Predicting Crime with Math

In several major U.S. cities, police are beginning to use math originally developed to predict earthquakes to fight crime. Or more specifically, the kinds of crime that occur in either series or sprees that tends to be concentrated within a defined region, such as burglaries or gang-related violence.

If you live in Los Angeles there are two things you might feel particularly worried about: earthquakes and crime. It’s nice to know, then, that mathematics can help to keep you safe from both. A software system called PredPol, that has been developed by the mathematician George Mohler, the anthropologist Jeff Brantingham, and others, is now being rolled out across multiple jurisdictions of the Los Angeles Police Department and in other cities too. Officers on the ground use it every day.

PredPol stands for “predictive policing”. It works by calculating the probability that crimes will be committed in a particular area on a particular day, based on real-time data from the previous couple of days. Police officers are then given prediction maps telling them where the probability is high, so they can put in place extra patrols and hopefully prevent at least some of those crimes from happening.

The initial results of using the math-based approach to directing police activity would appear to have some promise. After being introduced in the Los Angeles Police Department’s Foothill Division in 2011, after having seen its violent crime rates rise significantly in 2010 as the city reduced its budget for policing the area, the incidence of crimes fell by 13% in the first four months after the division implemented the PredPol software, while crime elsewhere in the city rose by 0.4%. That reduction came as the city further reduced its policing budget in 2011.

More recently, the LAPD Foothill Division saw a 20% reduction in the level of predicted crimes in the year from January 2013 to January 2014. According to PredPol’s marketing, other cities where the predictive crime software has been introduced have reported similar experiences.

+Plus Magazine‘s Marianne Freiberger describes how the math behind the software works.

To get a feel for how this works, let’s concentrate on something that’s rife in LA and other big cities too: gang crime. Fierce battles over territory are central to gang violence, and whatever one gang does to another, retaliation is sure to follow. That latter point is what makes gang violence similar to earthquakes: acts of violence come with follow-ups, just as earthquakes come with aftershocks.

Earthquakes can be described mathematically as self-exciting processes. They are events that happen over time as a result of all sorts of complex factors we don’t really understand. So we might as well treat them as random processes. What we do know, however, is that once an earthquake has happened, the chance of another one (an aftershock) goes up, at least in the immediate future. That’s the self-exciting part.

There’s a mathematical technique for dealing with self-exciting sequences of events, called a Hawkes process, which you can also apply to the rivalry between two gangs. The idea is to treat acts of violence between the gangs (it doesn’t matter which way around) as a sequence of events in time. What you are after is a rate function r(t), which essentially measures the chance that a crime happens at time t, with that chance depending on what happened previously (because the process is self-exciting). We usually think of a rate as the number of events over a given time interval, for example the number of crimes we expect to happen per day. In this case, however, the time interval is made infinitesimally small. So you can think of the rate function r(t) as the instantaneous rate at which we expect crimes to happen at time t.

The idea now is to express the rate function as a sum: the first term of the sum is the background rate of crime: that’s the rate at which unprovoked attacks between the two gangs happen, ignoring any retaliations. The other terms in the sum correspond to the amount by which any previous violation between them raises that background rate. These reflect the self-exciting part (the retaliations). The longer ago a particular attack happened, the smaller its contribution to the rate function at time t….

Once you have the parameters, you can use the rate function to simulate crimes between two gangs as sequences of events in time. The crimes happen randomly by chance, but that chance isn’t the same for all times t, rather it’s given by the rate function. By seeing how the simulated patterns of crimes compare to real data you can assess how well your model does at describing reality (there are standard statistical methods for making that comparison). And once you’re happy the model does reasonably well, you can use it to predict what will happen in the real world and, hopefully, intervene.

The same math works for property crimes like burglary as well, where criminal offenders will often case a defined region before their crime sprees to identify their targets of opportunity before working through them in a relatively short period of time.

On a final cautionary note, we can’t help but think in reading through PredPol’s success stories of when the software is introduced in some areas of a city but not others, which see crime fall where PredPol is directing police activity but also see general crime rates rise elsewhere, that the use of the software might in fact be partially responsible for those increases. Much like squeezing a water balloon in one spot causes it to expand everywhere else, as those seeking to conduct criminal activities moved away from where the police presence has been concentrated. In that sense, what the results suggest is that simply providing an effective police presence provides a deterrent to crime, which is something that doesn’t require an investment in software.

But that’s not the whole story. It is also important to recognize the budgetary environment in the city of Los Angeles during the period when the software was being introduced, where the money to fund policing activities across the whole city was being reduced. That crime in the area where resources were concentrated was reduced while only expanding rather modestly everywhere else suggests that the predictive crime software does provide a real benefit in making more efficient use of the city police department’s limited resources.

That’s a winning story in anybody’s anti-crime playbook.

References

Freiberger, Marie. Crimes and earthquakes. +Plus Magazine. [Online Article]. 11 August 2015. Accessed 29 August 2015.

International Association of Crime Analysts. (2011). Crime pattern definitions for tactical analysis (White Paper 2011-01). Overland Park, KS. [PDF Document].

McNeely, Jim. Your Home Security – Never Before Revealed – How Burglars Case Homes. [Online Article]. 13 July 2011. Accessed 29 August 2015.

Image Credit: National Science Foundation: UC Mathematical and Simulation Modeling of Crime Project.