To account for all factors affecting specific killing sites and travel distances would be impossible, as the human decision process is dynamic and virtually boundless.
As reported by Laukkanen and Santtila, however, “87% of serial rapists in the UK committed their crimes confirming the so-called circle hypothesis.” This technique predicts that the criminal will live within a circle drawn around the two crime loca-tions furthest from each other. Because difficult-to-solve rapes and homicides tend to occur within similar distances from the perpetrator’s residence (Santtila et al., 2007), we assume that similarly most serial murders occur within the circle.
Most serial killers and other types of criminals are also shown to not travel far from their base location, with crimes committed dropping as a function of distance (Santtila et al., 2007) - a theory we will refer to by its common name as distance decay. Surround the crime scenes, there also usually exists a buffer zone in which the criminal is not likely to live when avoiding detection.
2 The Problem
As contracted by the police, we are given crime scene locations and times of a specific criminal. Based on this data, we must predict the criminal’s base location and their next crime’s location and time.
Although humans may appear random, we must fit a geographical profile to the crime scene data for prediction purposes in a way such that our model can be used for any type of serial criminal with any personality type. This geographical profile generates a prioritized search for the police. In our plots, note that red specifies high search priority and blue low. We use bilinear interpolation to smooth the grid for ease of interpretation for the police.
3 Assumptions
As given such location and time data from the police, we assume that the police suspect all points used in a single run of our model are crime sites from a single criminal. We also assume, to simplify locating the criminal, that the specified killer has precisely one home base which we are trying to find.
To simplify the problem, we assume we are dealing with serial killers. Although this does not seriously affect our results in any way (Laukkanen & Santtila, 2006;
Santtila et al., 2007), we use case studies and research to support location statistics for serial killers and later extend our model results to other types of serial criminals.
Also, despite the fact that different criminal personalities will commit crimes at dif-ferent distances, we assume a form of the circle theory. Research states that 87% of serial rapists, which can be extended to other types of serial criminals, reside within a circle proposed by the circle hypothesis (see Background - Locating a Criminal) (Santtila et al., 2007); we assume our grid, although not necessarily encompassing the full circle around the furthest two points, will act similarly in encompassing our killer’s location for ease of modeling purposes. We also assume a distance decay, as
supported by research for typical stable criminals. Thus, we currently do not con-sider commuters who may travel far distances to commit crimes, though in the future we would like to include an option for different personality traits to compensate for distance preferences and account for such commuter criminals. We do not inherently assume a buffer zone, as will be later discussed, but instead test different models and see the accuracy of such an assumption.
Some factors which may affect a criminal’s choice of crime site include population density, opportunity, and landscape. In our model, we neglect these differences and assume a homogeneous geographical area for simplicity. If actually implemented, we would like to include these descriptors in our model, similar to CrimeStat (Levine, 2006). Thus, unfortunately our model may predict the criminal’s base location to be in an inhabitable area and may predict future crimes to be committed where there is, in actuality, little opportunity for such crimes.
For more discussion on our future extensions to dispose of such simplifying as-sumptions, see Future Research.
4 Metrics and Functions
4.1 Simple Spatial Metrics
When assuming we are given information including locations of N crime scenes, it is useful to look at spatial metrics to help define global traits of the locations. We define the following metrics:
Mean Center (“Center of Mass”)
This is the point output when taking the average values of latitudes (y) and longitudes (x):
It is possible to include a weight, wi, for each point to skew the center more toward it, which may be useful when specific points are more important than others.
Center of Minimum Distance
This is the point where the summed distances to all of the crime scenes is minimal, or the minimum considering every (x, y) point of
N
Standard Deviational Circle (SDC)
Here, a circle is drawn at the mean center, assuming wi = 1, such that the radius is one “standard distance” (related to standard deviation) to cover 68% of the data points (assuming randomness) and 95% for a radius of two standard distances. The larger the radius, the more spread out the data is considered to be.
Standard Deviational Ellipse (SDE)
While the standard deviational circle ignores skew of the data, the ellipse allows stretching in two dimensions along any two perpendicular lines on our plane.