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
here is where is where is most likely to be victims of burglarly or car theft tonight
So just by looking at these, just these really simple equations
putting in the numbers of the system
and reacting to what the maths tells you
they've reduced burglarly by up to 32% in certain areas of the States.
It's like a pre-crime, this is like "Minority Report".
Yeah, yeah, "predictive policing" that's what they call it. Yeah.
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So, as the authority's get smarter,
and the police get smarter and start using mathematics
so, you know.... fight crime,
could criminals start using mathematics to plan crime?
...chuckles...
Well...
I hope not.
...ummm....
I hope not.