Geographic profiling is the problem of estimating the residence (or place of work) of a criminal offender given the locations of crimes committed by the offender.  We have developed an agent-based, Bayesian method for geographic profiling that calculates a prior distribution of residences using housing/population density and a prior distribution of foraging parameters using historic crime data.  The method attempts to take into account how criminals interact with their heterogeneous environment.

The movie on the right is geographic profiles for 244 arrested burglars in Los Angeles.  The circles are the crime locations and the square is where the burglar lived.  Note that there is no probability mass in the ocean!!!  This is not true for methods currently used by police based upon kernel smoothing.

Highly clustered event sequences are observed in crime data, as burglars will attack a cluster of neighboring houses within a short period of time or an initial act of gang violence will incite a wave of retaliations.  The most widely used tool to display and analyze these patterns are crime hotspot maps, density plots of recent criminal activity (see figure on the right).

Our research incorporates observed patterns of criminal behavior into macroscopic models of crime in a systematic way.  One successful approach so far is to model crime as a Poisson process of “background” events, each of which can trigger a sequence of aftershocks analogous to those in seismology.  Such a point process is referred to as “self-exciting”.  Predictions can then be made by flagging the neighborhoods in the city where the conditional intensity takes on its highest values:

To the right is the conditional intensity for burglaries in the San Fernando Valley of Los Angeles for three consecutive days.  The intensity acts as an evolving risk surface that could be used by police to direct patrols.  The five percent of cells with the highest estimated risk are flagged as the most likely burglary targets.  Red indicates newly flagged cells and yellow indicates the removal of a flag.  Our findings indicate that forecasting strategies based upon self-exciting point processes significantly outperform recently proposed strategies based upon Crime Hotspot Maps.

Currently we are developing hybrid SPDE-point process models of burglary that incorporate small scale dynamic interactions between individual criminals and their environment.  Here criminals search for their targets by diffusing (with a bias) towards neighborhoods of higher attractiveness.  The advantage of a hybrid approach is that all of the tools for parameter estimation, model evaluation, and forecasting developed for standard self-exciting point processes can still be applied.  Click here to see a movie of such a point process, where the conditional intensity is found by solving a system of nonlinear PDEs between events similar to models of chemotaxis.