Cracking Down on Crime with Applied Mathematics

On Thursday, October 4, Dr. Andrea Bertozzi, professor of mathematics and director of applied mathematics at University of California Los Angeles, presented her findings on the mathematics of crime. Dr. Bertozzi and her team have created models that predict patterns of crime in inner cities; using computer programs supplied with various algorithms and equations, they can help to identify urban crime hotspots.

Bertozzi’s research started about seven years ago when she began working with the Los Angeles Police Department (LAPD) to develop predictive policing to reduce crime. LAPD wanted to know where and when crime was going to happen before it occurred, and Bertozzi has worked to deliver that answer. Her computer programs and algorithms have enabled LAPD and surrounding police departments to alert each shift to hotspots in the area before they depart. With targeted patrols made possible by applied mathematics, crime rates dropped dramatically, falling by as much as twenty-seven percent in some areas.

What equations and models does Bertozzi use to predict these high crime areas? Dr. Bertozzi started with the routine activity theory, which is based on the assumption that crimes occur where offenders encounter targets and victims who lack effective security, and that, when this opportunity is presented, crimes are more likely to be repeated in that area. By tracking where crimes occur, mathematicians can get a physical basis for crime pattern formation and can create analytical models based on criminal behavior.

Bertozzi uses a discrete model to simulate the actions of individuals in society who are committing crimes. Using 2-D grids, she can create abstract representations of cities, populate the grid with burglars, and define the field that indicates attractiveness with an equation that relates the intrinsic attractiveness and dynamic attractiveness of a location. The higher the attractiveness value, the more likely it is that a break-in will occur. Additionally, once this crime occurs, the attractiveness of the surrounding houses goes up, as does that of the burglarized house.

Natural decay in attractiveness occurs over time, however, and Bertozzi’s team has created yet another equation to predict the rate of decay with the help of a linear stability theory that plots the area over which risk diffuses. The diffusion of risks binds local crimes together and gives the analyst hotspots.

Not only can Bertozzi predict where and when crimes will occur, but she can also predict if police will be able to successfully suppress a crime hotspot by taking into account spatial heterogeneity, or the variation in offender and target area distributions. By using census data, crime density estimations, and high-resolution imaging and processing, Bertozzi can produce crime maps with accurate edges to advise police forces.

To end her presentation, Bertozzi discussed her biggest hurdle in mapping crime – gang networks in Los Angeles. These gangs have complicated rivalry networks, but, by relating the background rate of violence of a gang, the time since the most recent inter-gang incident, the likeliness of retaliation, and the average retaliation duration, Bertozzi can model rival intensity. She has also been able to use this information to help crack unsolved gang crimes and provide the police with suspect gangs. Bertozzi and her team of post-doctoral, graduate, and undergraduate students are currently analyzing data and constructing models to help the police better solve unsolved crime and will continue to perfect their crime mapping techniques.

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