# Catching criminals with maths

How crime science, and the maths it uses, is helping the police fight crime

As a result of decades of empirical research, crime science has emerged as the leading multidisciplinary approach to develop new ways to tackle crime and terrorism. As opposed to traditional criminologists, crime scientists commonly use a broad spectrum of different disciplines and sciences to achieve their aim of cutting crime. Using knowledge from chemistry, geography and physics, to architecture, public health, psychology and information technology, crime science has been able to offer new solutions to the most pressing issues that impact on the health and security of millions of people. Among all the fields and disciplines used, applied mathematics, statistics and econometrics are perhaps the most common tools used by crime scientists.

We decided to share with you some of the most interesting applications we have found of mathematics in crime science. This is obviously by no means an exhaustive list of applications, nor is it sorted in order of relevance (or any other order for that matter!).

A very good example that shows how relevant mathematics can be in cutting crime is the use of highly advanced quantitative applications to improve the services provided by police around the world. Research developed during the last few decades in the field of crime analysis has reached a very important conclusion: some specific types of crime always happen in the same places, giving rise to the formation of crime hotspots. When more detail is incorporated into the analysis, it seems that crime also occurs more often in some places and times of the day than in others. This is not a coincidence. Crime scientists have shown that delinquency is not really a random phenomenon: it actually clusters in time and space.

Based in University College London’s Department of Security and Crime Science, three investigators have developed an incredible amount of cutting-edge work in this field: Shane Johnson, one of the leading researchers in predictive policing in the world; Toby Davies, who focuses on predicting crime networks; and Kate Bowers, who is using big data to understand crime.

Does this surprise you? Well, consider to what extent you and the people around you have set routines on a daily basis. Most people behave repeatedly in the same way according to the precise setting or location they find themselves in. Needless to say, this makes all of us really predictable and as a result of this, the likelihood of a crime happening is the intersection of probabilities between victims and motivated offenders’ routines.

Crime scientists devote a substantial amount of time and resources developing mathematical models and algorithms, supported by existing criminal and environmental data, that will result in different types of interventions to help police officials reduce the number of crime incidents. This has given rise to new software, such as PredPol, that helps to allocate resources—such as policemen and patrols—in the best possible way.

The results of applying maths into crime are astonishing: using PredPol, the Santa Cruz Police Department reported a 27% decrease in the crime rate in the first month!

Although the film Minority Report may be a bit exaggerated, the rationale portrayed there is the same: if properly understood in its causes and consequences, crime can be predicted—and so it can be prevented.

Find out more about maths and policing here.

Often when we speak of organised crime, we tend to focus our attention on the traditional Mafias of Italy or the brutal drug cartels of Colombia and Mexico. With these stereotypical images in mind, it is no wonder that we perceive organised crime groups as close-knit families that only “do business” with men similar to themselves. However, thanks to the mashup of mathematics and social science, the past couple of decades have given rise to a new type of crime research, opening our eyes to the relationships that form organised crime networks.

Example of a sociogram in which the individuals (either people or groups of people) are represented by the nodes, and their connections (such as a friendship) is represented by the edges.

Initially introduced in the 1930s by JL Moreno, the notion of using a sociogram to illustrate relationships is one of the underlying principles of social network analysis (SNA). The idea of SNA is to study society by looking at the relationships that occur, instead of only looking at individual demographics. These relationships are then illustrated using a sociogram, where the dots (nodes) denote actors in the network, and the lines that go in-between the nodes denote the relationship (see image). Social network analysis is an old method of studying human behaviour, used for decades in anthropology, psychology, sociology, biology, and more recently crime science.

In the studies of crime and terrorism, the method of social network analysis has allowed us to understand who connects with who, why they connect, and what benefit they get from such ties. Using this method, we have learnt that organised crime is effectively changing with the rate of globalisation, going from the traditional hierarchical structures of our stereotypes (the Mafia and drug cartels) to more complex cell structures.

How does mathematics come into this? Well, the underlying principles of SNA are supported by mathematical equations. With these equations, we can take a closer look and see who is the most influential player in a network, why they are so influential, and who they are connected to. Thanks to the statistics behind SNA, we have discovered that the most important player in a criminal network is not necessarily the person with the greatest number of connections. Instead, crime scientists and law enforcement practitioners are increasingly becoming more aware of the importance of network centrality, and the value it brings to understanding network connectivity. Indeed, the eigenvalues and geodesic distances of actors in a network have taught us that it is not how many people you know that matter, but who you know, and indeed who knows you. Traditionally, criminologists would point to the person with the most connections and suggest that intercepting and arresting that actor would have a significant negative impact on a criminal network. However, as years have gone by and many kingpins of the organised crime underworld have been arrested, it has become evident that this may not be the case. This has led to a focus on centrality values instead.

Therefore, as social scientists, we are relying on the numbers and theories brought to us by mathematics. Without these, we would still be chasing the godfather, instead of his brokers.

Find out more about organised crime networks here.

Gangs are a big problem in some cities, such as Los Angeles, with groups fighting constantly over the control of what they consider to be their territory. Through statistical analysis of the fights between gangs, mathematicians discovered two behaviours. The first is that certain gangs tend to have constant fights with only a subset of the other gangs while ignoring the rest; and the second is that gangs tend to be a highly retaliatory group, with the first attack between two gangs unchaining a series of fights between them. Much like earthquakes, a gang fight increases the chances of any subsequent attack and, therefore, they can be modelled as a self-exciting process.

The Hawkes model takes into account past events to update the probability of attack. Much as an earthquake, the next attack is more likely to occur when there has been a recent attack.

The selective fighting between gangs makes the analysis of past events much more accurate and the self-exciting part of the model improves the forecast of subsequent attacks. Thus, modelling the probability of a gang fight based on previous events helps the police avoid violent confrontations and public displays of violence by allocating visible resources in areas where a fight is more likely to happen.

A similar technique has also been applied to certain types of terrorist attacks, where the probability that a new attack occurs is best modelled as a function of past events.

Find out more about gang rivalry here.

Certain attributes of a crime scene reveal information about the event. Thanks to mathematical models, we are able to estimate the time at which a crime was committed, identify the suspect’s fingerprints and analyse the pattern of bloodstains and bone fractures to reconstruct the crime scene. We will show some of the most common and simplest examples of the applications of mathematics in forensic science:

#### Estimation of time since death: Newton’s law of cooling

Knowing the time of death is a highly important factor in forensic science: it can exonerate a suspect or focus suspicion on the accused, it can refute or support witness and suspect statements, and it can be a key factor in either making or breaking a case.

To solve this problem, forensic scientists use temperature data to determine and establish an approximate time of death. These temperature estimates are based on Newton’s law of cooling, which states that the rate of loss of heat by a body is directly proportional to the temperature difference between the body and its surroundings: $T(t) = T_{e} + (T_{0}-T_{e})\exp{(-kt)}$, where $T_{0}$ is the victim’s body temperature prior to death (normal human body temperature is said to be constant at $T_0 \approx 37\mathrm{C}$) , $T_{e}$ is the room temperature and $k$ is a constant of proportionality that depends on the body’s characteristics.

For further information about the time of death read Time of Death: A critical part of the timeline. For more applications of Newton’s law of cooling visit one of our previous articles: The mathematics of tea-making.

#### Bloodstain pattern analysis

Bloodstain pattern analysis (BPA) is another tool used in forensic science that involves the study and analysis of bloodstains at a crime scene in order to recreate the actions that caused the bloodshed. The science of BPA applies scientific knowledge from other fields such as biology, physics and mathematics.

Many criminal investigations involve a death resulting from a gunshot, knife wound, etc, and it is crucial to the investigation to know the exact position of the victim or the murderer. The main objective of a BPA is to give answers to questions such as where the blood came from, how the victim and the perpetrator were positioned, how many potential perpetrators were present and whether the victim was moved or the scene changed in any way.

Blood pattern analysis

In the figure above, we show the general procedure used by forensic scientists to make a blood pattern analysis: in case A, the drop of blood hits the surface at an angle of $\alpha=90^{o}$, giving a shape that is a circle of diameter $W$; and in case B, the blood drop strikes the floor with $\alpha \ne 90^{o}$, giving a shape that is a teardrop made up of an ellipse (diameter $W$ and length of splatter $L$) and an extension resulting from the droplets’ momentum. For each droplet, the forensic scientist measures $W$ and $L$ to determine the angle of impact $\alpha$ ($\sin \alpha = W/L$), then extends the long axis of the ellipses (the dashed lines in figure C), to find the area of convergence: the zone where all the lines intersect. Once they locate this area, they measure the distance $D$ between an individual bloodstain and the area of convergence. Finally, using simple trigonometry (see figure D) they can determine the point of origin $H$ of the blood spatter ($H=D\tan\alpha$). The value of $H$ must then coincide for all the drops of blood.

For further information about BPA read here.

When a series of crimes have been committed, they might be linked to the same criminal when they have the same style or modus operandi: they then become known as serial criminals. Usually, a list of suspects can be constructed using information from each one of the linked crimes: this information might include the height of the criminal, if he or she is left-handed, the age, how strong or weak the criminal is, etc. The list of suspects might be quite extensive and is usually constructed from previous criminal records.

The location of the first seven Whitechapel murders, most of them ascribed to the serial killer known as Jack the Ripper

Geographic profiling involves analysing the spatial patterns of the linked crimes to assign a probability for each suspect based on the fact that criminals tend to act in areas with which are familiar (they are more likely to know how to get there, how to avoid being observed, how to hide and how to escape). Therefore, suspects who live or work closer to the regions in which the serial offences were committed are assigned a higher probability of being the criminal.

The pioneer in geographic profiling is Kim Rossmo, a Canadian criminologist who first used the power of a computer and the location of several crimes to construct a probability for different criminals applying what we now know as Rossmo’s formula.

Instead of spending considerable police resources and time investigating every single suspect, ranking them by the probability obtained by geographic profiling means that the police can start by looking first at the prime suspects.

Find out more about Geographic Profiling here.

So, as you can see, the uses of maths are everywhere! This summer we had the opportunity to interview Andrea Bertozzi, an expert in fluid dynamics, robotics, image processing and crime. Yes, crime.

[Pictures: 1 – adapted from Flickr.com –  Crime Scene by Catalina OlavarriaCC-BY 2.0; 2 – adapted from Flickr.com – Lunar Tunes by Marian Gonzalez, CC-BY 2.0; 3 – adapted from Flickr.com – Police by triplefivedrewCC-BY 2.0 ; 4 – adapted from Flickr.com – Dexter by mareina,CC-BY 2.0 ;  5 – adapted from Flickr.com – Street Gang by FouquierCC-BY 2.0; 6 – adapted from Flickr.com – Murder by Silla Rizzoli CC-BY 2.0; 7 – adapted from Flickr.com – LAPD North Hills Burglary Investigation 6/13/11 01 by Chris Yarzab,CC-BY 2.0; 8 – Jack the Ripper map belongs to the Public Domainother pictures by Chalkdust]

Hugo is a chemical engineer doing a PhD in Mathematics at University College London. He is currently working on non-Newtonian fluid dynamics. He is also interested in transport phenomena and rheology (the science of deformation).

David is doing a PhD in Security Sciences at UCL. He has a masters degree in Politics and Economy from the University of Essex, a masters degree in Security from UCL and a masters degree in Public Policy from Escuela de Gobierno y Transformación Pública (EGAP).

Rafael Prieto Curiel is doing a PhD in mathematics and crime. He is interested in mathematical modelling of any social issues, such as road accidents, migration, crime, fear and gossip.

Sanaz is doing a PhD in Security Sciences at University College London, where she is studying transnational organised crime networks. She has a background in clinical and forensic psychology, and has worked in offender mental health research.

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