Statistical modeling and analysis of repeat and retaliatory victimization

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Statistical modeling and analysis of repeat and
retaliatory victimization

• Andrea Bertozzi
• University of California Los Angeles

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

• 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, its always on the same
  day of the week I done been before cause I know
  there aint nobody there. (Subject No. 51)
• Wright and Decker Burglars on the Job (1996
  69)

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Crime Clusters in Space Time
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.
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Random Event HypothesisM. B. Short et al J.
Quant. Crim. 2009.

• 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

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REH and distribution of time intervals between
exact repeat events.

• The time T1 until the first event occurs
• Probability that first event occurs between times
  t and tdt
• Poisson process probability density function for
  time interval between events

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Probabilities of events with different rate
constants

• 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.

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Comparison of repeat probabilities using moving
window count Longbeach Burglary Data

• Fit to
• With N 3

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Interpretation of Data

• At first glance the good fit with N3 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 N1, N2, N3
  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.

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Fixed window method

• 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)

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Example with Long Beach data

• Comparison to REH shown as black line.
• D364

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Theory can be extended to higher order events
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Crime Clusters in Space Time
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.
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Self-exciting point process models in Seismology

• 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.

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Formula for conditional intensity

• 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
  yM).

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Parameter estimation

• 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 .

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Akaike Information Criterion

• Measure of goodness of fit of a statistical model
  used for model selection
• AIC2K-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
  2Knln(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.

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Gang networks and self-excitation
Rivalry network among 29 street gangs in
Hollenbeck, Los Angeles Tita et al. (2003)
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a general statistical structure

• event dependence is a common process driving
  repeat victimization across all crime types
• specific behavioral mechanismstreet
  smarts/street justicemay differ in detail, but
  outcome is the same
• Hawkes Process is a flexible representation of
  self-excitation

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Hawkes Process
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Mike Egesdal, Chris Fathauer, Kym Louie, and
Jeremy Neuman, Statistical Modeling of Gang
Violence in Los Angeles, submitted to SIURO.
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Overview of Hollenbeck Gangs
Here k0 is the expected number of retaliations
per attack, 1/w is the expected waiting time for
retaliation (in days)
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Point Process Crime Prediction
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Comparison with Crime Hotspot Maps
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.
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Current Research Insurgencies
n events Najaf, Iraq
inter-event times Najaf, Iraq
Data from Iraq Body Count, analysis by Erik
Lewis, UCLA
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Models with time dependent background rate

• 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.

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Parameter estimation using maximum likelihood

• Example linear background rate

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Data from Iraq Body Count

• 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)

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IBC data 2003-2007
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Number of events per day
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Karkh Hawkes (smooth) best fit
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Karkh - the data shown
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Najaf data linear model
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.
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AIC for Najaf data
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References

• 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.

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More references

• 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.