In due to the fact that current research methods

In this paper I will be delving into
the nature of crime within cities and how the occurrence and location of these
crimes may be mapped and predicted using various mathematical modelling tools.
I will be looking at how discrete agent based models can be used to shine light
on how crime hotspots emerge, diffuse and dissipate throughout time and space.

The nature of crime itself is ubiquitous
and for this reason there are necessary forces and laws put in place to help
counteract crime and aid in the prevention of such offences; every year
policing budgets are set with the aim of lowering the crime rate and attempting
to make a certain area safer for the public. However upon inspection of crime
location and the frequency of the crimes committed, we can see that the
distribution of said crimes is not uniform throughout time and space. I will be
analysing how different areas and neighbourhoods have contrasting crime levels
and what makes particular areas more desirable than others. Two main talking
points I will be covering are the phenomena known as ‘repeat victimization’ and
the ‘broken windows’ effect.

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Repeat
victimization encapsulates the theory that residential burglars prefer to
return to a house that has been burgled previously, or a house within the
vicinity of the aforementioned burgled house. This is due in part to the fact
that the criminals will have gained information regarding the different types
of property that may be stolen as well as the schedules of the targeted
inhabitants. CITE PAPER 1 In the paper cited, the author argues that
many support workers and individual police offices will have seen repeat
victimisations happen and how they have clustered in time. Unfortunately this
crime awareness has not effectively been incorporated into officer training and
even less into genuine crime prevention activity. Another problem with
identifying and preventing repeat victimisation is due to the fact that current
research methods which involve time based data, will lead to underestimates.
Furthermore the statistical systems which the Police use to track crime are not
particularly good at identifying repeat events. This makes it incredibly
difficult for law enforcement agencies to counteract this phenomenon. Repeat
victimisation remained near enough un-investigated until 1984 where Gottfredson
shone light on the high rate of repeat victimisation in the British crime
survey. The table below shows the highly skewed distribution of victimisation
that was found in the 1984 survey by Gottfredson.

From this table, we can see
clearly that 68% of the population were not affected during the yearlong survey
period. On the contrary the 14% of people who had reported being personally
victimised on at least 2 occasions in turn reported 71% of all the incidents.
Further analysis of the data allows us to see that the ~3% of the population
who were victimised 5 or more times accounted for 23.7% of all crime reported.
Further on in this paper I will be looking at agent based models are using
repeat victimisation as part of the attractivity factor which draws criminals
to a certain area and thus induces a hotspot to form.

The broken windows effect comes from the
academic theory proposed by James Q. Wilson and George Kelling in
1982 that used broken windows as
a metaphor for crime within neighbourhoods. They write “Social
psychologists and police officers tend to agree that if a window in a building
is broken and is left unrepaired, all the rest of the windows will soon be
broken. This is as true in nice neighbourhoods as in run-down ones.
Window-breaking does not necessarily occur on a large scale because some areas
are inhabited by determined window-breakers whereas others are populated by
window-lovers; rather, one unrepaired broken window is a signal that no one
cares, and so breaking more windows costs nothing.” CITE PAPER 2. There is
data to back up the broken windows theory, particularly in marginal
neighbourhoods where crime may be more prevalent than in a ‘safer’
neighbourhood. The broken windows theory was implemented in New York City, USA
a little over two decades ago. The main strategy being to flood the more
dangerous neighbourhoods with order maintenance enforcement, however the NYPD
decided it would be best to target the crime hotspots where violence was the
most popular crime. It was implemented in around 1990 and from the figure below
we can see a clear shift in the dynamics of crime in NYC. The respective
numbers of major crimes such as robbery, burglary and auto theft dropped 84%
from 1990 to 2009. CITE PAPER 3

 

  .

Discrete
Model

In this chapter I will be
reviewing the discrete agent based model developed in the paper ‘A statistical
Model of Criminal Behaviour ‘by Short Et al. The model itself and the goal of
said model is to analyse how crime hotspots form in certain areas and can move
and dissipate throughout neighbourhoods. The model incorporates two main
features; the houses at which burglaries occur and the criminal agents that
commit the burglaries. For ease of use the houses are imagined to be located on
a two dimensional lattice with equidistant spacing between houses. The goal of
modelling this way is to emulate a city as best as possible, we will further
denote the lattice spacing by the variable l, and the discrete time unit as
being DELTA T. The locations of the houses are given by their lattice site
coordinates s = (i,j), furthermore a measure of attractiveness for a given site
is included in the form As(t). This level of attractiveness is
modelled as being equal to the statistical rate of burglary at site s when
there is a burglar present.  In the model
we are not attempting to use underlying properties of the area as a measure for
the attractivity of a site; however it is based on the sociological phenomena
of the ‘broken windows effect’ as well as repeat and near repeat victimisation
which were touched upon briefly in my introduction.

We start by letting:

.

Where ­ represents the
static component of attractiveness of a site, this may vary through different
cities and neighbourhoods. We now let Bs(t) be the dynamic part of
the attractivity equation, this is the part associated with the sociological
phenomena highlighted in the introduction.

The criminal agents that are placed at the varying sites in
the lattice will be able to perform one of two actions at their respective
sites, they may burgle the house at which they are located or they may move to
another site which neighbours their current site. The act of burglary is said
to be a random event and is hence modelled by the following equation which represents
the probability the a burglar will indeed burgle the house at site s, between
the times t and,

.

To make the model as realistic as possible, the burglar is
removed from the site after they have burgled the house. The aim of this is to
emulate the action of genuine burglars who will flee the scene of the burglary.
To counteract this removal of criminal agents, they are also generated at a
rate  at each site on the lattice grid. If the
criminal agent decides not to burgle the site it is at then it will begin a
random walk to a neighbouring site that will depend on the level of attractiveness
of said neighbouring sites. This random walk process will become more evident
in later chapters where I will be implementing this process into a MATLAB
simulation.

The idea that criminals will decide to
burgle a neighbouring site stems from the ‘broken windows and repeat and near
repeat victimisation’ theories. Furthermore the notion that these criminal
agents will choose to move to an attractive neighbouring site is backed up by
the fact that journey to crime distributions generally show that the criminals
are less inclined the travel further away from their location to commit an
offence. This can be seen to be a trade-off where criminals face a choice; stay
local and commit a crime or travel further afield and potentially have a larger
gain.

The notion that the criminal agents will move from their
current site s, to a neighbouring site n is modelled as a probability and is
given by the following equation:

.In this equation here we are looking at
how the criminal agent will choose to move in the lattice grid, we evaluate
this by dividing the attractivity of a neighbouring site n, by the sum of the attractivity
of all the neighbouring sites. This can be seen later on in the MATLAB code
where we will denote the neighbouring sites by North, East, West and South for
better clarity.

To incorporate the theory regarding repeat
and near repeat victimisation, we will adjust the dynamic attractiveness Bs(t)
to depend on burglaries that have already occurred at a site s. To allow for
this, we introduce a new parameter which is added to the Bs(t) value
for the burgled site s. This means we are introducing a variable that will
allow us to effectively model how certain sites become more attractive to
criminals for a brief period of time.  We
now suppose that this heightened attractivity of the site will eventually tend
back to its original value via some decaying function. The new equation for
dynamic attractiveness now reads:

Where w sets the new timescale in which
repeat burglaries are most likely to occur and Es(t) is the number
of burglaries that took place at the site s during the time period starting at
t.

Lastly we can adjust this equation to
introduce new variables that will allow us to efficiently model the ‘broken
windows’ theory proposed by James Q. Wilson and George Kelling in 1982. The goal
of this modification is to show how criminal hotspots may spatially diffuse and
dissipate throughout the neighbouring sites.