Andresen, M.A. (2018). GIS and spatial analyses. In G.J.N. Bruinsma & S.D. Johnson (Eds.), Oxford handbook of environmental criminology (pp. 190 – 209). New York, NY: Oxford University Press.
Introduction
Though the spatial analysis of crime had its beginnings in early 19^{th} century France with the work of Michel Guerry (1832, 1833) and Adolphe Quetelet (1831, 1842)—see Weisburd et al. (2009) for a history of spatial criminology—environmental criminology came about when C. Ray Jeffery (1971) coined the term; C. Ray Jeffery was calling for a new school of thought within criminology to better understand the environment within which crime occurs. This call for a new school of thought was in spite of the fact that crime had been mapped for close to 150 years and was an important component of social disorganization theory. Shaw and McKay (1942), for example, had many maps of juvenile delinquency and related factors in Chicago. As pointed out by Brantingham and Brantingham (1981), previous research in criminology has focused on one of three dimensions: the law, the offender, and/or the target. However, there is a fourth dimension of spacetime that is the concern of environmental criminology and critical for understanding the environment within which crime occurs.
Because of the importance of the spatialtemporal dimension(s) within environmental criminology, the use and applications of geographic information systems (GIS) and spatial analysis are rather widespread. In this chapter I will cover some of the principles and advancements in the use of crime mapping and spatial analysis to study the spatial distribution of crime, primarily through the lens of environmental criminology. Books can (and have!) be written on many of the topics covered here, so this discussion cannot be considered exhaustive, but an introduction to the issues of how GIS and spatial analysis are used in that regard.
What are crime mapping and spatial analysis? And how is it done?
Simply put, crime mapping is the spatial representation of crime (in the context of criminal events) on a map. Consequently, in order to do so, one must have geographic coordinates for each criminal event to place it on a map. This may initially come across as an obvious statement in this context but it is an important factor in crime mapping for the accuracy of the information. For example, due to privacy concerns, a police agency may only provide the neighborhood within which a criminal event occurs. Or, more common, a police agency will provide X,Y coordinates or the 100block (street segment) to locate where the criminal event occurred in order to provide some level of privacy. Though such information may be accurate, the degree of that accuracy may be deceptive.
Most often, X,Y coordinates are assumed to as accurate geographic information as one can get. If this information was obtained using a global positioning system (GPS) unit, that may be the case. Though weather and GPS receiver quality impact accuracy of GPS receivers, most high quality GPS receivers are accurate within 3.5 meters, often within 1 meter.1 Such accuracy, however, cannot be assumed with criminal event data. Did the police officer on scene bring the GPS receiver to the exact location of the criminal event, or is the GPS receiver located in the police vehicle that may be one or two properties away? If a researcher or practitioner is interested in general spatial crime patterns, such issues are not of much concern, but if an analysis is concerned with minute details such as land use or inside/outside location of the criminal event, such an analysis must proceed with caution. Additionally, the X,Y coordinates provided may represent the geocoded location of the criminal event: the coordinates where a GIS places the address on a road network.2 The primary limitation of this method is that GIS geocoding procedures may assign geographic coordinates to a criminal event based on a linear interpolation in a street segment. That means that if a street segment has an address range from 101199 and the address is 150, the GIS places that address in the centre of the street segment. Though this will most often prove to be reasonably accurate, we must always use caution when interpreting the accuracy of spatiallyreferenced data, of any kind—for more details regarding geocoding (criminal event) data, see Ratcliffe (2001, 2004).
Once spatiallyreferenced criminal event data are available they need to be represented on a map. There are three primary ways in which spatiallyreference data can be presented: points, lines, and areas. Most often, criminal event data are represented as points (dot maps) or areas (census tracts or neighborhoods, for example), but maps considering lines (street segments) are becoming more commonplace—see Curman et al. (2015) and Weisburd et al. (2012), for example. Below, I show maps of points and areas because they are the most common in crime mapping.
Figure 1 shows a map of commercial burglary in Vancouver, British Columbia, Canada, 2013. These data are represented as points, but were derived from 100block (street segment) data. Consequently, no inference can be made at a spatial resolution smaller than the street segment; however, when viewed at the level of the city, this inaccuracy is unlikely to distort perceived patterns. As can be seen in Figure 1, 2013 commercial burglaries in Vancouver appear to be clustered, especially in the centralnorthern peninsula of the city that contains the central business district. It is important to note that this is not a particularly interesting “result” because commercial land use in Vancouver is also clustered. This highlights the importance of understanding the context in which criminal event maps are made. Generally speaking, if point level criminal event data are available, they should be used by the researcher because they will be the most spatially accurate. However, one difficulty with pointbased maps is that if the phenomenon being mapped has a high volume in a particular area it is difficult to see how many points are present because they are all on top of each other. This can be addressed using a number of different visualization techniques, with one of the more common aggregating points within an area—another method is kernel density estimation, discussed further below.
Figure 1. Commercial burglary, Vancouver, Canada, point data
Figure 2 shows commercial burglary counts within census tracts in Vancouver. The cluster of commercial burglaries in the central business district is still quite apparent with most of the city’s census tracts having very low counts of this crime for the year. Figure 3, on the other hand, shows the same commercial burglary data aggregated to dissemination areas—dissemination areas are similar in size to the census block group in the United States, most often with 400 to 700 persons. Though the same general spatial pattern is present, there are notable differences that emerge when using this smaller spatial unit of analysis. This points to two related, but distinct, issues in spatial analysis: the ecological fallacy and the modifiable areal unit problem. The modifiable areal unit problem, discussed here, refers to changes in statistical results when the spatial units of analysis change. The ecological fallacy essentially means that what is true of the whole is not necessarily true of all its parts; this is evident in Figures 2 and 3 with large areas in Figure 2 being represented differently in Figure 3. Each are discussed in further detail below.
Figure 2. Commercial burglary, Vancouver, Canada, census tracts


Figure 3. Commercial burglary, Vancouver, Canada, dissemination areas
Modifiable areal unit problem and the ecological fallacy
The modifiable areal unit problem (MAUP) emerges because raw pointlevel data are aggregated into areabased units—see Brian Lawton’s chapter in this volume that also discusses this issue. MAUP tends to be problematic because these areabased units do not have “natural” boundaries that make sense for the analysis at hand. Censusbased boundaries, for example, are defined by roads and how many homes census workers can visit in a day. Though it is possible that these areas will represent actual neighborhoods, many will not. In any case, these areabased units are modifiable because they can be redefined by arbitrarily changing their boundaries. The MAUP, though often attributed to Openshaw (1984a) for its identification, was noticed in statistical analysis as early as the work of Gehlke and Biehl (1934) who found that correlation coefficients increased in magnitude as the size of the areabased units increased.
The MAUP can emerge in one of two primary ways. The first is referred to the zoning problem and occurs when there is a spatial shift in currently defined boundaries. In this case, the size and shape of the areas do not change, they just represent different places. The easiest example of the zoning problem would be if a grid of one square kilometer cells was shifted 500 meters in one direction. In this case, there would be a change in which pointlevel events would be aggregated to each cell, potentially altering any statistical results.
The second, and more commonly referred to, version of the MAUP is the scale problem. With the scale problem, statistical results change when the size of the areabased units change. A prime example of the potential for the scale problem was shown in Figure 2: because of the moderate changes when changing from census tracts to dissemination areas any statistical analyses undertaken could produce different results. Of course, simply because there is the potential for a problem does not mean that a problem will emerge. The difficulty, however, is that with two exceptions there is no way predict and/or correct problems that are associated with the MAUP. The first exception was identified by Gehlke and Biehl (1934) with the magnitude of correlation coefficients increasing with the size of the areabased units. The second exception is just related to sample sizes and statistics; if the areabased units under analysis are smaller representing the same area (i.e. there are more areabased units) you are more likely to find statistically significant results because you have a larger sample size.
Despite the fact that the problem with MAUP is only a potential, research has emerged that shows how problematic the MAUP can be. Openshaw (1984b) found that through modifications considering the zooming problem alone one can get correlation coefficients for the same two variables to range from 0.99 to 0.99. As such, not only can the magnitude be incorrect, but so can the estimated coefficient sign. Moreover, Fotheringham and Wong (1991) extended the work of Openshaw (1984b) and considered the MAUP in a multivariate (regression) context. The authors investigated the reliability of regression coefficients in the presence of the MAUP and, similar to Openshaw (1984b) in the context of correlation, found unreliability. With regard to the scale problem, Fotheringham and Wong (1991) were able to have the magnitude of regression coefficients change by a factor of nine when reducing the number of areabased units from 800 to 25, covering the same area. Though such a reduction in the number of areabased units representing one place may not appear to be realistic, in the maps of Vancouver in Figure 2, there are 993 dissemination areas, 106 census tracts, and 24 official neighborhoods—all of these areabased units have socioeconomic and sociodemographic data available. In other words, such drastic changes in the number of areabased units are quite possible. In the end, Fotheringham and Wong (1991) were able to generate almost any result desired by changing the areabased units in a regression context. As such, they showed that statistical results are even more unpredictable than found by Openshaw (1984b).
With such results in a sensitivity analysis investigating the MAUP, it is easy to become disenchanted with spatial analysis. However, it is important to continue doing such analyses to investigate spatial patterns in hopes of reducing crime. Within the criminological literature the impact of the MAUP has not proven to be as depressing. Wooldredge (2002) analyzed 1932 individuals, 129 census tracts, and 48 official neighborhoods in Cincinnati in a regression context with over 30 variables. Overall he found that the differences were not qualitatively different most of the time. And Hipp (2007) spoke to the importance of considering the most natural areabased unit of analysis with regard to theoretical relationships. Specifically, the area used to represent a variable (or construct) may vary from variable to variable. In my own research that has considered multiple spatial units of analysis (usually census tracts and dissemination areas) I have not found any particularly interesting/disturbing changes in statistical results. Most often, with this possibility discussed above, dissemination area results tend to have more statistically significant variables, likely because of the larger sample size.
Overall, researchers and practitioners need to be aware of the MAUP and its potential impact on anything they study. If possible, it is advisable to use more than one spatial unit of analysis, particularly if one is attempting to inform policy. Because of the availability of so many forms of spatial analysis, not taking advantage of these statistical methods because of the MAUP would be worse than the possibility of being wrong.
The ecological fallacy is actually something that can be avoided by those who are careful with their inferences. The ecological fallacy is committed when a researcher interprets statistical results at a level finer than the analysis was undertaken. For example, one commits the ecological fallacy if an analysis is undertaken at the neighborhood level and inferences are made on the individuals who live in those neighborhoods. In order words, it is assumed that what is true of the whole is true of all its parts. Though this must be true, on average, otherwise the statistical result at the neighborhood level would not have emerged, it is most certainly not true for all individuals within that neighborhood. Relatedly, the atomistic fallacy occurs when inference is incorrectly made in the other direction: what is true of the part is expected to be true of the whole. For the same reasoning as the ecological fallacy, such a relationship may be true in some cases but cannot be assumed to be true all of the time.
Robinson (1950) was the first to identify and analyze what we now call the ecological fallacy. He defined an ecological correlation as a statistic that included multiple subjects that could be divisible, a neighborhood, for example; an individual correlation is a statistic that involves subjects that cannot be divisible, such as individuals. Robinson (1950) was able to show the conditions within which individual correlations and ecological correlations were the same (no ecological fallacy), but he found this to be the exception not the rule. Openshaw (1984b) found that the ecological fallacy was important for areabased census data, but the degree of the ecological fallacy depended upon the statistical analysis undertaken. Generally speaking, Openshaw (1984b) found that the differences between any ecological and individual correlations could not be known beforehand.
In some of my own research, when comparing the changes in spatial patterns over time, a colleague and I found that smaller areabased units that were contained within larger areabased units (dissemination areas within census tracts, for example) never had the same degree of change over time in more than 60 percent of cases (Andresen and Malleson, 2013). In fact, in a small number of cases, the type of change found at the level of the census tract was not found for any of the dissemination areas within that census tract! Only when the dissemination areas were aggregated into a census tract did the identified change (marginally) occur.
As stated above, however, despite the problems associated with the ecological and atomistic fallacies, they are easy to avoid. Simply do not make any statistical inferences at levels different from the level used in your analysis: do not make inferences about people when you analyze neighborhoods, and vice versa. The ecological and atomistic fallacies are always a potential in any method of spatial analysis, but we can avoid them if we make statistical inference with care.
Methods of spatial analysis
There are a number of ways to conduct spatial analyses (of crime) and some are more “spatial” than others. The primary distinction would be between statistical and visualization methods. However, there are a number of statistical methods that generate output that can be visualized. Each will be discussed below.
Kernel density estimation
I have already discussed and shown two visualization methods above: pointbased and areabased maps. A “spatial analysis” could simply be the mapping of crime and comparing that map to some other phenomenon that varies spatially. However, one of the most common forms of spatial analysis in the context of visualization is the kernel density map. Kernel density estimation involves placing a grid over the entire study area and essentially counting the number of events that fall within a certain distance of each grid cell. Depending on the size of the grid cells and the distance being used, there will be a high or low degree of “doublecounting” of points. This is done for the purposes of smoothing data over the entire study area to avoid the high density of points (sometimes referred to as blobs!) that make inferences difficult. The difficulty with kernel density estimation is that it generates a value for each grid cell over the entire study area. Consequently, a kernel density map may show “crime” in an area that experienced no criminal events; however, the same is true for any areabased representation such as those shown in Figure 2 and Figure 3.
Figure 4. Commercial burglary, Vancouver, Canada, kernel density map
The kernel density map for commercial burglary in Vancouver is shown in Figure 4. It should be clear that the representation of the greatest volumes of commercial burglary are in the same places. Figure 4, however, is more visually appealing than the maps in Figures 1, 2, and 3 because of the smooth contours on the map, and is not constrained by the boundaries used and, so, gives a more accurate picture of where crime clusters. One of the advantages of the areabased maps (Figures 2 and 3) is that risk can be measured by calculating a rate: the number of commercial burglaries per number of commercial establishments, if such data are available. These calculations cannot be made using pointbased maps. However, if pointlevel data are available for the number of commercial establishments, a dualkernel density map can be generated that considers both burglaries and establishments, providing an equivalent to a rate.
Journey to crime
One of the simplest, and oldest, methods to spatially analyze crime is to measure the distance travelled by offenders to the crime site, often referred to as the journey to crime—sometimes distance to crime is preferred because some scholars say the “journey” involves more than just the distance travelled. This literature on the journey to crime dates back at least 85 years (White, 1932) and has found two general results: the journey to crime is short (most often less than two kilometers) and the journey to violent crime is shorter than the journey to property crime—see Townsley and Sidebottom (2010) for a review and Andresen et al. (2014) and Ackerman and Rossmo (2015) for some more recent analyses.
There are three primary methods one may calculate these distances. The first method is to calculate the “as the crow flies” distance. This is simply the linear distance between the home location of the offender and the criminal event location—Wiles and Costello (2000) used the location the offender slept the night before the crime as the origin of the journey to crime with similar results. Though it is possible that an offender may travel in a straight line to a criminal event location (walking down the street, for example), this is unlikely to occur in many contexts. The second is “Manhattan distance” and is measured as the distance along the two sides of a triangle with the hypotenuse representing the “as the crow flies” distance between the offender’s home and the criminal event location. The Pythagorean Theorem is used to calculate this distance and has been found to be representative of an area that has a Manhattan grid street network. The limitation of this method, though often superior to “as the crow flies”, presents itself in more recent street networks in North America, particularly suburban neighborhoods, and older European cities that do not have a gridbased street network. As such, the third method for calculating the journey to crime is using a network or routefinding algorithm. Such algorithms have to use assumptions (shortest distance, for example), but they will be more precise than either of the first two methods outside a gridbased street network and just as precise within a gridbased network.
Though it is possible to calculate the journey to crime using locations on a map, a ruler, and the scale, the advent of GIS and spatial analysis methods that go along with it make it possible to undertake research in this area that would not have been feasible only a little more than a decade ago—for example, Andresen et al. (2014) calculate the journeys to crime for over 80,000 criminal events. A GIS makes “as the crow flies” and Manhattan distance calculations relatively simple and quick. Moreover, the use of network analysis and routefinding algorithms within a GIS are relatively straightforward. What this does is allow for the use of larger and larger datasets that can then be parsed into specific research questions that could not be answered previously because of a lack of data and/or the computational power to do so. For example, Andresen et al. (2014) had enough criminal events to break down the journey to crime by twelve crime types and individual years of age from 8 – 68. The availability of these data and the ability to use a GIS make significant insights possible.
A related avenue of research that considers distances travelled is the mobility triangle literature that considers the interaction between the offender’s journey to crime and the victim’s journey to victimization. This literature goes back over 90 years (Burgess, 1925) and has primarily been interested in identifying how close offenders and victims live to one another. Quite often, two of the three locations (offender’s home, victim’s home, and criminal event location) are in the same neighborhood—see Andresen et al. (2012) for a review. Distances, and subsequent areas, calculated for mobility triangles tend to be “as the crow flies” for simplicity in calculations. However, just as with the journey to crime, advances in GIA and spatial analysis allow for more nuanced insight into this strongly geographical phenomenon.
Spatial autocorrelation and spatial regression
Moving beyond the conceptually simple calculations of distance and area (that can be complex from a computational perspective), I will now turn to more complex spatial analyses that are statistical in nature and refer to the spatial relationships between places. These spatial analyses assume spatial relationships exist (and they almost always do!), considering spatial autocorrelation: how similar are places within some context (crime, for example) to other places that are nearby? In other words, can we make predictions about one place based on the values of places that are nearby?
Before spatial autocorrelation can be calculated, the concept of near has to be better defined. Near can be measured based on distance, often in the case of pointbased data, or based on contiguity. Areas are considered to be contiguous if they share a border: Rook’s contiguity is present if two areas share a border of some length, and Queen’s contiguity is present if two areas share a border even if it is at a corner. Queen’s contiguity is most commonly used in environmental criminology (and the social sciences, more generally) because Rook’s contiguity is considered rather strict—two neighborhoods would still be considered as contiguous if they met at an intersection diagonal from one another. Contiguity can also be ranked according to how near the areas actually are to one another. First order contiguity means that two areas actually share a border, second order contiguity means that two areas have one other area in between them that shares a border with each of them, and so on. More often than not, first order Queen’s contiguity is sufficient to account for spatial autocorrelation in the spatial statistical methods discussed below.
Once near has been specified, the degree of spatial autocorrelation can be measured. For simplicity, I will use areas (census tracts and dissemination areas) and first order Queen’s contiguity to measure spatial autocorrelation. Though there are two spatial statistics to measure the degree of spatial autocorrelation across a study area, Geary’s (1954) C and Moran’s (1950) I, Moran’s I is far more commonly used because its interpretation is more “natural” and along the lines of more traditional correlation. Moran’s I ranges from 1 (perfect negative spatial autocorrelation) to +1 (perfect positive spatial autocorrelation). Positive spatial autocorrelation means that near places have similar values with one another and negative spatial autocorrelation means that near places have dissimilar values with one another—a black and white checkerboard is an example of perfect negative spatial autocorrelation. In most criminological contexts, but not all, positive spatial autocorrelation is present.
With regard to the maps in Figure 2 and Figure 3, the Moran’s I values are 0.51 (census tracts) and 0.36 (dissemination areas), with both being statistically significant. As such, at both the census tract and dissemination area levels there is positive spatial autocorrelation. Moreover, it is important to note that the Moran’s I value for census tracts is greater than the Moran’s I value for dissemination areas. This is consistent with the statements above regarding the MAUP: geographically larger areas have increases in the magnitudes of correlation coefficients.
The degree of spatial autocorrelation and whether it is positive or negative can sometimes be of interest on its own. However, most often, the presence of (positive) spatial autocorrelation is viewed as a nuisance because it poses difficulties for inferential statistics. One of the assumptions in classical regression is the independence of the residuals terms: knowing the value of one residual does not provide enough information to predict the value of the next residual term any better than a random guess. However, in the presence of positive spatial autocorrelation in our variable(s) of interest most often leads to positive spatial autocorrelation in the residuals of a regression.3 This means that knowing one residual’s value can provide enough information to predict the value of the next residual’s value better than a random guess. The assumption of independence is violated. This leads to one primary problem: in the presence of positive spatial autocorrelation, the underestimation of standard errors. As with any case of underestimating standard errors, statistical significance values appear to be higher and one may consider a potential explanatory variable statistically significant when it is not—it is only statistically significant because of the positive spatial autocorrelation.
Spatial regression is a spatial statistical technique that was developed to address such an assumption violation. Spatial regression techniques filter out the spatial autocorrelation in order to identify the underlying theoretical relationships without those relationships being muddled by the presence of spatial autocorrelation. There are two primary forms of spatial regression: spatial lag models and spatial error models. A spatial lag model only filters out the spatial autocorrelation from the dependent variable, whereas a spatial error model filters out the spatial autocorrelation from both the dependent and independent variables. The choice of spatial regression model is identified using statistics (both GeoDa4 and R5 provide the statistics to make the appropriate model choice), but can be thought of as follows. If the spatial autocorrelation is only present in the dependent variable, there is no sense in filtering out any spatial autocorrelation from the independent variables; this implies a spatial lag model. However, if there is nothing to filter out, running a spatial error model when only a spatial lag model is necessary should not have any impact on the results. A spatial error model will be necessary when the nature of the spatial autocorrelation is different in the dependent and independent variables such that filtering out the spatial autocorrelation from one will not remove it from the other. Though the various statistical outputs should be used to identify the appropriate spatial regression model to be used, in my experience it makes little difference in the statistical results if a spatial lag model or a spatial error model is used.
Local spatial analysis
The methods of spatial analysis discussed thus far are classified as global spatial analysis methods. They are global in the sense that the statistics represent the study area globally, as a whole. For example, as mentioned above, commercial burglary in Vancouver has a statistically significant level of positive spatial autocorrelation for census tracts and dissemination areas. Also, in either a spatial lag or spatial error regression model, the estimated parameters represent the entire study area: a oneunit increase in an independent variable represents a β increase in dependent variable regardless of where you are in the city—it is an average representation for the city as a whole.
But are these global representations realistic? Positive spatial autocorrelation means that high crime areas are close to high crime areas and low crime areas are close to low crime areas. Is it not possible that there are places in Vancouver that do not have any clustering of crime, high or low? Is it not possible that there are low crime areas that border high crime areas (negative spatial autocorrelation) and vice versa? In the context of a regression, is it reasonable to think that a oneunit change in one variable, say the unemployment rate, has the same impact of β in all areas of the city? Would going from 2 percent to 3 percent unemployment in a wealthy neighborhood of Vancouver have the same impact as going from 14 percent to 15 percent unemployment in a disadvantaged area of Vancouver? The first is a 50 percent relative change in an area that has very little unemployment and the latter is a 7 percent change in an already disadvantaged area. One could argue that the first change would be significant but the second would hardly be noticed. Based on this brief discussion, it is easy to question the usefulness of global statistical analyses. However, we must remember that theoretical developments and social policy have to be made for populations as a whole, but it is important to remember that they will not apply in all contexts and we can empirically investigate where these global relationships will hold and where they will not.
Local spatial analysis methods focus on understanding local spatial relationships rather than global spatial relationships. In one sense, they can be considered searching for exceptions to the rule, with the corresponding global spatial relationship being the rule. The end result will be a (potentially) large number of spatial patterns to investigate and a better understanding of the nature of spatial patterns (of crime) across an area.
Fotheringham (1997) discussed three reasons why we should not expect global spatial relationships to hold when we consider local relationships—he referred to this as spatial nonstationarity. First, he stated that there will be variations by random chance. Though important to note, this is not particularly interesting. One could consider this along the lines of standard errors for regression coefficients. If regression coefficients only varied spatially within two standard deviations of the estimated parameter, that would not be particularly interesting. Second, some things may matter more in particular places than others. A prime example could be situations similar to the discussion above regarding the effect of a one percent change in the unemployment rate in different socioeconomic status areas of the city. And finally, spatial nonstationality may be found because of statistical misspecification. In other words, it may be found by accident because we failed to include all relevant variables (omitted variable bias), used an inappropriate statistical method (ordinary least squares instead of a countbased model), or the included variables are not measured properly. In other words, researcher error. Though this third reason is very important, for any type of analysis, the second reason is of primary interest to most spatial/geographical scholars. Where do our expectations hold, and where do they not hold? Can we explain why our expectations hold here and not there? These are not only interesting questions to ask and answer, but critically important for understanding the spatial distribution of crime and, hopefully, being able to reduce crime. Additionally, because local spatial statistics vary spatially, they can be mapped to help identify interesting spatial relationships.
Anselin (1995) is probably the most wellknown piece of research regarding local spatial analysis. In this article, he outlines local indicators of spatial association (LISA), that represents a category of local spatial statistics that have a corresponding global spatial statistic: local Moran’s I and Moran’s I, for example. As stated above, Moran’s I ranges from 1 (perfect negative spatial autocorrelation) to +1 (perfect positive spatial autocorrelation. Local Moran’s I is measured the same way, but a (local) statistic is calculated for each spatial unit in the study area, not just one (global) statistic representing the entire study area. This allows for the identification of spatial autocorrelation being different for different places within the same study area. In addition to allowing for the presence of no statistically significant spatial autocorrelation, local Moran’s I identifies four types of local clusters: HighHigh, HighLow, LowHigh, and LowLow. HighHigh and LowLow represent positive spatial autocorrelation with the former being areas with high values (high crime in the current context) surrounded by areas with high values and the latter being areas with low values being surrounded by areas with low values—hot spots and cold spots. HighLow and LowHigh represent areas with negative spatial autocorrelation: areas with high values surrounded by areas with low values and areas with low values surrounded by areas with high values, respectively. Though positive spatial autocorrelation is the norm for most sociodemographic and socioeconomic variables, particularly at the global level, negative spatial autocorrelation is often found in a handful of places at the local level.
Figure 5. Commercial burglary, Vancouver, Canada, local Moran’s I
Figure 5 shows the local Moran’s I clusters for commercial burglary in Vancouver. It should be clear that the dominant form of local clustering is HighHigh in the eastern portion of the central business district peninsula, extending moderately to the south and to the east. Considering the crime maps shown above, particularly those in Figure 2 and Figure 3, such a set of local crime clusters should not be a surprise. However, there are two things to note. First, though Figures 2 and 3 identified a strip of higher crime areas along the central southern border of the city, this does not emerge as a local crime cluster. As such, commercial burglary is higher in that area, but not so much as to establish a cluster of commercial burglary activity. This also highlights the importance of testing for statistical significance rather than just visualizing spatial patterns and assuming that something matters. Second, though the area is small, there is the presence of a small dissemination area that exhibits negative spatial autocorrelation, LowHigh, in the center of the HighHigh local crime clusters. Other research in Vancouver has shown a greater presence of local negative spatial autocorrelation (Andresen, 2011), but this result shows that there are exceptions to the rule and this may prove to be a more interesting result than finding that commercial burglary clusters in Vancouver. Is there something different about this dissemination area? Does it simply have no commercial land use, or is it able to repel commercial burglary in spite of having commercial land use and being surrounded by other places with statistically significant clustering of commercial burglary?
Another commonly used local spatial statistic is the GetisOrd Gi* (Getis and Ord 1992; Ord and Getis 1995). The GetisOrd Gi* statistic identifies clustering (positive spatial autocorrelation) through a comparison of the values for each local area and its surrounding areas. Unlike local Moran’s I, the GetisOrd Gi* statistic only identifies HighHigh and LowLow local (crime) clusters. A map of the GetisOrd Gi* statistics is shown in Figure 6. It should be clear that the statistical results for the GetisOrd Gi* statistic and local Moran’s I are very similar. Both local spatial statistics identify the eastern portion of the central business district as a HighHigh local crime cluster of commercial burglary that extends slightly east and south of the central business district. Both local spatial statistics also identify some degree of statistically insignificant clustering in the center of the larger HighHigh local crime cluster. The presence of this area is interesting in and of itself, and is not just an artefact of one (local) statistical method. However, the GetisOrd Gi* statistic, unlike local Moran’s I, does identify a local crime cluster (HighHigh) in that central southern strip of Vancouver.
Part of the reason for the differences in the GetisOrd Gi* and local Moran’s I results is quite simple: the default statistical significance level for local Moran’s I is 95 percent, whereas GetisOrd Gi* presents all results for 90, 95, and 99 percent statistical significance as a default. Of course, I could have presented the 95 percent statistical significance results for both local spatial statistics, but I did not in order to show that one does need to be careful when considering default program settings and then comparing statistical output. Regardless, using the same level of statistical significance does not meaningfully change the comparison of the local spatial statistical results. The GetisOrd Gi* statistic is always more inclusive for HighHigh local crime clusters, but the results are qualitatively the same. Incidentally, at the 90 percent statistical significance level, local Moran’s I does identify a local crime cluster in the central southern strip of Vancouver, but it is the dissemination area immediately to the west identified in Figure 6.
Figure 6. Commercial burglary, Vancouver, Canada, GetisOrd Gi*
The last local spatial statistical method I will discuss here is geographically weighted regression. Geographically weighted regression was developed in an effort to identify local parameters in a regression context (Brunsdon et al., 1996; Fotheringham et al., 2002). Just as local measures of spatial autocorrelation provide a statistic for each spatial unit under analysis, so does geographically weighted regression but for regression parameters. This allows for the investigation outlined above: does an independent variable matter to the same degree for all areas in the analysis? It is possible that the global regression coefficient is the same in all places, though unlikely; more realistic is that there will be a range of values with some areas not even achieving statistical significance. In fact, some criminological research has found that the signs on the coefficients switch from positive to negative, place to place (Cahill and Mulligan, 2007).
A geographically weighted regression was performed using commercial burglary as the dependent variable and the residential population as the independent variable. Of course, there are a number of other variables that should be included in a more formal analysis of the spatial patterns of commercial burglary, but this analysis is for illustrative purposes. The estimated parameter in the global regression is 0.007; commercial burglary is a low count crime type and the residential population is measured using individuals. The white areas on the map represent dissemination areas that have that a value of 0.007 +/ 2 standard errors. It should be clear that this represents a small portion of the city. The hatched areas all represent dissemination areas that have estimated parameters that are greater than the global parameter, with the grey and back areas having estimated parameters that are less than global parameter. Of particular interest are the black areas that represent the dissemination areas with estimated parameters that are less than zero. Not all of these dissemination areas have estimated parameters that are significantly different from zero, but the point is that a lot of the area of the city has results quite different from the global regression results. This set of results, yet again, shows the importance of considering spatial relationships, in general, and varying spatial relationships, specifically.
Figure 7. Commercial burglary, geographically weighted regression parameter estimates for residential population, Vancouver, Canada
Where to go from here?
This chapter has discussed the importance of GIS and spatial analysis, generally and in the context of environmental criminology. Spatial data have been shown to present a new set of obstacles and limitations to overcome, but also a set of new opportunities to test theory, develop crimerelated policy, and simply to better understand the (spatial) patterns of crime. The question now is: where do we go from here?
Part of the answer is to keep doing what we have been doing. More spatiallyreferenced data are becoming available all the time and there are still many unanswered questions and replications that need to be done. If there is any work more pressing than others, I would argue for more environmental criminologists to undertake local spatial analysis methods. There is a fair bit of research in this area, but it is relatively small to other forms of spatial analysis. Specifically, in the context of local crime clusters, there is very little research that investigates the properties of these clusters (see Andresen (2011) for an exception) that may prove to be particularly instructive. I am thinking specifically of better understanding negative local spatial autocorrelation. Another avenue of future research is the interaction of spatial and temporal analyses. This is not new to environmental criminology—see the chapter in this Handbook that discusses nearrepeat victimization, for example—but spatialtemporal statistical methods are relatively underutilized. Such methods identify clusters in both space and time (simultaneously) that occur and cannot be explained by random variation in the data.
Overall, this is an exciting time to be working in environmental criminology using GIS and (local) spatial analysis methods. The availability of spatiallyreferenced data for both criminal events and explanatory variables is only increasing and providing more opportunities to test the nuances of the theories within environmental criminology and develop crime prevention policies.
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