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Annals of GIS
ISSN: 1947-5683 (Print) 1947-5691 (Online) Journal homepage: https://www.tandfonline.com/loi/tagi20
A Bayesian spatial shared component model
for identifying crime-general and crime-specific
hotspots
Jane Law, Matthew Quick & Afraaz Jadavji
To cite this article: Jane Law, Matthew Quick & Afraaz Jadavji (2020): A Bayesian spatial shared
component model for identifying crime-general and crime-specific hotspots, Annals of GIS, DOI:
10.1080/19475683.2020.1720290
To link to this article: https://doi.org/10.1080/19475683.2020.1720290
© 2020 The Author(s). Published by Informa
UK Limited, trading as Taylor & Francis
Group, on behalf of Nanjing Normal
University.
Published online: 30 Jan 2020.
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A Bayesian spatial shared component model for identifying crime-general and
crime-specific hotspots
Jane Law
a,b
, Matthew Quick
c
and Afraaz Jadavji
b
a
School of Planning, University of Waterloo, Waterloo, Canada;
b
School of Public Health and Health Systems, University of Waterloo, Waterloo,
Canada;
c
School of Geographical Sciences and Urban Planning, Arizona State University, Tempe, AZ, USA
ABSTRACT
The spatial patterning of crime hotspots provides place-based information for the design, allocation,
and implementation of crime prevention policies and programmes. However, most spatial hotspot
identification methods are univariate,analyse a single crime type, and do not consider ifhotspots are
shared amongst multiple crime types. This study applies a Bayesian spatial shared component model
to identify crime-general and crime-specific hotspots for violent crime and property crime at the
small-area scale. The spatial shared component model jointly analyzes both violent crime and
property crime and separates the area-specific risks of each crime type into one shared component,
which captures the underlying crime-general spatial pattern common to both crime types, and one
type-specific component, which captures the crime-specific spatial pattern that diverges from the
shared pattern. Crime-general and crime-specifichotspotsareclassified based on the posterior
probability estimates of the shared and type-specific components, respectively. Results show that
the crime-general pattern explains approximately 81% of the total variation of violent crime and 70%
of the total variation of property crime. Crime-general hotspots are found to be more frequent than
crime-specific hotspots, and property crime-specific hotspots are more frequent than violent crime-
specific hotspots. Crime-general and crime-specific hotspots are areas that may be targeted with
comprehensive initiatives designed for multiple crime types or specialized initiatives designed for
a single crime type, respectively.
ARTICLE HISTORY
Received 30 August 2019
Accepted 18 January 2020
KEYWORDS
crime hotspot; spatial
pattern; Bayesian modelling;
multivariate; shared
component
Introduction
Crime offences exhibit non-random spatial patterns and
often concentrate at hotspot locations (Eck and
Weisburd 1995; Ratcliffe and McCullagh 1999; Anselin
et al. 2000). Identifying crime hotspots is central to the-
oretical development, crime prevention policy, and law
enforcement resource allocation. From a theoretical per-
spective, the spatial patterning of hotspots provides
exploratory insight into the sociodemographic and
built environment characteristics that may explain why
high and low levels of crime cluster at specific addresses,
street segments, and neighbourhoods (Sherman, Gartin,
and Buerger 1989; Hirschfield and Bowers 1997). From
a policy development and resource allocation perspec-
tive, hotspots are locations that may be suitable for
community-based crime prevention strategies, which
work to change the local social dynamics, institutions,
and organizations that influence criminal behaviour
(Herbert and Harries 1986; Tonry and Farrington 1995),
and/or geographically focused law enforcement inter-
ventions, such as hotspot policing or problem-oriented
policing, which aim to modify the places, situations, and
opportunities that facilitate crime events (Braga et al.
1999; Ratcliffe2004; Chainey, Tompson, and Uhlig
2008; Wang 2012).
The most common quantitative methods used to ana-
lyse crime hotspots are univariate and identify groups of
nearby points or areas that have high levels of one out-
come compared to other groups of points or areas that
have lower levels of the same outcome (McLafferty 2015;
Wang et al. 2017). Common hotspot methods applied to
area-based crime data include local Moran’s I, the Getis-
Ord G
istatistic, and the spatial scan statistic, for example
(Murray et al. 2001; Ratcliffe and McCullagh 1999;
Andresen 2009;NakayaandYano2010;Shiode2011;
Ceccato and Dolmen 2011; Li and Radke 2012;Wang
2012). However, because univariate hotspot identification
methods focus on only one outcome, they donot account
for the underlying data-generating processes shared
amongst multiple crime types and do not distinguish
between crime-general hotspots, or areas where there
are unusually high levels of two or more crime types,
and crime-specific hotspots, or locations with unusually
high levels of only one crime type (Knorr-Held and Best
CONTACT Jane Law jane.law@uwaterloo.ca School of Planning, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1,
Canada
ANNALS OF GIS
https://doi.org/10.1080/19475683.2020.1720290
© 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group, on behalf of Nanjing Normal University.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
2001; Quick, Li, and Law 2019; Yue et al. 2017).
Differentiating the spatial patterns of crime-general and
crime-specific hotspots advances understanding of the
possible risk factors associated with all, some, or only
one type of crime, and provides policy-relevant informa-
tion for the location, design, and implementation of com-
prehensive crime prevention initiatives that focus on
many crime types and/or specialized initiatives that
focus on a single crime type (Weisburd et al. 1993;
Haberman 2017).
This study applies a Bayesian spatial shared compo-
nent model to identify crime-general and crime-specific
hotspots of property crime and violent crime at the
small-area scale in the Regional Municipality of York,
Ontario, Canada. The spatial shared component model
jointly analyzes property crime and violent crime and
separates the area-specific risk of the crime types into
one shared component, which captures the crime-
general spatial pattern, and two type-specific compo-
nents, which capture the residual crime-specific spatial
patterns. The crime-general pattern represents the set of
processes simultaneously associated with multiple crime
types and the crime-specific patterns represent the set
of processes associated with only one type. Crime-
general and crime-specific hotspots are identified
based on the posterior probabilities of the area-specific
estimates of the shared and type-specific components
having high relative risk, respectively.
Following this introduction, the ecological theories
used to explain the spatial clustering of violent crime
and property crime are reviewed and the univariate and
multivariate methods used to identify crime hotspots in
past research are compared. Next, the crime data and
the shared component model are detailed, and the
results of this study are presented and discussed as
they relate to advancing the spatial analysis of multi-
variate data and informing place-based crime preven-
tion policy. In conclusion, the limitations of this research
are discussed and directions for future study are
recommended.
Theories explaining spatial crime clusters
Broadly, three ecological theories have been used to
explain the spatial clustering of crime at the local
scale: social disorganization theory, routine activity
theory, and crime pattern theory. Social disorganiza-
tion theory hypothesizes that high levels of crime are
due to low informal social control amongst commu-
nity members (Shaw and McKay 1942). Informal social
control is defined as the capacity of residents to rea-
lize common values and mobilize to control criminal
behaviour, and low informal social control within an
area is often characterized by high levels of residential
instability, socioeconomic disadvantage, and ethnic
heterogeneity (Sampson and Groves 1989). Widely
applied at the small-area and neighbourhood scales,
past research has shown that social disorganization
theory helps to explain the spatial clustering of many
crime types including violent crime and property
crime (Andresen 2006; Chamberlain and Hipp 2015).
Focusing specifically on the situations in which
offences occur, routine activity theory proposes that
crime offences result from the convergence of moti-
vated offenders, suitable targets, and a lack of capable
guardianship in space and time (Cohen and Felson
1979). Routine activity theory research has shown that
many crime types are associated with the same set of
geographically-varying characteristics (Felson 2008). For
example, Andresen (2006) observes that the per cent of
young people and the unemployment rate, which is an
indicator of motivated offenders, were positively asso-
ciated with property crimes (automotive theft and break
and enter) and violent crime at the small-area scale.
Likewise, Roncek and Maier (1991) show that bars and
taverns, which are places thought to bring together
motivated offenders and suitable targets with a lack of
capable guardianship, were associated with higher
levels of both property and violent crimes.
Building on the tenets of routine activity theory,
crime pattern theory highlights how the interactions
between offender activity patterns and features of
the built environment lead to clusters of high and
low crime at specific locations (Brantingham and
Brantingham 2010). Crime pattern theory proposes
that the spatial patterning of crime offencesissimul-
taneously influenced by activity nodes, activity paths,
and the environmental backcloth (Brantingham and
Brantingham 1993); activity nodes are locations used
by large populations for routine activities, such as
employment, education, or shopping; activity paths
are the routes between activity nodes, such as road
networks or transit routes; and the environmental
backcloth is composed of the broader social, political,
and physical contexts in which activity nodes and
paths are located (Deryol et al. 2016). While past
research has shown that high traffic activity nodes
and paths tend to have higher levels of crime than
low traffic nodes and paths (Wilcox and Eck 2011),
there are differences as to how specificcrimetypes
cluster in the urban environment. For example,
Kinney et al. (2008) observe that both property
crime (motor vehicle thefts) and violent crime
(assaults) cluster around regional shopping centres
whereas only property crimes tend to cluster around
schools and universities.
2J. LAW ET AL.
Crime-general and crime-specific hotspots
Social disorganization, routine activity, and crime pat-
tern theories provide the theoretical context for dis-
tinguishing between crime-general and crime-specific
hotspots as applied to violent crime and property
crime. Focusing first on crime-general hotspots, or
areas where high levels of multiple crime types are
co-located, the clustering of both violent crime and
property crime may occur in areas with low informal
social control. This aligns with past research showing
that the processes hypothesized by social disorgani-
zation theory are associated with a number of violent
and non-violent crime types (Chamberlain and Hipp
2015; Krivo and Peterson 1996). Routine activity the-
ory and crime pattern theory also support the pre-
sence of crime-general hotspots as property crime
and violent crime types have been shown to co-
cluster around many of the same built environment
features. For example, areas located in and around
a central business district may be hotspots for prop-
erty crime and violent crime because there are high
densities of commercial land uses with material
goods, which can be interpreted as potential prop-
erty crime targets, as well as high densities of people
engaged in employment and leisure activities, which
can be interpreted as potential violent crime targets
(Nelson, Bromley, and Thomas 2001).
Focusing on crime-specific hotspots, routine activity
theory, in particular, highlights how the distribution of
crime targets may lead to some locations being hot-
spots for only one crime type. For example, routine
activity theory contends that property crime-specific
hotspots may be located in non-residential areas that
have many attractive targets, such as in shopping dis-
tricts where there are many stores with physical goods
suitable for theft offences but relatively fewer features
likely to facilitate aggressive and violent behaviours
(LaGrange 1999;Quick,Li,andLaw2019). Violent
crime-specific hotspots, in contrast, may be more likely
to occur in residential neighbourhoods that have low
informal social control, weak social ties, and few attrac-
tive property crime targets. This is supported by differ-
ential opportunity theory, which hypothesizes that
between-resident social ties and neighbourhood socio-
economic characteristics influence crime type compo-
sition and crime frequency (Coward and Ohlin 1960;
Schreck, McGloin, and Kirk 2009), as well as past social
disorganization research suggesting that informal
social control has a stronger influence on violent
crime than property crime (Ouimet 2000; Haberman
2017; Sampson, Raudenbush, and Earls 1997).
Methods for analysing spatial crime hotspots
The most common quantitative methods used to iden-
tify spatial crime clusters or hotspots for small-area data
are univariate testing-based approaches that analyse
a single outcome or crime type, assume a null hypoth-
esis of a random spatial distribution (or no spatial
autocorrelation), and identify clusters or hotspots as
groups of nearby points or areas that exhibit positive
spatial autocorrelation of high crime (Anselin et al.
2000). Perhaps the most frequently applied method to
intra-urban crime data in academic research, the local
Moran’s I statistic calculates a measure of spatial auto-
correlation for a given area and its nearby areas (as
specified in an adjacency matrix) and evaluates statis-
tical significance by comparing the local autocorrela-
tion measure to a reference distribution created
through random permutations of the study region
data (Anselin 1995). Like local Moran’s I, the Getis-Ord
G
istatistic calculates a local test statistic for a target
area and its nearby areas, and compares this observed
value to the expected value calculated based on the
distribution of the variable for the study region (Getis
and Ord 1992). Hotspots identified via the G
istatistic
are areas where the difference between the local value
and the expected local value are large and statistically
significant. Both local Moran’sIandtheG
istatistic have
been applied to aggregate crime categories, specifically
violent crime and property crime, as well as crime sub-
types, including homicide, assault, robbery, burglary,
theft, auto theft, and drug crimes (Ratcliffeand
McCullagh 1999; Malleson and Andresen 2015;Quick
and Law 2013; Frazier, Bagchi-Sen, and Knight 2013;
Haberman 2017; Cohen and Tita 1999; Kennedy,
Caplan, and Piza 2011; Li and Radke 2012).
The third cluster detection method used in
research to identify local hotspots of one crime
type is the spatial scan statistic. The spatial scan
statistic uses a scanning window that moves across
point locations or small-area centroids to calculate
a local test statistic based on the observed and
expected crime risks inside of the scanning window
at each location. The scanning window changes in
size (i.e. increasing in radius up to a constraint) and
identifies hotspots as the groups of small-areas or
points inside of scanning windows that have high
observed to expected ratios (Kulldorff1997). Like
local Moran’sIandtheGetis-OrdG
istatistic, the
spatial scan statistic has been applied to analyse
a number of different crime types, such as robberies,
drug offences, burglaries, and auto theft (Quick and
Law 2013;Shiode2011).
ANNALS OF GIS 3
Multivariate cluster detection
Studies that apply the aforementioned univariate cluster
detection methods to explore the similarities and differ-
ences between multiple crime types generally compare
the locations, shapes, and sizes of hotspots identified
from separate analyses. For example, Haberman (2017)
applies three univariate hotspot identification methods
to eleven crime types and, for each of the methods,
quantifies the proportion of street intersections classi-
fied as hotspots for one or more crime types. Analysing
each crime type separately, however, does not quantify
the degree to which multiple crime types are located in
the same or nearby locations (Mohan et al. 2011). Five
multivariate methods have been proposed to explore
the local co-clustering of two or more crime types: the
local co-location quotient, the spatial point pattern test,
the bivariate local Moran’s I, the bivariate and multivari-
ate join count statistics, and principal component
analysis.
Given two point datasets (e.g. crime type aand crime
type b), the local co-location quotient (LCLQ) quantifies
the degree to which these patterns are clustered in similar
locations by calculating the ratio between the number of
type apoints located within a fixed or adaptive band-
width distance from a type bpoint and the number of
type apoints within the same bandwidth distance
(Cromley, Hanink, and Bentley 2014). The LCLQ allows
for spatial non-stationarity in the global co-location quo-
tient proposed by Leslie and Kroenfeld (2011). Applied to
analyse the co-location of crime and land use features,
Wang et al. (2017)find that residential burglaries are
significantly clustered near to entertainment facilities
that are located close to, but not in, the CBD in Jiangsu
Province, China, and Yue et al. (2017) show that e-bike
thefts tend to be co-located with stores, banks, restau-
rants, and governmental facilities, but not industrial
plants, in Wuhan, China. Because the LCLQ is designed
to investigate the co-location of point-level data, how-
ever, it is not suitable for identifying crime-general hot-
spots for aggregated count or rate data at the area-level
(Cromley, Hanink, and Bentley 2014).
The second method used to assess the co-location of
two crime types is the spatial point pattern test. The
spatial point pattern test measures the similarity of two
geographically referenced point datasets by sampling
a subset of points from one dataset (i.e. one crime
type), simulating a distribution of crime counts within
each area, and calculating the percent of areas for which
the second dataset (i.e. the area-specific counts of
a second crime type) falls within the confidence intervals
generated from the sampled data (Andresen 2009;
Malleson and Andresen 2015). The spatial point pattern
test provides global information as to the overall simila-
rities of two spatial patterns as well as local information
that enables the identification of areas that have similar
or dissimilar levels of the two crime datasets. Comparing
and contrasting a number of crime types, including
aggregated crime categories and specific subtypes,
Andresen and Linning (2012) use the spatial point pat-
tern test to show that some crime types have similar
spatial patterns, such as commercial robbery and indivi-
dual robbery, while other crime types do not have simi-
lar patterns, such as vehicle theft and robbery.
Extending the univariate local Moran’s I to two vari-
ables, bivariate local Moran’s I captures the relationship
between the one variable in a given area and the spatial
lag of a second variable (i.e. the average of the second
variable in adjacent areas) (Lee 2001; Anselin 2019). Co-
clusters of the two variables are identified as the areas
with high levels of the first variable and the high levels of
the spatially lagged second variable (Wang et al. 2017).
The bivariate local Moran’s I has been infrequently
applied in past crime research, perhaps because it does
not account for the in-place (or within-area) correlation
structures and overestimates the degree to which two
variables are co-located (Anselin 2019). Whereas the
bivariate local Moran’s I accommodate continuous vari-
ables measured within areas (i.e. crime rate), bivariate
and multivariate joint count statistics have been pro-
posed by Anselin and Xun (2019) to assess the co-
clustering of two or more binary variables for data that
exhibits in-place co-location, or where both variables
can have a value of one in a given area, and data that
does not exhibit in-place co-location, or where only one
of the variables can have a value of one in a given area.
Principal component analysis (PCA) has also been
applied to explore the co-location patterns of multiple
crime types. PCA is often used to reduce the dimension-
ality of multivariate datasets and to create principal
components, which are combinations of the two or
more observed variables, that are not correlated and
that maximize the variation of the observed variables.
Pope and Song (2015) use PCA to examine the spatial
patterning of 30 crime types at the small-area level,
creating four principal components that represent con-
traband crimes, violent crimes, property crimes, and
theft-related crimes. Conventional PCA methods, how-
ever, do not account for the spatial autocorrelation of
principal components between nearby areas.
Multivariate spatial modelling
The above univariate and multivariate methods are test-
ing-based insofar as they quantify the local clustering of
4J. LAW ET AL.
one dataset or the local co-clustering of two or more
datasets, but do not provide insight into the observed
(i.e. covariates) and/or latent (i.e. random effects terms)
data-generating processes (Robertson et al. 2010). For
example, while the LCLQ helps to identify locations
where two crime types are co-located, this method
does not consider how the patterning of each crime
type is simultaneously explained by crime-general pro-
cesses, as hypothesized by social disorganization and
routine activity theories, or type-specific processes, as
suggested by routine activity and differential opportu-
nity theory (see Crime-general and crime-specific hot-
spots). Multivariate model-based approaches, in
contrast, provide a framework for formalizing, testing,
and estimating the spatial and/or non-spatial data-
generating processes common to multiple variables
(Knorr-Held and Best 2001; Held et al. 2005).
Recognizing that the local patterning of many crime
types are similar, recent studies have used multivariate
models to examine the ways in which multiple crime
types are correlated over space and space-time. For
example, Liu and Zhu (2017) and Chung and Kim
(2018) use Bayesian models featuring the multivariate
conditional autoregressive prior distribution to capture
the correlations between two and three crime types,
respectively. Both of these studies observe that account-
ing for the between-crime type correlation structures
improves model fit compared to separate type-specific
analyses. Identifying crime-general spatial patterns
using a shared component modelling approach similar
to the method applied in this study, Quick, Li, and
Brunton-Smith (2018) show that burglary, robbery, vehi-
cle crime, and violent crime share two distinct crime-
general patterns, one that is common to all crime types
and a second that is common to the theft-related crimes,
and Quick, Li, and Law (2019) show that physical disor-
der, social disorder, property crime, and violent crime
share a crime-general spatial pattern and a crime-
general time trend.
Importantly, past studies applying multivariate mod-
elling approaches to local crime data do not consider
how these methods can be used to identify crime-
general and crime-specific hotspots. In model-based
methods, hotspots are often classified based on area-
specific risk estimates and can be applied to specific
model parameters via map decomposition (Richardson
et al. 2004; Haining, Law, and Griffith 2009), whereas
hotspots classified in testing-based approaches are
based on rejecting a null hypothesis of no local spatial
autocorrelation. Hotspot analysis methods are important
for applications to crime prevention policies and pro-
grams because they provide location-specific informa-
tion about where crime is highest and lowest (Chainey,
Tompson, and Uhlig 2008). Multivariate hotspot analyti-
cal methods, in particular, have the potential to identify
and differentiate between areas that should be targeted
with comprehensive crime prevention initiatives
designed for multiple crime types and areas that should
be targeted with specialized initiatives designed for only
one crime type (Weisburd et al. 1993; Haberman 2017).
Data and methods
Study region
The Regional Municipality of York is located north and
adjacent to Toronto, Ontario, which is Canada’slargest
city. In 2006, the study region had a population of
892,712, with nearly 95% of residents living in urban centres
such as the cities of Markham, Vaughan, Richmond Hill, and
Newmarket. The geographic unit of analysis for this study
was the dissemination area (DA). DAs are the set of smallest
census areas that cover the entirety of Canada and are
delineated such that residential populations are between
400 and 800. The geographical boundaries and population
data used in this study were retrieved from the 2006
Statistics Canada census. In total, York Region was com-
posed of 1,128 DAs that had an average population of 791
and an average area of 1.90 km
2
.
Crime data
Property crime and violent crime offence data for 2006 and
2007 were provided by the York Regional Police (YRP).
Crime types were classified using Uniform Crime
Reporting (UCR) survey codes; violent crimes included
offences such as assault, sexual assault, and murder (UCR
codes between 1000 and 2000), and property crimes
included offences such as break and enter, motor vehicle
theft, theft over $5,000, and theft under $5,000 (UCR codes
between 2000 and 3000). The location of each crime
offence was provided by YRP as a street address and each
address was geocoded in ArcGIS 10.0 with a 98% match
rate. This exceeds the minimum acceptable match rate of
85% proposed by Ratcliffe(2004). After geocoding, the
number of property crimes and violent crimes were
summed within each DA. The total counts of each crime
type were obtained by summing the 2006 and 2007 counts.
The standardized ratios of violent crime and property
crime are mapped in Figure 1 (see Appendix A for
descriptive statistics). Area-specific standardized ratios
were calculated as the observed crime counts within
each DA divided by the expected crime counts within
each DA. Expected counts were calculated as the overall
property crime or violent crime rate for the study region
multiplied by the residential population within each DA.
ANNALS OF GIS 5
Therefore, the expected crime counts account for the
level of crime that would occur in a DA if the crime was
distributed proportional to the residential population
and control for the different total counts of property
and violent crime in the study region. While there are
many possible measures of the population at risk, such
as daytime/evening populations for violent crimes or the
number of theft targets for property crimes, residential
population was the only data available for entirety of the
study region at the DA scale. Note that conceptualizing
and measuring the population at risk is challenging
because offenders and targets are mobile over space
and time and because quantitative estimates of these
moving populations are often not available or accurately
inferred using existing data (Kikuchi, Amemiya, and
Shimada 2012; Zhang, Suresh, and Qiu 2012).
Exploring the maps of the standardized ratios of prop-
ertycrimeandviolentcrimeshowsthatbothcrimetypes
had similar spatial patterns: areas with high-standardized
ratios (>2) were located in the southwest and southeast of
the study region, and areas with low-standardized ratios
(<1) were located along the eastern boundary. The simi-
larity of these two patterns is supported by a positive
pairwise correlation between the standardized ratios of
thetwocrimetypes(Pearson’sr= 0.71). Contrasting these
two patterns, areas located in the north and northwest
appear to have higher levels of violent crime than prop-
erty crime whereas only a few groups of areas have higher
levels of property crime than violent crime.
Multivariate spatial modelling
The shared component model used to identify crime-
general and crime-specific patterns and hotspots are
composed of Equations (1)–(4). Let Oik denote the
observed crime counts, where iindexes areas (= 1, . . .,
1128) and kindexes crime type (= 1, 2). For reference, Oi1
is the observed violent crime count in area iand Oi2is
the observed property crime count in area i. The crime
counts in each area are assumed to be independent
Poisson random variables conditional on means μi1and
μi2, respectively (Equation (1)). The Poisson model is
often used in Bayesian spatial modelling of count data
at the small-area scale, where overdispersion and spatial
autocorrelation are accounted for via random effects
terms (Besag, York, and Mollie 1991; Richardson et al.
2004; Haining, Law, and Griffith 2009). In Equation (2),
the Poisson means (on the log scale) are modelled as the
product of the type-specific expected crime counts (Eik )
and the type-specific relative risks (rik). The type-specific
expected crime counts are known quantities (as
described in Crime data) and the type-specific relative
risks, which are conceptually similar to the standardized
ratios (Figure 1), are unknown quantities and are esti-
mated in Equations (3) and (4).
Oik,Poisson μik
ðÞ (1)
log μik
ðÞ¼log Eik
ðÞlogðrikÞ(2)
Equation (3) estimates the relative risk of violent crime as
the sum of the expected violent crime counts (log Ei1
ðÞ),
a type-specific intercept (α1), a spatial shared compo-
nent (δθi), a set of type-specific spatially structured ran-
dom effects terms (si1), and a set of type-specific
unstructured random effects terms (ui1). Equation (4)
estimates the relative risk of property crime as the sum
of the expected property crime counts (log Ei2
ðÞ), a type-
specific intercept (α2), a spatial shared component
Figure 1. Standardized ratios of violent crime and property crime. The map insets show the predominately urban areas located in the
south of the study region. Standardized ratios greater than one indicate that the observed crime counts were greater than expected
and standardized ratios less than one indicate that the observed crime counts were less than expected.
6J. LAW ET AL.
((1=δÞθiÞ, a set of type-specific spatially structured ran-
dom effects terms (si2), and a set of type-specific unstruc-
tured random effects terms (ui2) (Knorr-Held and Best
2001; Held et al. 2005). In Equations (3) and (4), the
intercepts capture the average crime risks for the study
region and the shared components capture the correla-
tions between violent crime and property crime. The
sum of sik and uik captures the type-specific spatial pat-
terns that depart from the crime-general pattern and
represent latent processes associated with only violent
crime (Equation (3)) or only property crime (Equation
(4)). In particular, the type-specific spatially structured
random effects terms (sik) account for between-area
spatial autocorrelation and risk that is explained by spa-
tial processes amongst groups of nearby DAs and the
type-specific unstructured random effects terms (uik )
account for overdispersion and risk that is explained by
within-area non-spatial processes (Besag, York, and
Mollie 1991).
log μi1
ðÞ¼log Ei1
ðÞþα1þδθiþsi1þui1(3)
log μi2
ðÞ¼log Ei2
ðÞþα2þð1=δÞθiþsi2þui2(4)
The shared components in Equations (3) and (4) are com-
posed of a scaling parameter (δor 1=δ) and a set of shared
spatially structured random effects terms (θi). The shared
random effects terms capture the crime-general spatial
pattern common to both crime types and the scaling
parameters allow each crime type to have a unique asso-
ciation with the crime-general spatial pattern. The shared
component assumes that violent crime and property crime
are associated with one or more of the same data-
generating processes, which is supported by the crime-
general mechanisms highlighted by social disorganization
and routine activity theories, the positive correlation
between violent crime and property crime (see Crime
data), and the visual similarities observed in Figure 1.
For interpretation of the shared component, a value
of δclose to one indicates that violent crime and prop-
erty crime have a similar magnitude of association with
the crime-general pattern (i.e. if δ= 1, then 1/δ=1)
whereas a large positive value of δindicates that violent
crime has a stronger association with the crime-general
pattern than property crime (e.g. if δ= 2, then 1/δ= 0.5).
Areas with high risk due to the crime-general pattern will
have estimates of exp(θi) >1 at the 95% credible interval
(95% CI)
1
and areas with low risk due to the crime-
general pattern will have estimates of exp(θi) <1 at the
95 CI%. Likewise, areas with high risk due to the crime-
specific patterns will have estimates of exp(sik þuik )>1
at 95% CI and areas with low risk due to the crime-
specific patterns will have estimates of exp(sik þuik )<1
at 95% CI.
Hotspot identification
In Bayesian statistical models, all unknown para-
meters are estimated as probability distributions
(i.e. αk;δ;θi;sik;and uik)inEquations(3)and(4).
A common hotspot identification approach in the
Bayesian modelling framework uses these distribu-
tions to quantify the probability that one or more
model parameters exceed a researcher-specified
threshold. This is referred to as the posterior prob-
ability (Richardson et al. 2004). Following past stu-
dies using Bayesian modelling approaches to
identify local crime hotspots, crime-general hotspots
were evaluated based on the posterior probability of
the shared random effects being greater than one
(Pr(exp(θi) > 1 | data)) and crime-specific hotspots
were evaluated based on the posterior probability of
the type-specificrandomeffects being greater than
one (Pr(exp(sik þuik)>1|data))(Law,Quick,and
Chan 2014,2015;Lietal.2014). Unlike univariate
and multivariate testing-based hotspot identification
methods, this shared component approach accounts
for uncertainty in the area-specific risk estimates and
allows for hotspots to be classified based on differ-
ent thresholds that may reflect strategic priorities,
crime prevention capabilities, and resources avail-
ability, for example.
In this study, three thresholds were used to identify
and rank crime-general and crime-specific hotspots: 0.8,
0.9, and 0.95 (Richardson et al. 2004). Higher threshold
values represent stronger evidence that an area is
a hotspot. For example, if the risk due to the crime-
general spatial pattern is very high in area i(i.e. a large
proportion of the posterior distribution of exp(θi)is
greater than one), then Pr(exp(θi) > 1 | data) will likely
be greater than 0.8 and this area will be classified as
a crime-general hotspot. In contrast, if the risk due to
one of the type-specific components is close to one (i.e.
approximately 50% of the posterior distribution of exp
(sik þuik) is less than one), then Pr(exp(sik þuik)>1|
data) will be close to 0.5 and this area will not be
classified as a crime-specific hotspot.
Prior distributions and model fitting
All prior distributions specified for this modelling
approach are detailed in Appendix B, however, two prior
distributions are of note. First, each set of spatially struc-
tured random effects terms (θi,si1,andsi2) were assigned
an intrinsic conditional autoregressive prior distribution
(ICAR) with an unknown common variance (σ2
θ,σ2
s1,and
σ2
s2). The ICAR prior imposes a spatially smoothed risk
surface and captures spatial autocorrelation between
ANNALS OF GIS 7
adjacent areas as specified by a queen-contiguity adja-
cency matrix (Besag, York, and Mollie 1991). Second, the
logarithm of the scaling parameter (i.e. log(δ)) was
assigned a normal distribution with a mean of 0 and
a variance of 0.17. This prior assumes that both δand 1/
δare positive, which is supported by the positive correla-
tions between the two crime types (see Crime data). This
prior also assumes that the ratio of the two scaling para-
meters (i.e. δ/(1/δ)) is between 0.2 and 5 with 95% prob-
ability (Knorr-Held and Best 2001). This was confirmed
based on the results showing that this ratio was greater
than 0.2 and less than 5 at the 95% CI. Note that estimat-
ing one scaling parameter (δ) for one outcome variable
and assigning the inverse to the second outcome variable
improves model identifiability compared to estimating
separate scaling parameters for each outcome (Lawson
2009; Knorr-Held and Best 2001).
The Bayesian multivariate shared component model,
which was composed of Equations (1)–(4), was fit using
the Markov chain Monte Carlo (MCMC) algorithm in
WinBUGS v.1.4.3 (Lunn et al. 2000). Two MCMC chains
were initiated at dispersed starting values and the con-
vergence of model parameters was monitored by trace
plots and Gelman–Rubin statistics. For each chain,
20,000 iterations were discarded as burn-in and an addi-
tional 20,000 iterations, thinned by 10 to reduce auto-
correlation of the MCMC samples, were retained for
posterior inference.
Results
Table 1 shows the results of the multivariate shared
component model. The scaling parameters, which quan-
tify the relative association between each crime type and
the crime-general spatial pattern (the set of shared spa-
tially structured random effects terms), were both found
to be close to one. This indicates that violent crime and
property crime had a relatively similar influence on the
underlying crime-general pattern. One explanation for
this is that both property crime and violent crime were
analysed as standardized ratios and had similar scales
centred near one (see Appendix A). If the crimes were
analysed as counts without expected counts used as
offsets or as rates with a common population at risk or
denominator, for example, then the more frequent crime
type could have a larger scaling parameter and a greater
relative influence on the crime-general pattern (Quick, Li,
and Law 2019).
The magnitude of the empirical variances of the two
sets of type-specific random-effects terms helps to
understand the relative importance of the spatial and
non-spatial processes for understanding the crime-
specific spatial patterns. For both violent crime and
property crime, the largest empirical variances were
attributed to the shared component (Table 1). This indi-
cates that the crime-general spatial pattern was rela-
tively more important for understanding the overall
distribution of both crime types than the crime-specific
patterns. Of the two type-specific components, the
empirical variance of the unstructured random effects
terms was larger than the empirical variance of the
spatially structured random effects terms for both
crime types (Table 1). Note that when variance terms
are estimated to be near to zero, the corresponding
random effects parameters can be interpreted as having
little explanatory importance. This suggests that there
was little spatial autocorrelation exhibited by either
crime-specific pattern after accounting for the crime-
general spatial pattern.
Variance partition coefficients (VPC) provide an alter-
native approach to understanding how the crime-
general and crime-specific patterns explain the overall
patterning of violent crime and property crime
(Goldstein, Browne, and Rasbash 2002). For example,
the VPC quantifying the proportion of total variation of
violent crime explained by the crime-general pattern is
equal to the empirical variance of δθidivided by the sum
of the empirical variances of δθi,si1, and ui1. Likewise, the
VPC quantifying the proportion of total variation of
property crime explained by the crime-general pattern
is equal to the empirical variance of 1=δðÞθidivided by
the sum of the empirical variances of 1=δðÞθi,s2, and ui2.
In this study, the shared component representing the
crime-general spatial pattern captured 81% (95% CI:
74–89%) of the total variation of violent crime and 70%
(95% CI: 64–77%) of the total variation of property crime.
In contrast, the type-specific components explained less
than 30% of the total variation of both crime types, with
Table 1. Results of the multivariate shared component model. The posterior medians of model
parameters are shown with 95% CI’s in parentheses.
Violent crime Property crime
Intercept (exp(αk)) 0.71 (0.68, 0.73) 0.61 (0.59, 0.63)
Scaling parameter 1.02 (0.96, 1.08) 0.98 (0.93, 1.04)
Empirical variances of random effects terms
Shared component 0.56 (0.48, 0.65) 0.52 (0.44, 0.60)
Type-specific spatially structured component 0.01 (0.001, 0.05) 0.01 (0.002, 0.05)
Type-specific unstructured component 0.11 (0.07, 0.16) 0.21 (0.16, 0.25)
8J. LAW ET AL.
the unstructured type-specific random effects terms
accounting for more than 90% percent of this type-
specific variability.
Crime-general and crime-specific patterns and
hotspots
Figure 2 maps the crime-general spatial pattern and the
two crime-specific spatial patterns. The crime-general
pattern captures the underlying risk surface common
to both violent crime and property crime and the crime-
specific patterns capture the risk for each type that
diverges from the crime-general pattern. Visually, the
crime-general pattern is representative of the similarities
between the violent crime and property crime patterns
shown in Figure 1, with areas of high risk concentrated in
the southwest and in the north of the study region.
Focusing on the crime-specific patterns, violent crime
and property crime had a relatively similar number of
areas with a high relative risk (exp(sik þuik) > 1), with 506
areas for violent crime and 518 areas for property crime.
Property crime exhibited greater variation of type-
specific relative risks, with a minimum estimate of 0.46
(95% CI: 0.22–0.93) and a maximum estimate of 6.60
(95% CI: 1.00–10.82). This compares with violent crime,
which had a minimum type-specific risk estimate of 0.61
(95% CI: 033–1.10) and a maximum of 2.49 (95% CI: 1.53,
4.20). Visually, areas with high violent crime-specific risk
were dispersed throughout the study region whereas
areas with high property crime-specific risk were clus-
tered primarily in the southwest.
Crime-general and crime-specific hotspots are visua-
lized in Figure 3. Of the three hotspot types, crime-
general hotspots were the most frequent (34.8% of all
areas), followed by property crime-specific hotspots
(15.5%) and violent crime-specific hotspots (9.6%).
There were also a number of areas that had overlapping
hotspot classifications; approximately 10% of areas were
both crime-general and property crime hotspots, 7% of
areas were both crime-general and violent crime hot-
spots, and 15 areas (1.3% of all areas) were identified as
crime-general, violent crime-specific, and property
crime-specific hotspots (Figure 3).
Geographically, the crime-general hotspots were
often grouped in relatively large clusters, such as the
areas located in the north, in the southwest, and in
central areas of the study region (all with (Pr(exp(θi)>
1 | data) ≥0.95)). This can be explained by the shared
random effects terms being specified as a set of spatially
structured random effects terms that borrow strength
from adjacent areas to estimate a spatially smoothed risk
surface (see Prior distributions and model fitting). In com-
parison, violent crime-specific and property crime-
specific hotspots were typically composed of only
a small number of areas and were scattered throughout
the study region. This supports the results indicating
that the type-specific unstructured random effects
accounted for a larger proportion of variation than the
Figure 2. Crime-general (exp(θi)) and crime-specific spatial patterns (exp(sik þuik)).
ANNALS OF GIS 9
type-specific spatially structured random effects, and
accordingly, this study suggests that the non-spatial
within-area processes influencing only property crime
or only violent crime were relatively more important
than the type-specific spatial processes shared amongst
nearby areas (Table 1). Because this study controlled for
the different total counts of violent crime and property
crime via the expected crime counts (Equation (2)), the
larger number of property crime-specific hotspots can
be attributed to the shared component explaining
a smaller proportion of variation of property crime
(70% compared to 81% for violent crime) and, therefore,
a larger number of areas having unusually high (and low)
property crime-specific risks.
Discussion
This study has applied a Bayesian spatial shared com-
ponent model to identify crime-general and crime-
specific hotspots for property crime and violent crime
at the small-area scale. The multivariate shared compo-
nent model used in this paper jointly analyzes two
crime types and separates the area-specificrisksofthe
crime types into a shared component, which captures
the crime-general spatial pattern, and type-specific
components, which capture the crime-specificspatial
patterns that diverge from the crime-general pattern.
Crime-general and crime-specifichotspotswere
classified using the posterior probabilities of the shared
and type-specific components. This study found that
both violent crime and property crime had similar asso-
ciations with the underlying crime-general pattern, that
the crime-general pattern explained the largest propor-
tion of variation for both crime types, and that crime-
general hotspots were more frequent than both types
of crime-specific hotspots.
The shared component modelling approach illu-
strated in this study provides a framework for quantify-
ing the spatial and non-spatial correlation structures
between multiple variables and for differentiating the
data-generating processes that are crime-general, or
shared amongst multiple crime types, and crime-
specific, or unique to each crime type. This contrasts
with existing univariate and multivariate cluster detec-
tion methods, which test for local spatial autocorrelation
of one variable or more than one variable but do not
provide insight into if and how the processes associated
with crime patterns are crime-general and/or crime-
specific. For example, compared to the separate applica-
tion of univariate cluster detection methods to each
crime type and classifying crime-general hotspots as
the overlapping areas identified as clusters for both
property crime and violent crime, the shared component
model applied in this study estimates the crime-general
relative risk for all-areas in the study region; allows for
the crime-general pattern to be spatially autocorrelated
Figure 3. Crime-general hotspots and crime-specific hotspots. The areas outlined in red were classified as crime-general, violent crime-
specific, and property crime-specific hotspots at the 0.80 threshold.
10 J. LAW ET AL.
and to be differentially influenced by each crime type;
uses the crime-general relative risk estimates to identify
crime-general hotspots; and enables researchers and
analysts to classify hotspots based on multiple different
thresholds after accounting for uncertainty of the area-
specific risk estimates.
From a theoretical perspective, analysing crime-general
and crime-specific hotspots strengthens inference regard-
ing how the urban environment influences multiple crime
types simultaneously. The results of this study suggest that
the spatial patterns and hotspots of violent crime and
property crime arise from similar data-generating pro-
cesses at the small-area scale, as shown by the shared
component explaining the largest proportion of variation
of both crime types and by the crime-general hotspots
being more frequent than the type-specific hotspots.
While this has been observed in past social disorganization
and routine activity research that compares the results of
many univariate analyses (Ceccato, Haining, and Signoretta
2002;Andresen2006; Chamberlain and Hipp 2015), this
study provides quantitative evidence that the local pat-
terns of violent crime and property crime were relatively
more similar than different at the study region scale. Note
that, while including risk factors is not common in hotspot
analyses, this multivariate shared component model can be
extended to include covariates (Quick, Li, and Brunton-
Smith 2018;Quick,Li,andLaw2019) and future research
focused on explaining the differences between crime-
general and crime-specific patterns should look to include
covariates that operationalize social disorganization, rou-
tine activity, and crime pattern theories, and examine how
these risk factors influence the location of crime-general
and crime-specific hotspots.
Classifying crime-general and crime-specifichotspots
via a Bayesian spatial shared component model also pro-
vides information relevant to the design, allocation, and
implementation of crime prevention policies and pro-
grams at both the study region and small-area scales.
Because crime-general hotspots were found to be more
common than both types of crime-specific hotspots, the
results of this study suggest that law enforcement
resources at the study region scale be focused on com-
prehensive policies and programs that address the geo-
graphically situated processes associated with criminal
behaviour, broadly defined, rather than the geographi-
cally situated processes associated with only one crime
type. Comprehensive interventions may look to prioritize
addressing the factors common to multiple crime types,
specifically informal social control, as hypothesized in
social disorganization theory, and the convergence of
motivated offenders, suitable targets, and a lack of cap-
able guardianship, as outlined in routine activity theory.
Groff(2015) draws parallels between these two concepts
and suggests that interventions focused on changing
offender perceptions of what constitutes appropriate
behaviour in a given place may be effective for overall
crime reduction. A focus on comprehensive interventions
at the study region scale is also supported by the finding
that a majority of crime-general clusters had very strong
evidence of being a hotspot, with 60% of all crime-
general hotspots having posterior probability estimates
greater than 0.95 (Figure 3).
At the local scale, the 15 areas concurrently identified
as crime-general hotspots, property crime-specific hot-
spots, and violent crime-specific hotspots are areas
where law enforcement may look to initiate comprehen-
sive programs focused on all crime types. One example
may be a community consultation programme that
brings together law enforcement and residents to iden-
tify issues of local importance and increase participation
in crime prevention. These types of community-centred
initiatives may also help to uncover why high levels of
property crime and high levels of violent crime are cor-
related and strengthen resident-based informal social
control, which has been shown to be associated with
many different crime types (Braga and Schnell 2013).
Areas classified as only property crime-specific or violent
crime-specific hotspots, on the other hand, maybe sui-
table for interventions targeted to type-specific beha-
viours and contexts. One approach is situational crime
prevention, which attempts to prevent and deter crime
through reducing the physical opportunities for offend-
ing and increasing the likelihood that an offender will be
caught by the police (Clarke 1980). For example, target
hardening initiatives that increase security and surveil-
lance of material goods in stores or homes in areas
identified as property crime-specific hotspots may be
effective at reducing opportunistic property crime
offending in these areas, but may be less effective at
reducing aggressive and violent behaviours in areas
classified as violent crime-specific hotspots (Herbert
and Harries 1986; Clarke 1997).
Limitations and future research
One limitation of this study is that the crime data does
not account for crime events that are unreported to
police or all spatial dimensions of criminal behaviour.
For example, it is possible that datasets measuring
crime calls-for-service or crime victimization, or datasets
representing offender residences or crime harms may
exhibit different correlation structures and different
shared patterns than those identified in this study (Curtis-
Ham and Walton 2017). Future work may look to inte-
grate two or more of these datasets in a multivariate
shared component model, explore the similarities and
ANNALS OF GIS 11
differences amongst the different types of crime data,
and identify locations that have high risks of offences,
offenders, and harms.
A second limitation of this study is that only property
crimeandviolentcrimetypesweremadeavailabletothe
researchers by YRP, yet past research has shown that specific
crime subtypes exhibit different spatial patterns than these
two broader categories (Andresen and Linning 2012). As
such, the results of this study may not reflect the spatial
patterns and hotspots identified when analysing more spe-
cific crime subtypes. When data is accessible, future studies
should explore the crime-general and crime-specificpat-
ternsamongstall,orsubsetsof,multipledifferent crime
subtypes, however, this will likely require more complex
and potentially intractable models that better account for
sparse spatially correlated data, such as joint zero-inflated or
hurdle models (Feng and Dean 2012). Past research has also
suggested that different operationalizations of the popula-
tion at risk impact the identification of crime hotspots
(Andresen 2011) and so future studies should explore how
populations inferred via remote sensing of ambient day-
time/evening populations, tracking of individual or aggre-
gate activity spaces, or geolocated social media activity
impact the location of crime-general and crime-specifichot-
spots (Andresen 2011;Kikuchi,Amemiya,andShimada
2012;MallesonandAndresen2015;Downs2016).
Future research should also explore how local crime-
general and crime-specific spatial patterns change over
time. Extending the spatial multivariate shared component
model to space-time data at annual, seasonal, monthly,
daily, and hourly scales would allow researchers and practi-
tioners to identify emerging crime-general and crime-
specific hotspots and design location- and time-specific
policies for multiple crimes or only one crime.
A spatiotemporal shared component modelling approach
would also provide a framework for evaluating the effects
of place-based crime prevention policies and programs on
crime-general and crime-specificprocessesaswellasthe
corresponding diffusion or displacement of crime (Hall and
Liu 2009). Additionally, to overcome the limitations of data
aggregation within areas (Zhang, Suresh, and Qiu 2012),
studies should consider developing shared component
models for micro-scale or point-based data that can accom-
modate crime events recorded for street intersections,
street segments, or addresses.
Note
1. The 95% credible interval is the interval that contains the
true value of a parameter with 95% probability. In
Bayesian statistics, credible intervals are analogous to
confidence intervals in frequentist statistics.
Acknowledgement
Jane Law acknowledges the Natural Science and Engineering
Research Council of Canada (Grant # RGPIN-2014-06359) for
supporting this research. We gratefully acknowledge the York
Regional Police for the crime data. The analyses were based on
data and digital maps from the Statistics Canada Canadian
Census 2006. We express thanks to the anonymous referees
for their constructive comments.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Matthew Quick http://orcid.org/0000-0002-1112-9323
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Appendix A. Descriptive statistics for violent
crime and property crime (2006–2007) at the
dissemination area (DA) level
Appendix B. Prior distributions for Equations
(3) and (4)
Improper uniform priors were assigned to the two type-specific
intercepts (α1and α2). The two sets of type-specific unstruc-
tured random effects terms were assigned normal distributions
with means of zero with common unknown type-specific var-
iances, σ2
u1and σ2
u2.
The two sets of type-specific spatially structured ran-
dom effects terms were each assigned intrinsic condi-
tional autoregressive prior distributions (ICAR) with
a common unknown type-specificvariance,σ2
s1and σ2
s2.
The shared random effects terms (θi) were also assigned
an intrinsic conditional autoregressive prior distribution
(ICAR) with an unknown variance, σ2
θ(Held et al. 2005).
In the ICAR model, the posterior means of sik,forexam-
ple, are equal to the average of the posterior means of
the sik’s in adjacent areas and the variance is the equal to
σ2
sk/n,wherenis the number of adjacent small-areas
specified in the areal adjacency matrix. As such, the sik ’s
account for local spatial autocorrelation of crime between
nearby areas, and areas with many neighbouring areas
have more precise estimates of sik than areas with few
adjacent areas (Besag, York, and Mollie 1991). A queen-
contiguity adjacency matrix was used to define the spatial
structure of the study region.
For the variance parameters of the random effects
terms, Inverse Gamma (1.0, 0.01) distributions were
assigned for σ2
u1,σ2
u2,σ2
s1,σ2
s2,andσ2
θ(Held et al. 2005).
To ensure that this prior distribution did not substantially
influence the results of this study Inverse Gamma (0.1, 0.1)
and Inverse Gamma (0.01, 0.01) distributions were also
tested (Lunn et al. 2000; Ancelet et al. 2012). The results
for all sensitivity tests were nearly identical to the results
presented in this study.
Violent crime Property crime
Total count 10,970 38,959
Mean 9.72 34.53
Standard deviation 18.31 106.19
Minimum 0 0
Maximum 379 2,721
Standardized ratio
a
Mean 1.01 0.98
Standard deviation 1.27 2.14
Minimum 0 0
Maximum 19.64 39.88
a
The standardized ratios were calculated as the observed crime counts divided
by the expected crime counts (see Crime data).
ANNALS OF GIS 15
