Activity Space Estimation

The propositional definition of an activity space has changed little since the early presented

works of Horton and Reynolds (1971), and Johnston (1972). The methodologies employed to

study activity spaces, however, have been wide and varied (Chaix et al., 2012; Dijst, 1999;

Sherman et al., 2005). These range from the conceptual formulations that relate places to each

other as presented by Perchoux (2013), to the more rigorous mathematical formulations of

Sherman (2005) and Chaix (2012). In both cases, it is largely accepted that there exists an

important relationship between conceptual “anchor points” – typically the home or workplace – and other areas of daily activity. Furthermore, it is largely accepted that destinations individuals

indicate they prefer tend to follow a rule of distance decay: namely as distance increases, the

likelihood of engaging with a particular location diminishes (Golledge & Stimson, 1997).

Criminal events tend to follow similar patterns of distance decay (D. Canter et al., 2013; Emeno

& Bennell, 2013; O’Leary, 2011; Rossmo, 2000), and the assumed relationship between criminal events and day-to-day activities serves as the foundation upon which both routine activity and

crime pattern theories reside (Brantingham & Brantingham, 1995; Cohen & Felson, 1979).

These observations – distance decay and routine activities – underpin the various

operationalized formulations of activity space. Broadly speaking, these operational approaches

fall into one of several categories: (1) elliptical methods with the standard deviational ellipse

(SDE) being the standard, (2) network based measures such as the road network buffer (RNB)

from Sherman (2005), and finally (3) other polygonal definitions such as the minimum convex

hull. These categories can be thought to vary as a function of their complexity and the number of

assumptions made in their formulation. These three approaches relate to the foundational

generalized area which relates directly to the awareness space construct; the network based

measures have the most in common with crime pattern theory which emphasises crime in

relation to the common routes travelled by offenders. It is useful to inspect visual examples of

activity space surfaces, and Figure 16 provides location data for an example crime series to serve

as a reference; this crime series will be used in a number of examples throughout this section.

3.2.1 Convex Hull

The least complex of the three is the convex hull which is simply the smallest

encompassing polygon for a set of locations. This measure is described as “simple” because it makes no implicit assumptions about the relationship between the home and activity locations,

nor does it take into account any travel networks. Furthermore, by definition when ignoring the

possibility of spatial outliers, the convex hull will always include all observed activity locations.

The disadvantage of such an approach, however, is that it can imply regions of familiarity for

individuals where no such familiarity exists (Chaix et al., 2012, p. 448). The convex hull for the

reference data are provided in Figure 17.

3.2.2 Elliptical Methods

Elliptical methods are another approach to generating a simple geometric shape to describe

an activity space. There have been a number of methods explored within the literature, including

circular methods (D. Canter & Larkin, 1993), and standard deviational ellipses (Buliung &

Kanaroglou, 2006; Lee et al., 2016). Due to the mathematical requirements of defining an ellipse

or circle, elliptical methods make a number of implicit assumptions that convex hulls do not.

Specifically, they require some central point around which to draw the circle or ellipse as well as

some indication of range which represents either the radius of a circle, or the major and minor

axes of an ellipse. Typically, the centre of the ellipse is calculated as the geometric mean of all

observed activity locations (including the home). In the case of the SDE, the major and minor

axes are calculated as the standard deviation distance in the X and Y coordinate-direction for all

points from the mean centre. As Sherman et al. (2005) observe, the SDE does not necessarily

encompass all observed activity locations, and like the convex hull it does not make any implicit

assumptions regarding travel and how the observed activity locations may be connected.

3.2.3 Network Based Measures

Network based measures, unlike the previously mentioned measures of convex hulls and

SDEs, are an attempt to account for travel routes between observed activities. Such methods

typically attempt to estimate the shortest travel path between temporally adjacent activities and

then apply some buffer distance around this route. Like the convex hull, road network buffers

(RNBs) typically cover all observed activity locations. This is because the RNB space is defined

by three parts: origins, destinations and travel paths. The RNB has the most in common with

time-space geography principles of the three general approaches discussed thus far, but it comes

at cost. In instances where perfect travel information is not known, as is typically the case with

crime data, there is no guarantee that the route chosen reflects actual travel patterns. From a

practical perspective, RNB like calculations require extensive data on underlying travel networks

such as roads, busses, and trains that may not always be available, accurate or complete.

Network based measures like the road network buffer are all a subset of analysing means

of travel and pathways. This form of analysis is particularly difficult with crime data as it can

often be difficult or impossible to not only identify a specific time of occurrence but also identify

the means and route of travel. The RNB, for instance, assumes that individuals follow roads to

reach their destinations. The appropriateness of this assumption will obviously vary from data set

to data set as the underlying geography changes. To further complicate matters, any pathway

level analysis necessitates that a pathway be made up of a start and end point. Geographic

offender profiling (GOP) literature argues that such ‘anchor’ points are often an offender’s home

or equivalent (D. Canter et al., 2013; Canter & Hammond, 2006; Rossmo, 2000).

The relationship between the home and crime locations is not always clear even within the

marauder’ typology. And so if the home location is unreliably inaccurate or otherwise

unrelatable to a set of crimes, then what is left for establishing a pathway to study? The only data

points that remain would be the crime locations themselves, but it seems highly unlikely – except

in instances of extremely rapid ‘spree’ attacks – for any given crime location to be directly related to one another by direct travel.