Subtitles section Play video Print subtitles We're talking about the maths of crime Crime? Yeah! For people who don't know, you're a proper mathematician. And we're really getting into your research. Yup. We're gonna... In fact, even look at a paper that I've even done with one of my PhD students. So yes, I am a real person. This is your area of expertise? Ahh... yeah. Ahh... yeah. One of the things.... That's important to know about crime... or terrorism... things like that... is when it's going to happen. There's a bit of old maths that kind of helps us start off understanding that. and that's something call the Poisson Distribution, named after a guy called Poisson. Nothing to do with fish? :-) I don't think so, although some of my students call it the Fish Distribution, which is... *giggles* But the main point about the Poisson Distribution ...umm...its first practical application was looking in the Prussian army There were lots of soldiers who were dying from being kicked by horses over a number of years. By their own horses? By their own horses, yeah. Horses objecting to being used as an army horse, perhaps. And so... there was one guy called Bortkewitsch in 1898 who was tasked with looking into how frequently these horse attacks were happening... these horse kick attacks were happening Horse attacks? They sound more dramatic all the time! Ok... I know it does, and I'm sorry, and it is actually quite a serious thing. The point is... is that I like to think anyway... horse kicks are... generally independent, right, horses don't sort of... "collude" with each other and decide that they're going to kick up a ruckus on a particular day. So if you look at a timeline of incidents then you would sort of expect your incidents, your horse kick incidents to be kind of randomly distributed across this thing So maybe you'd have a couple very quickly after each other But what that means you can do is if you take a time interval, so a set number of years perhaps and you look at the chances of a particular number of incidents in that interval then it follows this really nice neat distribution which looks like this and this is called... this is your probability and this is your number of incidents and this is your Poisson Distribution so that means that there's an average number of incidents that you expect in a year, say and that average number of incidents is the most likely thing to occur and has the highest probability of all so it might mean that, you know, in 1890 you only have, you know, one incident perhaps and then in 1891 you have a huge number of incidents but also very low probability But... that most years you're going to expect to have something around the average rate of incidents It means that you can start looking at the time between different events and you can start coming up with sort of a susceptibility for events But there's one really crucial thing that this stuff is missing that the Poisson Distribution is missing which is that events, and crime, and terror attacks and things like that they're not completely independent, so... if one happens, the chances of another one happening very soon after really increase, and the Poisson Distribution can't take that into account. So the first people to look at events that weren't completely independent were scientists who were studying earthquakes Now you could say that perhaps earthquakes were random were completely random and Poisson distributed so each earthquake was independent of every other But the thing is, is that if you have one earthquake you're going to be really likely to have aftershocks Right, so a series of earthquakes in the same place, in quick succession of one another [Announcer] continual aftershock are keeping everyone nervous Scientists, and mathematicians developed something called "Hawkes Process" which I think might be named after Hawkes actually So they came up with something called the Hawkes Process which takes into account the fact that events aren't completely independent of one another So instead if you were looking at an earthquake you'd expect to have something much more like this One earthquake happened and then you'd expect a few more smaller earthquakes to happen within a really short space of time and then perhaps you'd go a little while you'd have one with no aftershocks and then another, but with another few, uhh, sort of, aftershocks tagged on quite quickly afterwards I mean things kind of take a bit more of this pattern But the thing that is nice is that uh... well, "nice" probably isn't the right word, uh... But... is that crime follows this same pattern. So if you take burglaries for example, anybody who's been burgled will know that your chances of being burgled again within a really short space of time hugely increases. This is something called "repeat victimization". And the reason is, is that burglars get to know the layout of your house they get to know, um, where you keep your valuables. They get to know all sorts of things about your local area. So your chance of being burgled again increases. But so does your neighbours', and your neighbours' neighbours', and neighbours' neighbours' neighbours' neighbours' and so on as you go along down the street. This Hawkes Process then, of seeing events as connected in time, means that you can then model what happens with burglary statistically. It goes beyond just sort of saying "Oh well, you know, obviously that happens" because you're actually able to describe it and capture it using numbers and using equations And as soon as you can do that, then you can start actually implementing genuine strategies back into the real world. So, for example, this is a paper that I wrote with one of my PhD students and this looks at, um, a very similar story about attacks from the IRA in Northern Ireland and you can see here, this is... the events as they go along This is really similar to this graph here. So you'll have one big event and then you'll have sort of a cluster of events afterwards. And then a gap for a little while and then another cluster of events going through. But what this means knowing that there's this model that sits behind the scenes is that you can actually assign numbers. There's a proper equation for this. So you have your kind of background rate, so this is... I don't know what that first symbol is! Oh, it's lambda, Greek lambda. Umm.. and that's a "mu"... another Greek letter So this one here... this... you're going to be talking about your "intensity" of attacks. How likely it is for an event to occur within a short space of time. So you have some sort of a background rate, so this is like your randomness, cuz there is still some element of complete randomness in this... But then, every time an event happens, you have a little "kick". So your chances of another event get a little "boost". And that's what this thing here does. But then finally, this "boost" it doesn't last for very long, so it looks like this... So your little "kick", your chance of another event happening boosts up and then dies away quite quickly in time. You're effectively... you're summing over all of the incidents that have happened in the past, and you're working out your "kick" from every possible incident . When a house gets burgled, or a bombing happens, or anything like that... numbers are being fed into equations that tell us what? Yeah, well, so they tell us, they tell us... they capture... sort of the process that's going on behind the scenes. But they do it in a way that's sort of free from emotion, and free from "hand-wavy-ness". So if you apply this to something like the Troubles in Northern Ireland and the frequency of IRA incidents there were 5 actual different phases of attacks and you can see here with this equation the different values of these different parameters at different points throughout the process. So you've got mu there, k-nought (the "boost") there, and omega, which is how quickly things died away back down to normal. And what's really interesting about this, is that this allows you to come up with a comparison between different processes, or different stages in a conflict and actually to quantify it. Hannah, is this all hindsight, or does this give, like, predictive powers? Or is this just something you apply afterwards, like "oh, yeah, I can see..." Well, so this example is all retrospective, but what I think is really exciting about these ideas is that you can also apply them in real time. So with burglary in particular, umm... if you're just looking at how the past influences the present and will influence the future which this allows you to do, by talking about intensity and susceptibility of burglaries what that means is that in real time you can pick up on a particular area, or even a particular street that is more likely to be the centre of our burglary hotspot going forward in time by using these methods. So there's a company in America called PredPol who were the first to take these equations and wrap it up neatly into sort of an iPad app, effectively. So that they can give it to different police forces across the U.S. and the police forces will then get a printout on basically a map with like a red square, saying