Andrea Bertozzi University of California Los Angeles

Thanks to contributions from Martin Short, George Mohler, Jeff Brantingham, and Erik Lewis.

repeat crime is much more likely to happen in a short interval of time after the first event

Short et al J. Quant. Crim. 2009

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 histogramfor 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 , 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 i.

Fit toWith 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 methodPick a fixed window time period DProbability 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 histogramfor 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 with magnitudes distributed independently of 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 AngelesTita 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

0( ) i

i

tt

tt

t k e

background rate of violence retaliation

strength

retaliationduration

rivalry intensity self-excitation

time since the mostrecent incident

simulated

actual

Mike Egesdal, Chris Fathauer, Kym Louie, and Jeremy Neuman, Statistical Modeling of Gang Violence in Los Angeles, submitted to SIURO.

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.

inter-event times Najaf, Iraq

n events 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.