Andrea Bertozzi
University of California Los Angeles
Thanks to contributions from Martin Short, George Mohler, Jeff Brantingham,
and Erik Lewis.
0.006
Short et al J. Quant. Crim. 2009
repeat crime is much more
likely to happen in a short
interval of time after the
first event
probability p(τ)
0.005
0.004
0.003
0.002
0.001
0
0
50
100
150
200
250
300
350
days between repeat crimes (τ)
400

burglars return to places to replicate the
successes of and/or exploit vulnerabilities
identified during previous offenses:
“I always go back [to the same places] because, once you
been there, you know just about when you been there
before and when you can go back. An every time I hit a
house, it’s always on the same day [of the week] I done been
before cause I know there ain’t nobody there. “ (Subject No.
51)
Wright and Decker Burglars on the Job (1996: 69)
On right, histogram of times between pairs of burglaries
separated by 200m or less. On the left, similar histogram
for Southern California earthquake (magnitude 3.0 or
greater) pairs separated by 220km or less.
Events occur entirely at random, defining a stochastic
process where each event occurs independently of prior
events.
 Mathematically, such a phenomenon can be modeled as a
Poisson process characterized by a rate parameter l,
representing the expected number of events per unit time.
 the probability that one burglary occurs within a time
interval t to t + dt is given by
 The probability that k burglaries occur is given by the general
Poisson distribution


The probability that no events occur within a time interval
dt, then, is given by

The time T1 until the first event occurs

Probability that first event occurs between
times t and t+dt

Poisson process probability density function
for time interval between events

Suppose we have different types of events
associated with different locations, e.g.
residential burglaries whose rates vary by
spatial location. Then the composite
probability is

Where wi is the fraction of homes exhibiting
rate constant li.
Fit to
With N = 3




At first glance the good fit with N=3 suggests
that the Long Beach data satisfies the REH.
However it turns out that only a fraction of
the total number of houses fit into the N=1,
N=2, N=3 bins as determined by “house
order” the total number of times burgled
during the time period of evaluation.
Suggests we need another method for
measuring repeat victimization.



Parameter free method
Pick a fixed window time period D
Probability distribution of time intervals
between victimization for order 2 homes
(homes that have exactly two events during
this window perios, assuming REH):


Comparison to REH shown as black line.
D=364
On right, histogram of times between pairs of burglaries
separated by 200m or less. On the left, similar histogram
for Southern California earthquake (magnitude 3.0 or
greater) pairs separated by 220km or less.

A space-time point process is characterized
by its conditional intensity given a history Ht

Epidemic Type Aftershock Sequence models
(ETAS) divide earthquakes into two
categories: background events and
aftershock events.
Background events occur according to a stationary
process m with magnitudes distributed
independently of m with probability j(M).
 Each of these earthquakes then elevates the risk of
aftershocks and the elevated risk spreads in space
and time according to the kernel g(t; x; y;M).


Parameter selection for ETAS models is most
commonly accomplished through maximum
likelihood estimation, where the log
likelihood function (Daley and Vere-Jones,
2003), is maximized over all parameter sets .





Measure of goodness of fit of a statistical model – used for
model selection
AIC=2K-2ln(L) where K is the number of parameters in the
model and L is the maximized value of the likelihood
function of the model.
The AIC methodology attempts to find the model that best
explains the data with a minimum of free parameters.
If model errors are normally and independently distributed,
then AIC is equivalent to 2K+n[ln(RSS)], RSS is residual sum
of squares (difference between data and model prediction)
where n is number of observations.
Preferred model has the lowest AIC value.
Rivalry network among 29 street gangs in Hollenbeck, Los Angeles
Tita et al. (2003)

event dependence is a common process
driving repeat victimization across all crime
types

specific behavioral mechanism—street
smarts/street justice—may differ in detail, but
outcome is the same

Hawkes Process is a flexible representation of
self-excitation
time since the most
recent incident
rivalry intensity
self-excitation
l (t )  m  k0 e
 t ti 
t t i
background
rate of violence
retaliation
strength
retaliation
duration
actual
Mike Egesdal, Chris Fathauer, Kym Louie, and Jeremy Neuman,
Statistical Modeling of Gang Violence in Los Angeles, submitted to SIURO.
simulated
Here k0 is the expected number of retaliations per attack,
1/w is the expected waiting time for retaliation (in days)
Percentage of crimes predicted vs percentage of cells flagged
for 2005 burglary (left) and 2007 robbery (right). Curve for CHM is
point wise max over a variety of hotspot map prediction methods
discussed in the criminological literature.
n events Najaf, Iraq
inter-event times Najaf, Iraq
Data from Iraq Body Count, analysis by Erik Lewis, UCLA





Iraqi data shows a clear temporal
dependence on background rate likely linked
to troop presence.
We consider several models for change in
background rate :
(a) step model,
(b) linear increase,
(c ) variable bandwidth kernel smoothing.

Example – linear background rate






Time period: March 20, 2003 – Dec. 31, 2007
15,977 events
Start date, end date, min and max # deaths,
town and/or district.
In the analysis no distinction is made
between different # deaths per event.
Do not distinguish between type of event
(e.g. IED or gunfire).
Only consider start date. (93% of events have
same start/end date)
A histogram of all 149 events in Najaf with 30 bins is plotted on the left. The
estimated fit with a linear background rate is plotted on the right (the jagged curve).
The linear fit without self excitation is shown as well.







M.B. Short, M.R. D'Orsogna, P.J. Brantingham, and G.E. Tita, Measuring
and modeling repeat and near-repeat burglary effects, J. Quant.
Criminol. 25 (2009).
G.O. Mohler, M.B. Short, P.J. Brantingham, F.P. Schoenberg, and G.E.
Tita, Self-exciting point process modeling of crime, preprint (2010).
Feller W (1968) An introduction to probability theory and its applications,
3rd edn., vol 1. Wiley, New York.
Daley, D. and Vere-Jones, D. (2003). An Introduction to the Theory of
Point Processes, 2nd edition. New York: Springer.
Statistical Modeling of Gang Violence in Los Angeles Mike Egesdal, Chris
Fathauer, Kym Louie, Jeremy Neuman, SIAM J. Undergraduate Research
Online, 2010.
Mark Allenby, Kym Louie, and Marina Masaki, project report, Tim Lucas
mentor, A Point Process Model for Simulating Gang-on-Gang Violence ,
2010 REU program at UCLA.
E. Lewis, G. Mohler, P. J. Brantingham, and A. L. Bertozzi, Self-Exciting
Point Process Models of Civilian Deaths in Iraq, preprint 2010.
Johnson, S. (2008). Repeat burglary victimisation: a tale of
two theories. IEEE Trans. Automatic Control , 4 , 215-240.
 Townsley, M., Johnson, S. D., & Ratclie, J. H. (2008). Space
time dynamics of insurgent activity in Iraq. Security Journal ,
21 , 139-146.
 Iraq Body Count. (2008). Iraq body count.
http://www.iraqbodycount.net.
 Akaike, H. (1974). A new look at the statistical model
identication. IEEE Trans. Automatic Control , AC-19 , 716-723.
 Akaike, H. (1973). Information theory and an extension of the
maximum likelihood principle. Budapest: Akademiai Kiado.
