When event occurrence varies in both space and time, it can be of great importance to determine whether there is also spatio-temporal dependency. Tests for spatio-temporal interaction are distinct from tests that identify the presence of purely spatial or temporal dependency (or, indeed, both): they focus on the likelihood of a further event occurring in a particular location, given the time and location at which a prior event has occurred. This information can be useful in policy-making. During outbreaks of rioting in a city, for instance, police leaders face decisions concerning the allocation of limited resources of police officers in real time. Insights into the spatio-temporal behaviour of rioting can help to answer questions such as whether police resources should remain at sites recently rioted or whether these resources would be better deployed elsewhere in the city, for example at perceived attractive targets that have not yet experienced rioting.
In this section, in order to investigate such questions, the level of spatio-temporal dependency in the 2011 London riots is determined through the use of a grid-based Knox statistic. Similarly to the test for spatial autocorrelation in Section 3.3, a model of the riots is constructed under the assumption that there is no spatio-temporal depen- dency. This enables the comparison between the empirical data and the data generated using a null model. Differences between the two can then be evaluated in order to de-
400
450
500
550
600
650
0
15
30
a)
SdEmpirical data
Simulation
400
450
500
550
600
650
δs(m)
0.0
0.1
b)
SIFigure 3.5: Results of the test for CSR. a) The values of Sd (in black) and S (g) d (in
white) are shown for each iterationg = 1, 2, ..., 499 for different grid sizes. In this case,
the points in white are so close together that the different iterations are indistinguish- able. This demonstrates further how strong the spatial clustering is in the empirical data. b) The values ofSI (in black) and S
(g)
I (in white) are shown for each iteration
g = 1, 2, ..., 499 for different grid sizes. In this case, the different iterations of the
termine whether the null model provides a reasonable account of the data generating process, or whether further analyses might be required.
It was demonstrated in Section 3.3 that the distribution of the locations of offences is highly likely to be spatially clustered. As discussed at the beginning of that section, there are two prominent explanations for the spatial clustering of event data that have been investigated in the literature for different types of crime: the flag hypothesis and the boost hypothesis. These two explanations can be distinguished between by deter- mining the level of spatio-temporal interaction between events. To explain, the flag hypothesis supposes that variations in the risk levels of different areas are due to static time-stable influences that can encourage crime. For example, in the case of residential burglary, it may be that houses with fewer visible security features are more likely to be targeted (since the burglar will perceive they are more likely to succeed) and there- fore experience a higher risk of burglary. This risk will be relatively constant over time (provided that the homeowners do not improve the level of security during this time), and so, such properties will likely experience a larger number of burglaries in any given time period, when compared to another house that has many visible security features. On the other hand, the boost hypothesis supposes that properties are more at risk as a direct result of it being targeted for a relatively short period of time after an offence has occurred. If the boost hypothesis was at play, a larger number of events would be expected in the locality of a prior event, above and beyond the spatial and temporal distribution of events within the wider region of study. The boost hypothesis implies spatio-temporal interaction; whereas the flag hypothesis attributes apparent space-time clusters to a heterogeneous distribution of risk in space combined with natural variation in crime trends. Understanding the extent to which both of these mechanisms play a role can lead to policy recommendations. For instance, if the boost hypothesis is sig- nificant in influencing future levels of risk, then, after a burglary, efforts could be made to reduce the underlying risk levels, ensuring that the risk of burglary does not get too large.
In the case of rioting, the analogue of the flag hypothesis suggests that time-stable features of different areas might also influence the risk of rioting at a given location. For example, if offenders participate in rioting due to the opportunity for them to loot high- value goods, then targets containing high-value goods are likely to be more at risk of
experiencing a riot than other targets, as potential rioters perceive the greater benefit of selecting that target over others. This rational choice perspective on the part of rioters— that they select targets based on the ability for those targets to fulfil their objectives— has been explored elsewhere (e.g. Martin et al. (2009)) and there have been several efforts to understand the features of targets that make them particularly attractive to rioters (Berk and Aldrich, 1972; Rosenfeld, 1997). In Chapter 4, the environmental features of different targets, and their role in attracting rioters will be explored further. It is noted for now that environmental features of regions can certainly play a role in the spatial clustering of riots; however, if the environmental features of these regions are static and time-stable, then the times at which events occur at these locations can be taken to be independent random events with times drawn from the temporal distribution of offences over the entire geographic region of interest. The null model for tests of spatio-temporal interaction supposes that this is indeed the case and, therefore, that events occurring at a given location do not influence the likelihood of future events proximate to that location, beyond the spatial and temporal distributions of the observed data.
The riots are modelled under the null hypothesis of spatio-temporal independence by randomly permuting the event times. Considering events (si, ti) for i = 1, ..., N ,
the set of times at which events occur, given by{t1, t2, ..., tN}, is permuted as follows:
choose a uniform psuedo-random integer,k(1)1 , between1 and N . Then swap the posi-
tion oft1withtk(1)
1 . Next, choose a uniform psuedo-random integer,k
(1)
2 , between2 and
N . Then swap the position of t2withtk(1) 2
. Continue for eachi = 3, .., N− 1 by choos-
ing a psuedo-random integer,ki(1), betweeni and N and then swapping the position of ti
withtk(1) i
. This results in the random permutation of event timesntk(1) 1 , tk (1) 2 , ..., tk (1) N o . The modelled riot data under the null hypothesis of spatio-temporal independence is then given by(si, tk(1)
i ) for i = 1, ..., N . The modelled riot data has the same spatial
distribution, given by the locations s1, s2, ..., sN, and the same temporal distribution,
given by the timest1, t2, ..., tN, as the original data; however, the association between
them is randomised and any interdependency beyond purely spatial and temporal fac- tors is removed.
As was the case with the CSR model in Section 3.3, comparing this single realisa- tion of the dataset under a null hypothesis with the empirical dataset is not particularly
instructive: the significance of any differences between the two datasets is impossible to determine. However, on the other hand, taking a full permutation of the event data is computationally intensive for large values ofN (there are N ! different possible permu-
tations). Therefore, a sample ofG = 499 from the possible N ! random permutations is
taken, leading to temporal permutations
n tk(g) 1 , tk (g) 2 , ..., tk (g) N o forg = 1, ..., 499, which
are then compared against the empirical data.
In order to compare the empirical data with the modelled data, a test statistic is required. A common statistic for identifying spatio-temporal interaction is the Knox statistic, SK (discussed from a methodological perspective in Knox (1964a) and first
employed as a test of spatio-temporal interaction in Knox (1964b)). SK is defined as
the number of pairs of events that occur within a given space-time window of each other. If the space-time window selected for the calculation of the Knox statistic is large enough, then the Knox statistic will be given by its maximum value,N (N−1)/2,
since all possible pairs will be included. Employing the same spatio-temporal grid as in Section 3.3, a grid-based Knox statistic is defined by taking the spatial window for significant pairs as all events occurring within first-order queen contiguity distance of the original event, as in Figure 3.4. The temporal window for significant pairs is taken to be one hour. This value is chosen so as not to exceed the resolution of the reported data, in which many of the offences are recorded as occurring to the nearest hour. Temporal resolutions of2, 3, 4, 5 and 6 hours were also tested in order to alleviate the
implications of the modifiable unit problem from a temporal perspective. These results were consistent with those for1 hour and for reasons of clarity are not presented here.
Of the 3, 914 offences associated with the London riots that were obtained from
the Metropolitan Police Service,2, 592 contained details of both the location at which
the offence took place as well as the time at which the event occurred. These were the events used in the analysis. Figure 3.6 shows the Knox statistics for the empirical data and for the simulated data for different spatial grid resolutions. The values of the Knox statistic associated with the empirical data are much larger than the values associated with the simulated data for all spatial grid sizes tested. In fact, the effect is extremely strong, with the values of the empirical Knox statistics being around four times the value when there is no spatio-temporal dependency. Since no simulated Knox statistic is larger than the empirical Knox in 499 simulations, the p-value is calculated to be
0.002, however, given the distance of the empirical result to the simulated result, it is
likely that a much smallerp-value could be found through the use of further iterations.
400
450
500
550
600
650
δs
0
10
20
30
40
S
K/1
0
3Empirical data
Simulation
Figure 3.6: Results of the Knox test. The empirical Knox statistic plotted against499
realisations of the simulated Knox statistic for a range of spatial grids.
It can be concluded that during the riots in London, there was significant spatio- temporal dependency in the event data. This finding implies that it was not just the suitability of certain locations in space, combined with the suitability of a particular time that led to riots, but that there was also strong evidence for event dependency: the occurrence of an event at a particular point in space and time increased the likelihood of observing a further event in proximity to the original event. In the remainder of this chapter, the precise nature of the interaction between proximate events is explored by considering the geographic patterns made by the riots.