Fighting crime with Maths

I was inspired to write this post by a video I watched on Numberphile’s YouTube channel, which I would highly recommend watching as I found it extremely interesting.

To summarise, in the video Hannah Fry talks about how the frequency of crime events can be explained mathematically using the Poisson Distribution and Hawke’s process (which takes into account the fact that events aren’t completely independent from one another). As soon as we establish how frequent crime happens in certain places, this information can then be used to predict in what areas a robbery, for example, will most likely happen on a certain night, using formulas derived using these probability distributions.

This led me to research in what ways mathematics is helping to fight crime.

Mathematical Modelling

In UCLA, researchers were able to build a mathematics model that allows then to analyse different types of criminal ‘hotspots’ – places where many crimes occur. They found that there are two types of hotspots: Super-Critical Hotspots where there are small spikes in crime that grow and ‘Subcritical Hotspots’ generated by a large spike in crime that pulls offenders into a central location. This finding has allowed policing actions to be customised to the type of hotspot as their actions will be very different for each. Additionally, using their mathematical model, the scientists are able to predict how each type of hotspot will respond to increased policing, as well as when each type might occur using bifurcation theory.

Inverse Problems

Inverse problems are mathematical detective problems. To solve an inverse problem, mathematicians need a physical model of the event to understand what causes lead to what effects. Then, given the known effects, mathematics can be used to give the possible causes. Furthermore, maths is also used to establish the limitations of the model and therefore the accuracy of the answer. Examples of inverse problems include remote sensing of the lad or sea from satellite images, using medical images for diagnosing tumours and interpreting seismographs to prospect for oil.

Inverse problems are extremely relevant to solving crimes; evidence which is left at the a crime scene must be analysed and then scientists must work backwards to deduce what happened and who did it.

Speeding

Mechanics can be extremely useful when analysing evidence when vehicles are involved, for example when analysing skid marks. Skid marks are caused by the speed of the car as well as other factors such as braking force, friction with the road and impacts with other vehicles.  Mathematically, mechanics can be used to model this event:

\[  s=\frac{u^2}{2\mu g}  \]

$s$: length of skid, $u$: speed of the vehicle, $g$: acceleration due to gravity and $\mu $: coefficient of friction times braking efficiency.

Say we wanted to know if the car was speeding, the equation can be rearranged to:

\[  u=\sqrt {2\mu gs}  \]

allowing the speed of the car to be determined.

Contaminants

Mathematics can also be used to find where a contaminant was released by modelling how the water and contaminant flow through the water network. As the contaminant flows, its its concentration decreases. This decrease can be described by $-kC$.

Firstly, $-kC$ depends on time and this change in concentration as time passes is shown by $C_ t$. Secondly, it is also dependant on the contaminants reaction with the space it is in (in this case the pipe) which is influenced by the volume and flow of the water. Let’s writes  $Q$ for water flow and $C_ x$ for the changes in concentration due to the surrounding space, and from this we can create a mathematical model for the overall loss in concentration of the contaminant:

\[  C_ t + QC_ x = -kC  \]

Furthermore, we must take into account that at each junction different solutions mix together:

\[  C^1_{in}Q^1_{in}+C^2_{in}Q^2_{in}=C_{out}Q_{out} . \]

The solution from two pipes is mixed at the junction to give a new concentration of the contaminant.

Source: plus.maths.org

To reconstruct what is occurring in the water network, mathematicians need to find the flow rates in the pipes and measure the contaminant concentrations at the junctions. Then, they can make a guess at the initial concentration, flow the model forward in time, and compare it with what was measured in the real network. The initial concentration is then adjusted until there is a match with the real network. This process is called nonlinear optimisation.

No more blurry images

By mathematically modelling the blurring process, we can remove some of the blur to get a clearer picture of the plate. The model involves a blurring function, g, that is applied to the original image to give the blurred image. The formula that describes this process is:

\[  h(x) = \int f(x-y)g(y)d^2y . \]

The variable x describes the various pixels in the image. Each pixel has its own value which gives information on its colour and brightness. The function f(x) gives gives the pixel value before blurring and h(x) is the pixel value after blurring. By running the formula backwards we can deblur the image, hence obtain f(x) from h(x). However, this can of course only be done if we know the blurring function, g.

This is only a short summary of some of the ways that maths can help solve crimes. There are many others that I haven’t even touched on, such as cryptography, however I felt this was a good selection of quite unknown applications of mathematics to crime. It’s incredible to see how mathematical modelling can have such a beneficial impact on society. M x

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