Can Math and Science Help Solve Crimes?

Researchers try to predict and prevent crime using sophisticated mathematical models.

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UCLA's Jeffrey Brantingham works with the Los Angeles Police Department to analyze crime patterns. He also studies hunter-gatherers in Northern Tibet. If you tell him his research interests sound completely unrelated, he will quickly correct you.

"Criminal offenders are essentially hunter-gatherers; they forage for opportunities to commit crimes," said Brantingham, UCLA associate professor of anthropology. "The behaviors that a hunter-gatherer uses to choose a wildebeest versus a gazelle are the same calculations a criminal uses to choose a Honda versus a Lexus."

Brantingham has been working for years with Andrea Bertozzi, UCLA professor of mathematics and director of applied mathematics, to apply sophisticated mathematics to urban crime patterns. They and colleagues have built a mathematical model that allows them to analyze different types of criminal "hotspots," where many crimes occur, at least for a time.

They believe their findings apply not only to Los Angeles, but to cities worldwide. Crime hotspots come in at least two different types. There are hotspots generated by small spikes in crime that grow (which the researchers call a "super-critical hotspot"), and a second type of hotspot that looks the same from the surface, but is not, in which a large spike in crime pulls offenders into a central location (which they call a "subcritical hotspot"). Policing actions directed at one type will have a very different effect from policing actions directed at the second type.

"This finding is important because if you want the police to suppress the hotspot, you want to be able to later take them out and have the suppression remain — and you can do that with only one of the two, in the subcritical case," Bertozzi said.

"Unless you are really looking for them, and our model says you should, you would not suspect these two types of hotspots," Brantingham said. "Policing actions directed at one type will have a very different effect from policing actions directed at the second type. Just by mapping crime and looking at hotspots, you will not be able to know whether that is generated by a small variation in crime or by a big spike in crime.

"If you were to send police into a hotspot without knowing which kind it is, you would not be able to predict whether you will just cause displacement of crime — moving it somewhere else, which is what our model predicts if it's a hotspot generated by small fluctuations in crime — or whether you will actually reduce crime," he said. "Many people have argued that adding police to hotspots will just push crime somewhere else, but that seems not to be true, at least in certain cases. You get displacement in some cases, but not nearly as much as many people thought."

Drug hotspots and violent crime hotspots have been suppressed, and analysts up until now have not been able to explain why.

In their mathematical model, the scientists are able to predict how each type of hotspot will respond to increased policing, and when each type might occur by a careful mathematical analysis involving what is known as "bifurcation theory."

"Although this is an idealized model for which all parameters must be known precisely in advance in order to make predictions, we believe this is an important step in understanding why some crime hotspots are merely displaced while others are actually removed by hotspot policing," Bertozzi said.

Predicting crime and devising better crime prevention strategies requires "a mechanistic explanation for how and why crime occurs where it does and when it does," Brantingham said. "We think we have made a big step in the direction of providing at least one core aspect of that explanation. We will refine it over time. You need to take these initial steps before you can develop new crime fighting strategies."

Their model, Bertozzi said, "is nonlinear and develops complex patterns in space and time." These features, she noted, are well known for related models in other areas of science.

They have been studying crime patterns in Los Angeles, using the last 10 years of data from the Los Angeles Police Department. They have been able to identify violent crime hotspots, burglary hotspots and auto theft hotspots, among other findings. They believe their analysis likely applies to a wide variety of crimes.