A Statistical Model of Criminal Behavior

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A Statistical Model of Criminal Behavior

• M.B. Short, M.R. DOrsogna, V.B. Pasour, G.E.
  Tita, P.J. Brantingham, A.L. Bertozzi, L.B. Chayez

Maria Pavlovskaia
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Goal

• Model the behavior of crime hotspots
• Focus on house burglaries

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Assumptions

• Criminals prowl close to home
• Repeat and near-repeat victimization

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The Discrete Model

• A neighborhood is a 2d lattice
• Houses are vertices
• Vertices have attractiveness values Ai
• Criminals move around the lattice

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Criminal Movement

• A criminal can
• Rob the house he is at
• - or -
• Move to an adjacent house
• Criminals regenerate at each node

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Criminal Movement

• Modeled as a biased random walk

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Attractiveness Values

• Rate of burglary when a criminal is at that house
• Has a static and a dynamic component
• Static (A0) - overall attractiveness of the house
• Dynamic (B(t)) - based on repeat and near-repeat
  victimization

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Dynamic Component

• When a house s is robbed, Bs(t) increases
• When a neighboring house s is robbed, Bs(t)
  increases
• Bs(t) decays in time if no robberies occur

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Dynamic Component

• The importance of neighboring effects ?
• The importance of repeat victimization ?
• When repeat victimization is most likely to
  occur ?
• Number of burglaries between t and ?t Es(t)

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Computer Simulations
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Computer Simulations

• Three Behavioral Regimes are Observed
• Spatial Homogeneity
• Dynamic Hotspots
• Stationary Hotspots

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Spatial Homogeneity
Dynamic Hotspots
Stationary Hotspots
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Computer Simulations

• Three Behavioral Regimes are Observed
• Spatial Homogeneity
• Large number of criminals or burglaries
• Dynamic Hotspots
• Low number of criminals and burglaries
• Manifestation of the other two regimes due to
  finite size effects
• Stationary Hotspots
• Large number of criminals or burglaries

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Continuum Limit

• In the limit as the time unit and the lattice
  spacing becomes small
• The dynamic component of attractiveness
• The criminal density

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Continuum Limit

• Reaction-diffusion system
• Dimensionless version is similar to
• Chemotaxis models in biology (do not contain the
  time derivative)
• Population bioglogy studies of wolfe and coyote
  territories

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Computer Simulations

• Dynamic Hotspots are never seen
• Spatial Homogeneity or Stationary Hotspots?
• Performed linear stability analysis
• Found an inequality to distinguish between the
  cases

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Summary

• Discrete Model
• Computer Simulations
• Spatial Homogeneity, Dynamic Hotspots, Stationary
  Hotspots
• Continuum Limit
• Dynamic Hotspots are not observed due to finite
  size effects
• Inequality to distinguish between Homogeneity and
  Hotspots cases

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Applications

• House burglaries
• Assault with a lethal weapon
• Muggings
• Terrorist attacks in Iraq
• Lootings