Introduction
Current thinking in crime pattern theory relies on two somewhat contradictory observations. On the one hand, it is well-known that a small fraction of locations in any one environment accounts for a large fraction of the crime (Sherman, Gartin, and Buerger 1989; Weisburd 2015). On the other, there is good evidence that crime events themselves occur with a high degree of spatial-temporal variability (Bowers, Johnson, and Pease 2004; Mohler et al. 2015; Short, D’Orsogna, Brantingham, and Tita 2009; Wang, Rudin, Wagner, and Sevieri 2013; Wang and Brown 2012; Wang, Gerber, and Brown 2012).
Both observations hold important implications for not only understanding the causes of crime, but also designing crime prevention strategies. The former observation tends to encourage the view that crime patterns are predominantly static, persisting from one time period to the next in a roughly constant spatial configuration. The implication is that there is a tight coupling between crime and place that remains reasonably stable over time (Weisburd, Groff, and Yang 2012). If true, then it is clearly advantageous for police to repeatedly target the same places to achieve crime reduction (Sherman and Weisburd 1995).
The latter observation, by contrast, encourages the view that crime patterns are predominantly dynamic, with hotspots emerging, spreading, and dissipating only to reemerge in new locations (Johnson, Lab, and Bowers 2008; Short, Brantingham, Bertozzi, and Tita 2010). The implication is that crime is coupled to place quite loosely via a probabilistic decision making process (Brantingham and Brantingham 1978, 1981; Maltz, Gordon, and Friedman 1990: 1). If true, then police may find advantage in anticipating how that probabilistic process is evolving in space and time and target a shifting series of locations on the landscape (Bowers et al. 2004; Mohler et al. 2011; Wang et al. 2013; Wang and Brown 2012; Wang et al. 2012). Recent work has looked at whether dynamic targeting of places by police has an impact on crime (Gorr and Lee 2015; Mohler et al. 2015; Telep, Mitchell, and Weisburd 2014). The present work seeks to show how both of these perspectives can simultaneously be true.
Our presentation gets straight to the point. We forego a review of the theoretical and empirical literature on crime and place, as there are several recent works that cover such information in great detail (see Eck and Weisburd 1995; Weisburd et al. 2016; Weisburd et al. 2012). We therefore focus singularly on the analysis of crime concentration and the dynamics of crime hotspots. The paper is structured as follows. In the first section, we introduce our analytical approach, which is motivated by recent studies on the micro-geographic patterning of crime and place (Weisburd 2015; Weisburd, Bushway, Lum, and Yang 2004; Weisburd et al. 2012; Wyant, Taylor, Ratcliffe, and Wood 2012). As in Weisburd (2015), we are interested in the concentration of crime in a small number of geographic locations. However, we focus on how measured crime concentration changes as both the temporal and spatial windows for counting crime change (Brantingham, Dyreson, and Brantingham 1976; Chainey, Thompson, and Uhlig 2008; Steenbeek and Weisburd 2016; Towns-ley 2008). We are also interested in measuring the stability of crime patterns in the context of changing spatial and temporal scales. We introduce a very simple measure that counts the percentage overlap in hotspot locations from one time period to the next when measured at different temporal and spatial scales (see Mohler et al. 2015). The approach is far simpler than other recent assessments of crime pattern stability (Johnson et al. 2008; Weisburd et al. 2004; Weisburd et al. 2012), but offers practical advantages in terms of ease of interpretation.
The second section turns to empirical assessments. We analyze crime patterns in Los Angeles during the years 2009–2015, and Chicago during 2008–2015. In both settings, we analyze assault, burglary, motor vehicle theft, and robbery, independently for each crime type. We offer theoretical motivation for choosing these crimes based on fundamental differences in the potential mobility of offenders and victims involved in each of these crime types (Tita and Griffiths 2005).
The third section presents our two principal findings. First, hotspots defined at smaller temporal and spatial scales capture the same amount of crime, while covering less total land area. In other words, crime appears to be much more concentrated when using smaller, short-term counting units compared with larger, long-term counting units. Second, hotspots defined at smaller temporal and spatial scales are much more dynamic than those defined at larger temporal and spatial scales. In other words, small, short-term hotspots spatially overlap much less from one time period to the next compared with larger, long-term hotspots. There is thus an apparent tradeoff with crime hotspot characterization. Smaller, short-term hotspots are better at identifying the highest crime concentrations in an environment, but those locations change substantially in placement at that short time scale. Alternatively, more stable crime patterns can be identified by adopting larger spatial and temporal scales, but at the cost of reduced crime concentration.
The final section discusses implications of the work. We discuss how the concentration-dynamics tradeoff impacts our understanding of crime causation. We then draw some general observations about the scale of analysis and policing and crime prevention. The punchline is that crime patterns do not exist only at one scale (Brantingham, Brantingham, Vajihollahi, and Wuschke 2009; Steenbeek and Weisburd 2016). This is not necessarily an indication of aggregation bias. Rather, it is indicative of behavioral processes operating at different scales. Policing and crime prevention efforts can benefit from calibrating to these scales.
Methods
Our methodological approach is divided into four principal parts. The first involves defining the spatial and temporal counting units for hotspot quantification. The second concerns measuring the global concentration of crime given those spatial and temporal units. The third concerns assessing the spatial stability (or lack thereof) of hotspots from one time period to the next. Our measure of spatial stability of hotspots is likely dependent upon macroscopic patterns of how smaller spatial units are organized into larger clusters (see Steenbeek and Weisburd 2016; Weisburd et al. 2012). The fourth part is therefore tabulating cluster sizes for each spatial and temporal scale.
We adopt a straightforward method for defining spatial and temporal counting units. Our spatial units are constructed as a regular square lattice or grid laid out over the entire jurisdiction. Specifically, we examine grids where each cell is 200 × 200, 400 × 400, or 800 × 800 m in size. Fixed grid counting units may be contrasted with categorical spatial units such as street segments (Davies and Bishop 2013; Weisburd et al. 2012), reporting districts, census tracts, or formally recognized neighborhoods (Wooldredge 2002). Our temporal units are similarly defined in discrete terms as fixed time windows measured in days, months, or years. These discrete spatial-temporal units lead naturally to a histogram method for counting crime. We count all of the crimes of a specified type occurring in each grid cell during each defined time period. For example, we will count all of the robberies occurring in each 200 × 200 m grid cell per day, or all burglaries in each 400 × 400 m grid cell per month. Note that common hot spotting methods such as kernel density estimation (KDE) are closely related to the histogram counting procedure suggested here, as both are non-parametric estimators for the probability density function of a point process.
After the number of crimes within grid cells have been counted, the resulting counts are ranked in decreasing order by crime count. For example, the Rank 1 cell will contain the greatest number of crimes among all cells, the Rank 2 cell will contain the second greatest number of crimes, and so on. It follows that the Rank 1 cell will capture the greatest percentage of crime compared to all other individual cells for that specific time period, the Rank 2 cell will capture the next greatest percentage, and so on. Starting at the top of the ranked list, we flag each cell in order until the collection of flagged cells in total represents a predefined cumulative percentage of the total crime over that time window. For example, we might flag cells until 5%, 10%, 25%, or 50% of all crime within that time period is captured by those cells.
For simplicity, we will use the term hotspot to refer to an individual grid cell flagged in this way. All of the cells not flagged in this way are not considered crime hotspots for that particular cell size and time period. Collections of flagged grid cells are referred to as clusters, when they form contiguous spatial blocks, or simply by the plural hotspots, when their spatial arrangement is not relevant. Note that our procedure is very closely related to that of Weisburd (2015), who counts the percentage of all street segments needed to capture a fixed percentage of crime (see also Weisburd et al. 2004; Weisburd et al. 2012). To facilitate comparison with Weisburd’s results we report results for hotspots capturing 25% and 50% of crime, respectively. We caution, however, that crime counts aggregated by street segments and areal units such as grid cells may not be strictly equivalent (see Steenbeek and Weisburd 2016).
It is a simple matter to convert the number of crime hotspots into a measure of crime concentration. First, the number of flagged hotspots is converted into an area by multiplying by the known area of each cell (e.g., 200 × 200m). Dividing this total hotspot area by the total land area of the jurisdiction yields the percentage of land area needed to capture a fixed percentage of crime. The smaller the percentage of land area sufficient to capture a target crime percentage, the more concentrated crime is in space. For example, crime is two times more concentrated if 0.1% of a jurisdiction’s land area captures 25% of crime, compared with 0.2% of the total land area capturing 25% of crime.
We measure hotspots stability in a similarly direct manner. For any given collection of crime hotspots, we measure the percentage overlap in hotspot locations from one time period to the next. For example, imagine a collection of one hundred hotspots each 400 × 400 m in size sufficient to capture 25% of recorded crime over the course of one month. Now imagine that we perform the same analysis for the following month, again yielding one hundred hotspots of the same size. We then compare the hotspot locations from month 2 with those present in month 1 and find that 50% of the locations are the same. Thus half of the pattern is stationary at this temporal and spatial scale, and the other half is dynamic. Our approach is similar to that of Andresen and Malleson (2011) wherein spatial units are compared across two time periods for the volume of crime present. Units are scored as stationary if the volume of crime is statistically equivalent across time periods. Our approach is considerably different from that of Weisburd et al. (2004) and Weisburd et al. (2012) who use group-based trajectory analysis to identify latent hotspot groups given the entire history of crime on street segments over a 16 year time period. They find that some latent groups display very stable crime patterns at an annual time scale over the study period, while others display secular variation in crime volume over time.
We also examine hotspot cluster sizes. To find the size of a hotspot cluster, we simply count the number of contiguous flagged hotspots present in a given time window. The counting procedure is as follows. Given a starting grid cell flagged as a hotspot, each immediately adjacent cell is joined to the same cluster if it is also flagged as a hotspot. These first-order neighbors are then used to look for unique adjacent cells that are also flagged as hotspots. These comprise second-order neighbors. The process is repeated until no unique hotspots can be joined to the component. We are primarily interested in how cluster size changes as a function of the spatial and temporal scale of measurement. To assess whether observed cluster sizes are different from what would be expected given random occurrence of crime, we simulate random hotspot placement for equivalent hotspot densities and then compute cluster sizes using the aforementioned method for these randomly placed hotspots.