-trivial role.
Next we investigate the implications of our findings for the prediction of crime. Here we compare the
self-exciting point process model to crime hotspot maps generated using a Gaussian kernel with 200m, 400m,
and 800m bandwidths and 2 week, 1 month, and 2 month time intervals. For every day k of 2005, each
model assesses the risk of burglary within each of M 2 cells partitioning an 18km by 18km region of the San
Fernando Valley in Los Angeles. Based on the data from the beginning of 2004 up through day k, the N
cells with the highest risk (value of λ) are flagged yielding a prediction for day k + 1. The percentage of
crimes falling within the flagged cells on day k + 1 is then recorded and used to measure the accuracy of
each model. We then repeat this process for robbery data within the same spatial region occurring during
the years 2006 and 2007.
In Figure 3, we plot the evolving risk surface corresponding to the self-exciting point process forecasting
strategy outlined above. Each day the flagged cells change, as new crimes occur and the conditional intensity
given by (3) is updated. By accurately selecting which cells are flagged, we find that a large percentage of
crimes can be predicted, even when flagging only a small percentage of cells. Directed police patrol keyed
to this risk surface would maximize crime deterrence given limited resources.
In Figure 4, we plot the percentage of crimes predicted averaged over the forecasting year against the
percentage of flagged cells for the self-exciting point process and the pointwise maximum over all bandwidths
and time intervals for the crime hotspot maps. Depending on the percentage of cells flagged, the self-exciting
point process predicts 20% to 95% more crimes on average than the crime hotspot maps. For example, with
5% of the domain flagged, the self-exciting point process predicts ∼45% of robberies on average, whereas
the crime hotspot maps predict ∼25%. The difference in accuracy between the two methodologies can be
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Figure 3: Evolving crime forecasts. (top) The conditional intensity λ for San Fernando Valley burglary given by
Equation (3) for three consecutive days in 2005. Here the cells are 500m by 500m and blue, yellow, and red correspond
to progressively higher values of λ. (bottom) The 5% of cells with the highest value are flagged as the most likely
burglary targets. For the second and third days, red corresponds to newly flagged cells and yellow corresponds to
cells where the flag from the previous day has been removed.
attributed to the crime hotspot maps failure to account for the background rate of crime. While a certain
percentage of crime risk stems from near-repeat effects and evolves on a daily basis, we find that a large
percentage is stationary and is isolated in neighborhoods with intrinsically higher crime rates (see Figure 3).
Accurate, city-wide models of crime can be constructed using self-exciting point processes. For the 18
km by 18km region of the San Fernando Valley under study, parameters of the point process are estimated
in 1-2 minutes and next day forecasts are generated in seconds on a laptop computer. Furthermore, self-
exciting point processes can be used to simulate future crime and thus daily, weekly, and monthly forecasts
can be generated rapidly using standard point process simulation algorithms 18, 21 . Software based on these
methods could be used by police to better focus patrolling routes 22, 23 and by the general public to shore up
the formal and informal local community institutions that yield beneficial collective efficacy effects 24 .
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Figure 4: Forecasting strategy comparison. Average daily percentage of crimes predicted plotted against percentage
of cells flagged for 2005 burglary (left) and 2007 robbery (right) using 200m by 200m cells. Here the self-exciting point
process is compared to the pointwise maximum over all crime hotspot maps tested. Depending on the percentage
of cells flagged, the self-exciting point process predicts 20% to 95% more crimes on average than the crime hotspot
maps. For example, with only 2% of the domain flagged the point process predicts 30% of the total crimes, whereas
the crime hotspot maps predict less than 15%.
Supplemental Information
Parameter estimation
Here we describe the maximum likelihood estimation of (3). For event k occurring in cell (l, m), the stochastic
declustering algorithm estimates the probability pk that k is a background event as,
The background intensity is then implicitly defined as,
µlm = (1/T ) ·
pk = µlm
. (4)
λlm(tk)
{tk

log-likelihood function can be written as,
l(K0, ω) =
lm
− µlm +
{k: i(k)=l, j(k)=m}
T
log λlm(tk; K0, ω) −
tk
g(t − tk; K0, ω)dt , (7)
and is maximized using Monte-Carlo optimization in the search space K0 ∈ [0, 1], ω −1 ∈ (0 days, 60 days].
Data
The residential burglary data used in this study consists of all reported residential burglaries, collected by
the Los Angeles Police Department, within an 18km by 18km region of the San Fernando Valley during the
years 2004 and 2005. The robbery data used in this study consists of all reported robberies, collected by the
Los Angeles Police Department, within the same 18km by 18km region of the San Fernando Valley during
the years 2006 and 2007. In addition to a geocoded address, each event is associated with a reported time
window over which it could have occurred, often a few hour or less span. We defined the event time to be
the midpoint of each time window. In total, the burglary data set contains 5,466 events and the robbery
data set contains 2,849 events.
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Acknowledgements
The authors would like to thank the National Science Foundation for support through grant BCS-0527388.
Correspondence and Requests for materials should be addressed to:
George Mohler
Santa Clara University
Department of Mathematics and Computer Science
Santa Clara, CA
gmohler@scu.edu
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