Regional Variation and Local Information in Crime Linkage Analysis
Jonathan Allen Kringen1
D. Kim Rossmo2
Marcus Felson3
Asian Association of Police Studies
2014 Annual Conference
Tokyo, Japan
1
Henry C. Lee College of Criminal Justice and Forensic Sciences, University of New Haven
Center for Geospatial Intelligence and Investigations, School of Criminal Justice, Texas State University
3 School of Criminal Justice, Texas State University
2
ABSTRACT
Probability-based methods have been shown to be useful in crime linkage analysis (the
determination of which crimes were committed by the same offender). However, little consensus
exists as to the value of specifically tailoring these methods to individual jurisdictions.
Considering regional differences in routine activities, incorporating local information into crime
linkage analysis may improve the effectiveness of linkage strategies. Using simulation methods,
this study addresses the impact of regional variation in factors affecting criminal opportunity on
crime linkage tools.
Crime linkage is an important task in the investigation of a serial crimes (Bennell, Jones,
& Melnyk, 2009; Burrell, Bull, & Bond, 2012). Knowledge of multiple crimes committed by the
same offender allows investigators to pool information to pursue their investigation. This results
in several practical advantages. The additional information can be used to more efficiently
allocate investigative resources (Woodhams, Hollin, & Bull, 2007) which can yield greater
productivity (Bennell et al., 2009; Grubin, Kelly, & Brundson, 2001; Labuschagne, 2012).
Further, the additional information can increase the likelihood of identifying and apprehending
an offender as well as reduce the amount of time necessary to complete each task (Burrell et al.,
2012). Additionally, knowledge of a crime series can allow the use of specific investigative tools
such as geographic and behavioral profiling (Rossmo, 2000). Finally, the additional information
can strengthen the evidence in the case (Woodhams, Bull, & Hollin, 2007), potentially resulting
in more successful trial outcomes (Labuschagne, 2006).
Despite these benefits, research has demonstrated that informal linkage decisions made
by investigators are problematic. These linkage decisions are often based on limited information
and result largely from subjective impressions of individual investigators (Canter, 2000). These
impressions frequently differ between investigators (Maltz, Gordon, & Friedman, 1990), and
investigators often poorly perform when asked to discern linkages between crimes (Wilson,
Canter, & Butterworth, 1996). To overcome these limitations, objective analytic techniques that
estimate crime linkages have been proposed (Green, Booth, & Biderman, 1976; Grubin et al.,
2001).
An important consideration in developing analytic techniques to estimate crime linkages
concerns which information should be used. For information to be useful for crime linkage
analysis, it must meet two specific criteria. First, it must be effective at distinguishing between
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linked and unlinked crimes. Second, it must be available to investigators. For example,
confessions, eyewitness testimony, physical and/or forensic evidence such as fibers, fingerprints,
or DNA all may be highly effective at discerning linkages (Grubin et al., 2001). However, these
types of evidence may be rare (Ewart, Oatley, & Burn, 2005; Hazelwood & Warren, 2003). Even
when these types of evidence are present at a crime scene, the evidence is not always collected
making it unavailable to investigators (Davies, 1991). Thus, although these types of information
are effective at distinguishing linkage, they often fail to meet the second requirement of
availability.
Without these types of evidence, crime scene characteristics may be the only information
investigators have to conduct linkage analyses (Bennell & Woodhams, 2012). Thus, crime
linkage techniques based on crime scene information are necessary (Burrell et al., 2012; Mokros
& Alison, 2002; Rossmo, 2000). These types of techniques rely on analyzing the patterns of
information from crime scenes as similarities between crime scene factors can help determine the
likelihood of a crime series (Rossmo, 2000). These factors may include location in space and
time, modus operandi (MO), and signature (Rossmo, Kringen, & Allen, 2012). While research
has demonstrated the value of each of these factors in classifying crimes as linked or unlinked
(Davies, Tonkin, Bull, & Bond, 2012), most research has consistently demonstrated that location
in space exhibits the greatest potential for use in crime linkage analysis (Bennell & Canter, 2002;
Bennell & Jones, 2005; McCarthy, 2007; Burrell, Bull, & Bond, 2012).
Beyond its ability to discern linkages, location information is also important considering
its routine availability to investigators. Location is typically recorded by police and can be
recorded in a reliable fashion (Bennell & Jones, 2005). This allows for greater confidence in
analyses based on the spatial patterns that emerge. Thus, based on ability to discern linkages,
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availability to investigators, and high reliability, location in space remains the most promising
single factor for crime linkage analysis.
Despite general agreement on the value of location in space and other crime scene factors
to estimate crime linkages, less attention has been paid to formalize the techniques for practical
crime linkage analysis. Techniques that have been used in the past have rarely been vigorously
tested to demonstrate their predictive validity (Funder & Colvin, 1991), and less research has
focused on comparing linkage analysis performance for emerging techniques (Dowden, Bennell,
& Bloomfield, 2007). As a result, several questions concerning how to best proceed in
developing practical linkage tools remain. One such question concerns the utility of general
linkage techniques versus techniques developed specifically for certain crime types or tailored
for specific jurisdictions (Bennell & Jones, 2005). The present study attempts to provide insight
into this question by determining the relative advantage of incorporating local information in a
recently proposed crime linkage analysis technique.
LITERATURE REVIEW
Two central assumptions are important considerations for crime linkage analysis. The
first assumption is known as the consistency hypothesis. This hypothesis asserts that an
individual offender’s behavior is relatively consistent from crime to crime (Canter, 1995). The
second assumption is the distinctiveness hypothesis. This hypothesis asserts that offenders’
behaviors are heterogeneous and vary largely between individual offenders (Goodwill & Allison,
2006; Salfati & Bateman, 2005). Taken together, these two assumptions suggest there should be
at least some differences in the characteristics associated with crimes committed by different
offenders and some similarities in the characteristics of crimes committed by the same individual
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offender. This consistency within offenders and the variation between offenders should help link
an individual offender’s crimes to each other while distinguishing them from crimes committed
by other offenders (Burrell et al., 2012).
Empirical evidence for both consistency and distinctiveness has been found for spatial
location of crimes (Markson, Woodhams, & Bond, 2010; Lundrigan, Czarnomski, & Wilson,
2010; Santilla, Laukkanen, & Zappala, 2007, & Tonkin, Grant, & Bond, 2008). Since location of
target may be the most crucial decision an offender makes and is the decision that an offender
has the most control over, it follows that this aspect of behavior will be more consistent than
other, context-dependent behaviors (Bennell, & Jones, 2005; Harbers, Deslauriers-Varin,
Beauregard, & Van Der Kemp, 2012). Consistent with these findings, several studies have
demonstrated that the spatial characteristics of crimes are able to distinguish between linked and
unlinked crimes. Bennell and Canter (2002) showed evidence that linkage in commercial
burglaries could be discerned using the distances between crime sites. While other factors
studied (including entry behavior, target characteristics, and property stolen) all had predictive
validity, inter-crime distance was the best predictor of serial linkage accurately predicting 80%
of serial commercial burglary. Later research demonstrated similar results for serial residential
burglary (Bennell & Jones, 2005), serial commercial robbery (Woodhams & Toye, 2007), serial
burglaries (McCarthy, 2007), serial personal robbery (Burrell, Bull, & Bond, 2012), and serial
auto theft (Davies, Tonkin, Bull, & Bond, 2012).
Bayesian Likelihood Ratio for Crime Linkage
While multiple studies demonstrate that location in space can be useful at discerning
linkages, few studies have tested or offered specific techniques for performing crime linkage
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analysis. One exception is a proposed a Bayesian method for linking serial crimes (Rossmo et al.,
2012). The technique involves estimating a likelihood ratio (LR) of the probability of linkage to
the probability of non-linkage to determine the relative chance that crimes are linked. The
likelihood ratio proposed can be based on any available crime scene data and can incorporate
spatial location information using inter-crime distances as follows:
𝐿𝑅𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =
𝑃(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒|𝑙𝑖𝑛𝑘𝑒𝑑)
𝑃(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒|𝑢𝑛𝑙𝑖𝑛𝑘𝑒𝑑)
The numerator in the likelihood ratio can be viewed as measures of consistency of spatial
behavior within a crime series. A high probability for the numerator indicates a high likelihood
that the distance between two crimes would occur if they were linked. In contrast, the
denominator in the likelihood ratio can be viewed as measures of distinctiveness. A high
probability for the denominator indicates the observed inter-crime distance is common for the
crime type regardless of offender. In contrast, a low probability in the denominator indicates the
inter-crime distance is rare among different offenders. As the consistency measure of the serial
behavior (the numerator) increases, holding uniqueness (the denominator) constant results in a
larger likelihood ratio. However, as uniqueness (the denominator) increases, holding consistency
(the numerator) constant, the likelihood ratio decreases. In this way, larger likelihood ratios
indicate a greater chance of linkage.
Initial empirical work involving a sample of 162 cases consisting of 4,192 crimes showed
support for the likelihood ratio technique for crime linkage (Rossmo et al., 2012). The data
consisted of a variety of crime types with robbery, sexual assault, burglary, and serial murder
forming the majority (76.5%) of the data. Additional validation of the method was accomplished
5
through Monte Carlo simulation testing the Bayesian technique on over seven million simulated
crime series (Kringen, 2014).
General versus Local Information
The Bayesian linkage technique requires estimating both the probability of particular
inter-crime distances when crimes are linked and the probability when they are unlinked. This
task is accomplished using empirically-derived probability distributions based on the spatial
locations of known serial crimes. Importantly, the validation studies for the Bayesian technique
derived these distributions using data on distances between linked crimes from a variety of
jurisdictions. The data were standardized and aggregated to form a single distribution for a
particular type of crime. Thus, this approach assumed that jurisdictional characteristics did not
substantially impact the distributions of inter-crime distances.
This “general information” approach was validated as the Bayesian technique was
capable of accurately classifying crimes as linked and unlinked using the derived probability
distributions. However, there are multiple theoretical reasons to believe that inter-crime distances
may vary substantially between jurisdictions for both linked and unlinked crimes, and that this
local variation may be useful for crime linkage analysis utilizing a “local information” approach.
Crime pattern theory (Brantingham & Brantingham, 1993a) suggests the value of local
information. According to the theory, criminal behavior is a “complex form of subjective spatial
behavior in which movement patterns depend on underlying spatial mobility biases, knowledge,
and experience” (Brantingham & Brantingham, 1984, p. 332). Thus, movement patterns are
guided by characteristics of the behavioral environment, especially the physical settings in a
given area. Crime pattern theory asserts that offenders use a spatially structured, hierarchical
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decision process when navigating the physical environment in search of criminal opportunity
(McCarthy, 2007). This process is based on an offender’s awareness space or area of familiarity,
and offenders prefer to commit their crimes within this space (Brantingham & Brantingham,
1993a).
Three important features define an offender’s awareness space. These three factors are
nodes, paths, and edges (Brantingham & Brantingham, 1993b). Nodes are the locations or the
centers of activity where an offender engages in non-criminal acts. These include locations such
as an offender’s home, an offender’s work or school, locations where an offender shops, and
locations where an offender regularly goes for entertainment. Because an offender spends time at
each of these locations, they develop familiarity with the areas. Paths are the routes that connect
nodes, and offenders develop some awareness along paths. Together, nodes and paths form an
activity space in which an offender operates (Brantingham & Brantingham, 1993b).
Routine activity theory (Cohen & Felson, 1979) also suggests the value of local
information. Routine activity theory focuses on the patterns that arise from regularities in
everyday life. Routine activities are the basic activities that individuals engage in on a regular
basis. While these activities are non-criminal, they distribute offenders, targets, and guardians
over space and time thus affecting the locations where crimes occur. Because routine activities
disperse individuals in patterned ways, offenders and targets intersect in distinct patterns. Thus,
the underlying opportunity structure for crime is non-random, and locations of crime exhibit
distinct patterns.
Both crime pattern and routine activity theories suggest that underlying environmental
features, including physical (e.g., the locations of housing and work areas), social (e.g.,
customary times for regular activities like beginning work or having dinner), and behavioral
7
(e.g., the use of public versus private transportation) have a strong impact on how criminal
opportunity develops. Considering many such factors can vary remarkably between jurisdictions,
the patterns of locations of criminal events may vary as well. While crime pattern theory
suggests that differences in jurisdictions might result from local variation in the ways offenders
develop awareness spaces, routine activities suggests that differences between jurisdictions
might result in local variation in the convergence patterns of motivated offenders and suitable
targets. Either way, both theories suggest that the locations of crimes will be largely influenced
by local environments. Thus, local information may substantially impact the performance of the
Bayesian linkage technique.
RESEARCH METHODS
To determine the effect of local information on the performance of the Bayesian crime
linkage technique, this study analyzed the performance of the technique using receiver operating
curve (ROC) analysis. Monte Carlo simulation methods were used to conduct ROC analysis over
multiple samples to determine the general performance of the technique. Each technique is
presented in the following sections.
Diagnostic Tests and ROC Analysis
Crime linkage analysis can be conceptualized as a diagnostic task where the goal is to
classify crimes as either linked or unlinked based on the available information (Bennell &
Canter, 2002). Crime linkage analysis is a specific type of diagnostic test where there are two
possible outcomes to be classified. These types of diagnostic tests are known as two alternative,
yes-no tests (Swets, 1988). For any pair of crimes, there are two actual possibilities (linked or
8
unlinked), and two corresponding predictions, resulting in four possible decision outcomes for
each observation/prediction pair. These decision outcomes are presented in Table 1.
Table 1: Decision Outcomes
Actually Linked
Actually Unlinked
Predicted Linked
Hit
False alarm
Predicted Unlinked
Miss
Correct rejection
The four decision outcomes are known as hits, misses, false alarms, and correct
rejections. A hit (or true positive) occurs when a linkage prediction is correct. A miss (or false
negative) occurs when two linked crimes are predicted to be unlinked. A false alarm (or false
positive) occurs when a linkage prediction is incorrect. A correct rejection (or true negative)
occurs when two unlinked crimes are accurately predicted as such.
The probability of certain types of linkage decisions are used to measure the accuracy of
a diagnostic test, and these probabilities are calculated using the frequencies of the four decision
outcomes (Swets, 1988). The calculation of the probabilities for each decision outcome are
presented in Table 2 with the letters A, B, C, and D indicating their frequency within each cell.
9
Table 2: Decision Outcome Probabilities
Predicted Linked
Actually Linked
Actually Unlinked
A
B
𝐴
𝐴 + 𝐶
𝑃(𝐴) =
C
Predicted Unlinked
𝑃(𝐶) =
𝑃(𝐵) =
𝐵
𝐵 + 𝐷
D
𝐶
𝐴 + 𝐶
𝑃(𝐷) =
𝐷
𝐵 + 𝐷
The hit rate, or true positive rate, is known as the sensitivity of the test. The correct
rejection rate, or true negative rate, is known as the specificity of the test. Whereas the sensitivity
of the test indicates the probability that a crime classified as linked is actually linked, the
specificity of the test indicates the probability that a crime classified as unlinked is actually
unlinked. Since the probabilities in each column of Table 2 sum to one, two pieces of
information can be used to summarize all the information; therefore, sensitivity and specificity
are commonly used as measures of the performance of classification systems (Swets, 1988).
Importantly, the sensitivity of a test (the true positive rate) and the specificity of a test
(the true negative rate of a test) are both functions of the decision threshold used to classify
observations. Thus, the decision threshold (also known as a cut score) plays an important role in
the overall performance of a test. Because the performance of a test is a function of the cut score,
analyses of test performance based on the use of any particular cut score determine the impact of
the cut score rather than the overall capacity of the classification system.
Receiver operating curve (ROC) analysis overcomes this limitation and evaluates the
performance of a test without the effect of a cut score. The ROC is the curve that emerges by
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plotting the specificity along the x-axis of a graph in decreasing order4 and the sensitivity along
the y-axis in increasing order. A diagonal line splitting the graph from (0,0) to (1,1) (i.e., y = x)
indicates that both true positives and false positives increase at an equal rate across possible cut
scores. Therefore, this line represents a test that is uninformative. ROC curves are indicative of
predictive capacity when they increase more rapidly along the y-axis than the x-axis. An example
of a ROC curve generated from a crime linkage analysis is presented in Figure 1.
Figure 1: Example ROC curve
Test performance as indicated by ROC analyses can be quantified by calculating the area
under the curve (AUC). Because the line y = x is uninformative (indicating that false positives
increase at the same rate as true positives), an AUC of 0.5 or less indicates that the model
4
This is equivalent to plotting P(false positive) along the x-axis in increasing order.
11
performs no better than chance. However, AUCs greater than 0.5 denote predictive capacity. An
AUC between 0.50 and 0.70 is considered a predictive yet “poor” model, 0.70 to 0.80 a “fair”
model, 0.80 to 0.90 a “good” model, and 0.90 to 1.0 an “excellent” model. For this study,
predictive capacity of the Bayesian linkage technique applied to various scenarios was measured
using AUC.
Monte Carlo Simulation
Monte Carlo simulation methods are computational techniques where repeated samples
are analyzed. By sampling from a variety of possible scenarios, Monte Carlo simulations can
provide detailed information on the general performance of a technique as well as provide
information on the variability in the technique’s performance between scenarios. As the number
of scenarios (i.e., samples) tested increases, the information provided by the simulation becomes
more precise. In this way, Monte Carlo simulations are well-suited to determining the efficacy of
crime linkage techniques, as precise estimates of a particular technique’s performance can be
obtained based on a wide variety of scenarios.
There are a minimum of three steps required to conduct a Monte Carlo simulation. The
first step requires the researcher to define the parameters of the simulation and to generate
simulated data. This is known as defining the data generating process (DGP) and generating
pseudo-data. In the second step, the researcher samples a set of pseudo-data, applies the
technique to the pseudo-data, and evaluates the performance of the technique for that sample.
This process is repeated over as many samples as deemed necessary, and the performance results
are recorded for each sample. The third step involves aggregating the performance data from all
12
samples and evaluating the overall performance of the technique by analyzing the performance
data.
Data Generating Process and Simulating Pseudo-Data
The first step in a Monte Carlo simulation involves defining the DGP and generating
pseudo-data. This process reflects a belief that there is some process that models measureable
observations (Mooney, 1997). For the present study, the assumptions of the DGP were derived
from routine activity theory and the concept of distance decay as applied to the journey to crime.
These assumptions included:
1) Offenders are present who vary in motivation, reside at a given location, and are more
likely to seek criminal opportunities near home.
2) Targets are present with varying levels of suitability.
3) Guardians are present who vary in capability.
These assumptions were converted into a DGP through the following steps:
1) A sample space was created to represent a single jurisdiction.
2) A number of offenders was randomly drawn to reside within this jurisdiction.
3) Each offender was assigned a level of motivation (defined by a randomly-drawn
number).
4) Each offender was assigned a primary location within the jurisdiction to serve as his
home (randomly-drawn coordinates).
13
5) Each offender was assigned a journey-to-crime profile (randomly-drawn probability
density for distances travelled).
6) All points within the jurisdiction were assigned a level of risk (defined by a
randomly-drawn number) to represent variation in the distribution of targets and
suitability as well as guardians and capability.
The data were then generated on a jurisdiction by jurisdiction basis, with each jurisdiction
representing a single sample for the Monte Carlo simulation. Within each jurisdiction (sample), a
random number of offenders was generated. For each offender, a number of attempted crimes
was randomly generated. Using each offender’s home location and their journey-to-crime
profile, a set of coordinates for each of these attempts were generated. When the risk value for
the given coordinates exceeded the motivation level defined for the offender (i.e., the
circumstances were right) a crime was recorded. Based on this process, the data recorded for a
given jurisdiction included the locations of crimes and the specific offender who committed each
crime.
Each jurisdiction simulated included both linked and unlinked crimes. From the set of
linked crimes, a single location was defined as “suspected linked” for investigative purposes.
Treating this location as suspected linked by investigators allowed calculation of the distances
between this location and the locations of all other crimes. For the purpose of analysis, each of
these distances was then coded as either linked or unlinked, with linked indicating that the crime
was committed by the same offender that committed the reference crime.
14
Evaluation of the Linkage Technique in Each Jurisdiction
To evaluate the performance of the linkage technique in each jurisdiction, the Bayesian
linkage technique was applied twice to the data for the jurisdiction. The first time, the linkage
likelihood ratio was calculated using general distance distributions derived by Kringen (2014).
The second time, the linkage likelihood ratio was calculated using distance distributions
calculated from within each jurisdiction. The local distance distributions for linked crimes were
generated by treating a sample of other linked crimes in the jurisdiction committed by other
offenders as solved. Thus, the inter-crime distances for these linked offenses formed the linked
distance distribution for that jurisdiction. The local distance distributions for unlinked crimes
were generated using all other crimes in the particular jurisdiction.
As a result of this process, each jurisdiction’s information included a list of whether the
crime was linked to the reference crime known to investigators, the likelihood indicated by the
general information approach, and the likelihood indicated by the local information approach. To
isolate the relative capacity of each approach (general or local), ROC analysis was conducted on
each jurisdiction for both the likelihood ratio given by the general approach as well as the
likelihood ratio given by the local approach. The AUCs for each approach (hereafter AUCgeneral
and AUClocal) were recorded for the given jurisdiction.
Aggregation and Analysis
The process of repeated sampling jurisdictions and calculating AUCgeneral and AUClocal
for each jurisdiction was repeated a total of 250,000 times. This allowed evaluation of the
performance of each of the approaches for calculating the linkage likelihood ratio over a wide
variety of possible scenarios. Additionally, this number of samples provided adequate detail to
15
compare the performance evaluation for each approach. This performance information was
collected and descriptively analyzed to determine which approach performed better.
FINDINGS
The Bayesian linkage likelihood ratio calculated using general information approach was
able to isolate linkages. Over the 250,000 simulated jurisdictions, the mean of AUCgeneral was
0.75 with a standard deviation of 0.16. The median was 0.74. Rating the performance of the
technique based on general information, 18.0% of samples resulted in “fair” predictions, 17.5%
resulted in “good” predictions, and 23.1% resulted in “excellent” predictions. Figure 2 shows the
distribution of AUCgeneral.
Figure 2: Distribution of AUCgeneral across 250,000 samples
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The performance of the Bayesian linkage likelihood ratio calculated using the local
information approach was substantially better than the general information approach. Over the
250,000 jurisdictions the mean of AUClocal was 0.91 with a standard deviation of 0.08. The
median was 0.93. Rating the performance of the technique based on local information, 8.2% of
samples resulted in “fair” predictions, 26.8% resulted in “good” predictions, and 62.7% resulted
in “excellent” predictions. Figure 3 shows the distribution of AUClocal.
Figure 3: Distribution of AUClocal across 250,000 samples
The local information approach performed better than the general approach in 95.8% of
cases. Both approaches performed equally well in only 3.8% of cases, with the general approach
outperforming the local information approach in only 0.4% of cases.
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DISCUSSION
As found in previous research, the Bayesian likelihood ratio based on inter-crime
distance was effective in identifying linkages regardless of whether the method relied on general
or local information. However, over the 250,000 simulated jurisdictions, linkage likelihood ratios
calculated using local information substantially outperformed those calculated using general
information. Importantly, 41.4% of jurisdictions analyzed resulted in “poor” or “uninformative”
results using the general information approach. In contrast, only 2.2% of analyses using local
information were classified as “poor” with only 0.1% being classified as “uninformative.”
With 62.7% of analyses conducted using the local information approach resulting in
“excellent” predictions, there is substantial evidence to suggest that the local information
approach warrants greater attention. This implies that crime linkage analysis techniques should
be developed in a manner that allows local information on past crimes, including knowledge
about behavior of previous serial offenders in the jurisdiction, to be incorporated in attempts to
classify unsolved crimes as linked or unlinked.
While this analysis demonstrates the potential advantage of using jurisdiction-specific
information on the spatial locations of crimes, the implications of the findings are instructive for
other factors. Location in time, MO, and signature all have the capacity to improve linkage
assessment over distance alone. Just as the jurisdiction-specific information on crime location
results in better assessment of linkage, jurisdiction-specific information on temporal location,
MO, and signature may provide similar improvement. Further research is needed to investigate
this possibility.
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