Predictive policing

mathowl (64)in #mathematics • 6 years ago (edited)

Hi Steemicians. If you are a movie fan you must have seen Minority Report (and if you haven't you should definitely watch it). In the movie crime is predicted using three mutated humans called Pre-cogs. Would-be murders are hunted down and locked up before they can commit a crime. In our reality we also have a predictive policing tool called Predpol. It is not run by mutated humans but by amazing mathematics. Predpol is able to predict crime hubs. So if you send extra police to that area the criminals are deterred to commit a crime. It is proven to reduce crimes in certain areas by 15-30%[1]! Today I will give a short introduction to the math behind it.

Crime dynamics

Before we get to the math we first need to make some assumptions on crime dynamics. We will restrict to house burglary. In certain cities there are neighourhoods where there is a lot of crime. In these neighbourhoods people who get burgled have been burgled before. Furthermore, their neighbours also have a high likelyhood to get burgled[2]. So why does happen? A burgler who robs the same house or same neighbourhood can get around easier since he knows the area. Also, in certain neighbourhoods the inside layout of houses is very similar which is beneficial if you want to get in and out quickly. So it makes sense to hit the same location twice. Hence, we will make the following assumption on the dynamics of crime: crime in the past effects crimes in the future.

Source

Hawkes process

The Hawkes process, which is named after Alan G. Hawkes, is a mathematical model for so-called self-exciting process. This means that if an event occurs the process get 'excited' in the sense that the occurence of a subsequent event is increased. If you substitute process by crime in the last sentence then you can see why it might be applicable in crime predicting. Hawkes processes also have applications in the field of seismology and financial trading[3]. If you want to check out the formal definition of a Hawkes process you can check these sources [3,4].

Implementing the Hawkes process for crime

The Hawkes process comes together with certain parameters that you need to determine using the crime data. The nice part is that the mathematics of Hawkes processes is not too difficult so you can run real-time updates on a small laptop. This is exactly what PredPol did. The software tells police officers in which areas there is a high-probability of crime so that the police can check it out.

Does this sofware give any new information. By formulating everything in a 'Hawkes process setting' you can analyse subtle behaviour which gives you predictive power. Since PredPol is not a free-to-use software they of course don't tell what these subtle things are. However, in [5] these type of predictive subtleties are found for terror attacks by the IRA in Northern Ireland. So I am sure that PredPol also found something similar.

The perfect crime

Suppose that the criminals have a bit of understanding of how PredPol works but they cannot actually get the sofware and the data they need. What can they do to stop the police from showing up when they want are about to burgler a house? Easiest is to hit neighourhood with low crime. But for neighourhoods with low crime you cannot ask your criminal buddies if the houses are easy targets or if there is alot to get.

From a mathematical perspective the best you can do it so model your criminal behaviour on a process where there is no correlation between the past and future. More specifically, you can use the Poisson distribution to do this since it gives the probability of a given number of events when these these events occur independently of the time since the last event [6]. This means that the police has no extra benefit from using PredPol since your behaviour does not follow a Hawkes process. Hope I won't help anybody by telling this D:

Source

Sources

[1] http://www.predpol.com/
[2] http://www.sciencemag.org/news/2010/02/mathematics-clumpy-crime
[3] Patrick J. Laub, Thomas Taimre, Philip K. Pollett, Hawkes Processes, arXiv:1507.02822
[4] http://mathworld.wolfram.com/HawkesProcess.html
[5] S. Tench, H. Fry, P. Gill, Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army, Volume 27, Issue 3 (Mathematical Modelling of Crime and Security), June 2016 , pp. 377-40
[6] Frank A. Haight, Handbook of the Poisson Distribution. New York 1967, John Wiley & Sons.

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Thank you!

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