WEEK 2.2

O R I G I N A L P A P E R

Is it Important to Examine Crime Trends at a Local ‘‘Micro’’ Level?: A Longitudinal Analysis of Street to Street Variability in Crime Trajectories

Elizabeth R. Groff • David Weisburd • Sue-Ming Yang

Published online: 1 January 2010 � Springer Science+Business Media, LLC 2010

Abstract Over the last 40 years, the question of how crime varies across places has gotten greater attention. At the same time, as data and computing power have increased,

the definition of a ‘place’ has shifted farther down the geographic cone of resolution. This

has led many researchers to consider places as small as single addresses, group of

addresses, face blocks or street blocks. Both cross-sectional and longitudinal studies of the

spatial distribution of crime have consistently found crime is strongly concentrated at a

small group of ‘micro’ places. Recent longitudinal studies have also revealed crime con-

centration across micro places is relatively stable over time. A major question that has not

been answered in prior research is the degree of block to block variability at this local

‘micro’ level for all crime. To answer this question, we examine both temporal and spatial

variation in crime across street blocks in the city of Seattle Washington. This is accom-

plished by applying trajectory analysis to establish groups of places that follow similar

crime trajectories over 16 years. Then, using quantitative spatial statistics, we establish

whether streets having the same temporal trajectory are collocated spatially or whether

there is street to street variation in the temporal patterns of crime. In a surprising number of

cases we find that individual street segments have trajectories which are unrelated to their

immediately adjacent streets. This finding of heterogeneity suggests it may be particularly

important to examine crime trends at very local geographic levels. At a policy level, our

research reinforces the importance of initiatives like ‘hot spots policing’ which address

specific streets within relatively small areas.

E. R. Groff (&) Department of Criminal Justice, Temple University, 550 Gladfelter Hall, 1115 W. Berks Street, Philadelphia, PA 19122, USA e-mail: [email protected]

D. Weisburd Hebrew University, Jerusalem, Israel

D. Weisburd Administration of Justice, George Mason University, Manassas, VA, USA

S.-M. Yang Criminal Justice, Georgia State University, Atlanta, GA, USA

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J Quant Criminol (2010) 26:7–32 DOI 10.1007/s10940-009-9081-y

Keywords Crime concentration � Hot spots � Micro � Spatio-temporal

[V]ariation in crime within communities is probably greater than variations across

communities. The very meaning of the concept of a bad neighborhood is an open

empirical question: whether the risk of crime is randomly or evenly distributed

throughout the neighborhood, or so concentrated in some parts of the neighborhood

that other parts are relatively safe (Sherman et al. 1989, p. 29).

Scholars have long been interested in how crime varies over space and the topic has

received increasing attention over the last 40 years. 1

Seminal studies examining crime

across larger geographic units such as states (Guerry 1833; Loftin and Hill 1974; Quetelet

1831[1984]), cities (Baumer et al. 1998), and even neighborhoods (Boggs 1965; Bursik and

Grasmick 1993; Bursik and Webb 1982; Byrne and Sampson 1986; Chilton 1964;

Kornhouser 1978; Reiss and Tonry 1986; Schuerman and Kobrin 1986; Skogan 1986;

Stark 1987) establish the foundation for the continuing interest. More recent studies point

to the potential theoretical and practical benefits of focusing research on micro crime

places (Eck and Weisburd 1995; Sherman 1995; Sherman and Weisburd 1995; Taylor

1997; Weisburd 2002). Cross-sectional micro level studies suggest that significant clus-

tering of crime at place exists, regardless of the specific micro unit of analysis defined

(Brantingham and Brantingham 1999; Crow and Bull 1975; Groff and LaVigne 2001;

Pierce et al. 1986; Potchak et al. 2002; Roncek 2000; Sherman et al. 1989; Weisburd and

Green 1994; Weisburd et al. 1992). Longitudinal work examining the developmental

trajectories at micro levels (Weisburd et al. 2004a, 2009b) has consistently identified

tremendous crime concentration at specific places. These micro level findings provide

evidence of significant intra-neighborhood variance in crime that is lost when neighbor-

hoods are examined as homogenous units.

A major question that has not been answered in prior research concerns the geography

(i.e., the form and the degree) of street to street variability of temporal crime trends at

micro levels of analysis. 2

The initial study identifying developmental crime trends across

micro places (Weisburd et al. 2004a, b) did not use full geographic tools. To more pre-

cisely answer this question, we examine both temporal and spatial variation in crime across

street segments in the city of Seattle Washington.

In this paper we explore the spatial distribution developmental trajectories. We begin by

using 16 years of crime incident data (1989–2004) to produce developmental trajectories

of crime in Seattle, WA (Weisburd et al. 2009a). We then use spatial statistics to examine

the geography of street segments in each group. Rigorous geographic analysis enables us to

more exactly describe the spatial relationships among temporal crime patterns and to

examine the open question of whether it is important to examine crime trends at a ‘micro’

level. We address the following research questions: (1) What is the distribution of temporal

crime trajectories across Seattle? (2) What is the spatial pattern of street segments within

the same temporal crime trajectory (i.e., clustered, dispersed or random)? and (3) Are street

segments of certain trajectories found near one another or are they spatially independent?

1 Due to space constraints only selected references are mentioned here. More complete overviews of the

history of place-based criminology can be found in the introductory chapters of Eck and Weisburd (1995) and Weisburd et al. (2008). 2

For an exception see Groff et al. 2008. They use the same methodology but explore the patterns of crimes committed by juveniles.

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Finally, we discuss the implications of our findings both to the theoretical development of a

‘geography of crime concentration’ and to policy and practice.

Geography and the Criminology of Place

In the 1970s a group of scholars began to develop theories that focus on why crime

happens where it does. The ‘opportunity’ theories they developed have been instrumental

in guiding the investigation of place and crime. Routine activity theory (Cohen and Felson

1979) and crime pattern theory (Brantingham and Brantingham 1991 [1981], 1993) both

emphasize the context of crime events and importance of human activity in understanding

crime patterns. As a group, opportunity theories hold that crimes occur when the normal

everyday activities of offenders and victims intersect with no guardian present. Thus,

routine activities are the key dynamic element because they directly affect the convergence

of the other elements necessary for a crime to occur. Opportunity theories focus on how the

built and social environments shape human activity and thus provide the foundation for

understanding why crime happens where it does. After all, it is the interactions between

humans and their environment which serve as the source of explanation of observed spatial

patterns (Aitken et al. 1989; Gold 1980; Golledge and Timmermans 1990; Timmermans

and Golledge 1990; Walmsley and Lewis 1993). As software improved and data became

more widely available, the work of these theorists inspired many empirical studies of micro

places (Weisburd and McEwen 1997).

Just as Wolfgang et al.’s (1972, p. 89) identification of the ‘chronic six percenters’ (i.e.,

6% of study participants were responsible for over 50% of the offenses committed) gal-

vanized research into individual level criminality, Sherman et al.’s (1989) finding that 3%

of the addresses in Minneapolis produced 50% of the calls for service sparked renewed

interest in micro level crime patterns and how they develop over time. There is now a

significant body of literature demonstrating the existence of clustering of crime at place

regardless of the specific micro unit of analysis defined (Brantingham and Brantingham

1999; Crow and Bull 1975; Groff and LaVigne 2001; Pierce et al. 1986; Roncek 2000;

Sherman et al. 1989; Smith et al. 2000; Weisburd and Braga 2002; Weisburd et al. 2004a;

Weisburd and Green 1994). Taylor and Gottfredson (1986) took the examination of micro

places one step further by making the connection between the physical and social envi-

ronments of micro level places and their crime levels. This body of work suggests the

salience of Sherman et al.’s call for the development of a ‘‘criminology of places’’ (emphasis in original, Sherman et al. 1989, p. 30).

Building a place-based criminology necessitates pursuing a deeper understanding of the

developmental aspects of crime at micro level places (Weisburd et al. 2004a, 2009a).

Weisburd et al. (2004a, b) used group-based trajectory analysis (Nagin 1999, 2005; Nagin

and Land 1993) to classify crime on street segments from 1989 to 2002 in Seattle,

Washington. They identified 18 trajectory groups that reflect distinct longitudinal crime

patterns. Echoing Wolfgang et al. (1972) and Sherman et al. (1989) they found approxi-

mately 4–5% of the street segments in each year account for 50% of all crime. At the other

end of the spectrum, each year between 47 and 52% of street segments had no crime at all.

In a related study of crimes committed by juveniles, Weisburd et al. (2009b) applied

trajectory analysis and once again found both concentration and stability in the patterns.

The results of Weisburd and colleagues in Seattle are consistent with other micro level

longitudinal examinations which focused on a subset of places. In Baltimore, Maryland,

Taylor and National Consortium On Violence Research (1999) reported a high degree of

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stability in both crime and fear of crime across ninety street blocks using data collected in

1981 and 1994 (see also Robinson et al. 2003; Taylor 2001). Spelman’s examination of

high schools, public housing communities, subway stations, and parks in Boston over

3 years found about 10% of locations account for 50% of the calls for service to the places

he examines. Importantly, those ‘worst’ places remain fairly stable over the 3 years of his

study.

While these longitudinal micro level studies examine the temporal distribution of crime

across places, they leave the question of the geographic distribution of those trajectories

largely unexplored. Weisburd et al. (2004a, b) took a first step in this direction. They

examined the distribution of trajectories via kernel density maps. The maps showed stable

trajectories tended to be found in all areas of the city but were concentrated in areas with

less residential density and higher income. Additionally, the maps revealed overlap in the

downtown area between areas with the highest concentrations of increasing trajectories and

those with decreasing trajectories; leading them to speculate that some of the same pro-

cesses may underlie both. In a later study of crimes for which a juvenile was arrested,

Weisburd et al. (2009b) used point maps of the locations of the streets in the highest rate

trajectories to show they were found all over Seattle. They once again discovered evidence

of strong street to street variability in that subset of crimes (Groff et al. 2008). Together

these studies suggest the importance of more fully describing the variation in crime across

micro places.

Toward a Geography of Crime Concentration

The previously outlined developments provide ample evidence for the existence of geo-

graphic concentration and spatio-temporal stability. What remains is the need for a more

in-depth examination of the structure of geographic concentration across micro places; specifically addressing the degree of block to block variability. Sherman et al. (1989, p. 28)

made the case for studying the geography of crime concentration by arguing that the study

of the ‘‘variation across space is one of the basic tools of science.’’ They noted that crime

concentration itself does not necessarily mean clustering. What if the high crime places are

randomly distributed across space? The theoretical and policy implications of such a

finding would be very different from a finding of clustering.

Sherman et al. (1989) tested for clustering in the distribution of calls for service by

comparing the observed distribution to the expected distribution under a Poisson model

and uncovered significant clustering. Other researchers have used spatial statistics to show

how concentration in the distribution of crime incidents exists in ‘hot spots of crime’

(Spring and Block 1988). Since then, the identification of geographic ‘hot spots’ of crime

has been the topic of many studies. 3

More recent work has expanded to include the

temporal structure of hotspot areas (Grubesic and Mack 2008; Johnson et al. 2008). These

authors call for more work to identify hot spots by the temporal trajectory of their crime;

‘‘upwards, downwards and time-stable’’ (Johnson et al. 2008, p. 43). 4

Other researchers have described the structure of concentration by measuring the dis-

tance between hot places. As part of the design of a patrol experiment, Sherman and

Weisburd (1995) identified 420 clusters of hot spot addresses with 20 or more calls for

3 The literature around hot spots is immense. Two recent overviews provide the insight into the state of the

art (Chainey et al. 2008; Eck et al.2005). See Weisburd et al. (1992) for a theoretical introduction. 4

Weisburd et al. (2004a, b) came at this from the opposite direction. They first identified temporal trends in crime and then used kernel density maps to find hotspots of temporal trajectory patterns.

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service related to Part I offenses that were within one-half a block of one another. Other

work identified the mean distance between clusters of robbery hot spots in Minneapolis as

only 888 feet, which is approximately two blocks in most urban areas (Linnell 1988). The

same study revealed 90% of robbery hotspots in Minneapolis were on just seven arterial

roads. These studies provide ample evidence about the structure of crime concentration;

namely that concentration takes the form of hot spots (i.e., clusters) and tends to be found

near one another and in places with high amounts of human activity. They also begin to

describe the structure of the clustering.

Another open question about crime concentration pertains to whether areas such as hot

spots and neighborhoods are uniformly hot. Early evidence suggests these areas may be

heterogeneous rather than uniform. In other words, ‘bad neighborhoods’ may contain ‘good

streets’ and ‘good neighborhoods’ may be home to ‘bad streets’. Roughly 60 years ago,

Henry McKay noted the lack of offenders on some blocks within high crime neighborhoods

(Albert J. Reiss, Jr., personal communication as cited in Sherman and Weisburd 1995).

More recently, Weisburd and Mazerolle (2000) found activity in drug hot spots to be more

related to the hot spots themselves, than the levels of crime and disorder in the surrounding

neighborhood. Taylor and Gottfredson (1986) uncovered street block level variation in

crime and found it to be related to the social and physical environment. Urban planners

have long pointed to street to street variation in characteristics of the physical environment

and their relationship to crime (Jacobs 1961). Crime prevention through environmental

design (Jeffery 1971) and ‘defensible space’ (Newman 1972) explained why such variation

may be related to the physical environment (For an opposing view see Merry 1981). Hillier

(1999) made a compelling case for focusing on the built environment directly and mea-

suring its affect on crime and other social variables of interest. By quantifying both the

accessibility and the type street pattern he demonstrated that traditional street patterns (i.e.,

grid) are safer (Hillier 2004). More recent studies uncovered evidence of extensive street to

street variation in both the characteristics of the physical environment and measures of

criminal activity such as burglary (Groff and LaVigne 2001) and auto thefts (Potchak et al.

2002). As a whole, these studies point toward the existence of bad places in good

neighborhoods as well as good places in bad neighborhoods. Our research takes the next

step and provides a direct, quantitative examination of that question.

Achieving a More Complete Description of Crime Variation Across Places

The concentration and stability of micro crime places suggest they are an important unit of

study for understanding crime at place. But those characteristics do not put to rest a key

concern in assessing the importance of such small geographic units in the crime equation.

If the focus on micro places adds to our study of crime, then it should represent a type of

‘reductionism’; the understanding of small parts will lead to an explanation of the whole

which is not provided by higher units of analysis. While prior studies have shown that

crime is concentrated at micro units of analysis, they have not examined whether this

variability is distinct from what would be observed had they focused on higher geographic

units such as communities or neighborhoods. For example, do macro level studies simply

mask concentrations of crime that are found in high crime communities? More specifically,

is there street to street variability in crime trends at micro places, or do examinations of

micro crime places simply divide up larger area trends?

The answers to these questions have implications for both theory and crime prevention

policy. If geographically proximal street segments have the same or similar temporal crime

patterns, it would suggest there is no need for micro-level examination of places. It would

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also provide support for neighborhood-level crime prevention initiatives rather than micro

level ones. If, however, street segments of the same trajectory are spread throughout the

city and/or street segments spatially adjacent to one another vary in their temporal crime

pattern, then further examination of micro level patterns is supported. In this case, more

narrowly focused efforts on individual street segments may provide more crime prevention

impact. Our focus in this study is on providing more comprehensive evidence of the

variation in crime across micro level places.

Analytic Strategy

The incorporation of spatial methods into criminological research has increased rapidly

since the 1990s (Messner and Anselin 2004). Researchers have taken seriously the error

introduced by failing to account for spatial effects when analyzing inherently spatial data

and have responded by incorporating a range of spatial data analysis techniques. 5

This

research continues that trend by using spatial statistics to describe geographic patterns of

crime trajectories across street segments.

Study Area

Our study focuses on Seattle Washington. Seattle was a logical choice because the city had

an extended and unbroken series of electronic crime incident data. Seattle is located on the

west coast of the United States. It is bounded on the west by the Puget Sound and on the

east by Lake Washington. This unusual geography has significant ramifications for human

activity patterns. The western border of the city is permeable only to water traffic (via a

ferry system). Automobile and bus traffic can enter Seattle from the east using one of two

bridges. It is only on the two shortest borders (on the north and the south) that typical levels

of porosity in boundaries are found. The lack of permeable boundaries has the benefit of

reducing concerns about spatial edge effects. Once inside Seattle, there are additional

natural barriers in the form of waterways. The southern section of the city is split northwest

to southeast by the Duwamish Waterway (three bridges cross it). The northern section of

the city is split from the central by a waterway consisting of Salmon Bay, Lake Union,

Portage Bay, and Union Bay. Seattle is a mature city. There was no additional development

of residential or commercial areas requiring changes to the street network between 1989

and 2004. There were two changes to the built environment of note over the study period; a

major baseball stadium was opened in 1999 (Safeco Field) and a new cultural center

opened in the Seattle Center downtown in 2003.

Unit of Analysis

We use the street segment, both sides of a street between two intersections, as our unit of

analysis ( n = 24,023).6 The average length of a street segment in Seattle is 387 feet. The

5 The volume of research explicitly examining spatial dependence or spatial error in models is far too large

to detail here (as examples see Baller et al. 2001; Chakravorty and Pelfrey 2000; Cohen and Tita 1999; Cork 1999; Jefferis 2004; Morenoff and Sampson 1997; Roman 2002). 6

This unit of analysis is slightly different from the ‘hundred block’ measure used in the original Seattle study. See the final report for more information on the ‘hundred block’ definition (Weisburd et al. 2004b). More detailed information on the creation of geographically defined street segment is available (see Weisburd et al. 2009a).

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majority of the streets (roughly 64%) are between 200 and 600 feet. Using our definition,

very few streets (less than 2%) ended up longer than 1,000 feet. In addition to the theo-

retical and technical reasons discussed earlier, we have several other substantive reasons

and a technical one for using street segments as our unit of analysis including: (1) routine

activity theory is essentially a micro-level theory and thus informs patterns observed at the

micro level (Eck 1995; Sherman et al. 1989); (2) using micro places such as individual

addresses, intersections and street segments minimizes the aggregation in the analysis and

consequently, the risk of ecological fallacy (Brantingham et al. 1976); (3) when consid-

ering policing strategies as they relate to place, a key factor is how much of the variation in

crime involves factors the police are able to address (Taylor 1998) and whether the policy

implications from such will be immediately actionable. On the technical end, street

segments reduce spatial heterogeneity among the units of observation that has been

shown to exist when larger areal units are used (e.g., block groups and census tracts)

(Smith et al. 2000).

Data

We use computerized records of crime incident reports to represent crime for the period

from 1989 to 2004. Incident reports are generated by police officers or detectives after an

initial response to a request for police service. In this sense, they represent only those

events which were both reported to the police and deemed to be worthy of a crime report

by the responding officer and thus provide a measure of vetted crime. We include all crime

events for which a report was taken except those which occur at an intersection. 7

In

addition, we exclude records that lack a specific address, occur on the University of

Washington campus or at a police precinct or police headquarters, and those written for

crimes that occur outside city limits. We geocode the remaining records and are left with

1,697,212 incident reports over the time period. 8

Dependent Variable

We apply group-based trajectory analysis (Nagin and Land 1993; Nagin 1999, 2005) to

cluster street segments into groups with distinct developmental trends over the time period

studied (see ‘‘Appendix 1’’ for technical details). Our approach follows closely the

methodology of an earlier study using 14 years of data and hundred blocks (Weisburd et al.

2004a, b). In our study we use 16 years of data and redefined the unit of analysis. Despite

the use of two additional years and a redefinition of the unit of analysis, the trajectory

results are roughly similar to the earlier study (Weisburd et al. 2004a). Twenty-two

7 There are two main reasons for excluding intersection crime. First, since events at intersections could be

considered ‘part of’ any one of the participating street segments, there is no satisfactory method for assigning them to one or another. However, it is also the case that incident reports at intersections differed dramatically from those at street segments. Traffic-related incidents accounted for only 3.77% of reports at street segments, but for 45.3% of reports at intersections. 8

All geocoding was done in ArcGIS 9.1 using a geocoding locator service with an alias file of common place names to improve our hit rate. The geocoding locater used the following parameters: spelling sen- sitivity = 80, minimum candidate score = 30, minimum match score = 85, side offset = 0, end offset 3%, and Match if candidates tie = no. Manual geocoding was done on unmatched records in ArcGIS 9.1 and then in ArcView 3.x using the ‘MatchAddressToPoint’ tool (which allowed the operator to click on the map to indicate where an address was located) to improve the overall match rate. Research has suggested hit rates above 85% are reliable (Ratcliffe 2004). Our final geocoding percentage for crime incidents was 97.3%.

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distinct groups are predicted by the model (as opposed to 18 groups in the original study).

To simplify our description and focus our discussion more directly on patterns in the level

and direction of temporal changes in crime at place across time, we divide the initial

trajectories into eight patterns based on a visual inspection of the level of crime over the

time period and the overall direction of change (Table 1; Fig. 1a). These eight patterns

capture eight distinct types of temporal crime patterns and thus become our dependent

variable in the spatial analysis.

The first general pattern represents the street segments in our study that can be seen as

relatively crime free during this period (Fig. 1a). They experienced an average of less than two crimes per year over the entire time period. One trajectory started the period with three

crimes and declined to less than two and the other started low and increased to three crimes

per year. They account for about half of the street segments in Seattle. Approximately 30%

of the street segments are associated with what we have called the low stable pattern (Fig. 1b). As with the relatively crime free segments, these places evidence low and

essentially stable crime trends (between 3 and 12 crimes per year) and they reinforce the

simple descriptive finding that most places in the city have little or no crime. Two other

trajectory classifications represent places with much more serious crime problems, though

they also evidence strong stability in trends over time. What we term the moderate stable pattern, includes about 1.2% of the street segments (Fig. 1e). These street segments

average around 20 crime incidents throughout the study period and are basically stable

over the study period. Street segments in what we term the chronic high pattern can be defined as the most serious crime hot spots in the city (Fig. 1h). The average number of

crimes per segment is consistently more than 80 crime incidents per year. The difference in

level of crime is what made this trajectory group earn its own pattern. Only 1% of the street

segments ( n = 247) are found in this pattern. The trajectory patterns we have described so far all represent stable crime trends. The

four remaining trajectory patterns include only about one in five street segments in the city.

But nonetheless, they help recognize that crime trends at very micro levels of geography

are more complex than overall city trends would suggest. Two of the trajectory patterns

evidence decreasing crime trends during this period. We use the level of crime and the

trend to group these trajectories into the following patterns. The low decreasing pattern accounts for almost 10% of the street segments in the city and ranges from 8 to 18 crime

incidents on average (Fig. 1c). Importantly, by the end of the study period crime had

declined to less than half of the crime averages evidenced at the outset. Similarly the street

Table 1 Number of street seg- ments per temporal trajectory grouping

Group Number Percentage

1 Crime free 11,898 49.5

2 Low stable 7,688 32.0

3 Low decreasing 2,202 9.2

4 Low increasing 903 3.8

5 Moderate stable 292 1.2

6 High decreasing 572 2.4

7 High increasing 221 .9

8 Chronic high 247 1.0

Total 24,023 10.0

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segments in the crime pattern we term high decreasing evidence strong crime declines (Fig. 1f). Here the base levels of crime in 1989 are much higher ranging between 20 and 60

crime incidents on an average street segment. Only about 2.4% of the street segments in

our study are high decreasing. The last pattern of trajectories is particularly interesting in light of the overall crime

decline in Seattle. We term these trajectories low increasing (Fig. 1d). About 4% of the street segments fall in crime trajectories with a low increasing pattern. The high increasing pattern shows a similar trend but at a higher level (Fig. 1g). Roughly 1% of the street

segments fall in these groups.

Identifying Spatial Relationships in Group Distributions

Previous studies identifying spatial relationships among trajectory group members have

used a sequential approach, conducting a spatial analysis of the output of the develop-

mental statistic (Griffiths and Chavez 2004; Kubrin and Herting 2003; Weisburd et al.

2004a). We also follow a sequential approach but instead of descriptive analysis we

systematically apply a series of point pattern statistical techniques are used to analyze the

spatial patterns of street segments. 9

We use the Ripley’s K-function and a bivariate-K function to examine the second order effects (i.e., local relationships) related to spatial

dependence (Bailey and Gatrell 1995; Fotheringham et al. 2000). Together the two

Fig. 1 Temporal crime trajectories

9 In order to apply point pattern statistics to street segments we use the midpoint of each line/street to

represent the street segment.

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techniques provide a more nuanced picture of local variation than would be possible with

either alone. Unfortunately no single software package exists which combines the required

spatial statistics with a powerful cartographic display engine, so data analysis and display

were done using a variety of software packages including R, SPlancs, CrimeStat �

, GeoDa �

and ArcGIS �

9.3.

Ripley’s K describes the proximity of street segments in the same trajectory to one another. For each street segment, it counts the number of street segments of the same

trajectory that fall within a specified distance band and then repeats for subsequent distance

bands. In this way it characterizes spatial dependence among locations of street segments

with the same trajectory at a wide range of scales. In order to make more formal statements

about the point patterns, we compare the summary statistics calculated from the observed

distribution of street segments with those calculated from a model distribution (i.e.,

complete spatial randomness (CSR)). When used in this way the K-function is able to identify whether the observed pattern is significantly different than what would be

expected from a random distribution (Bailey and Gatrell 1995). Ripley’s K is calculated and then compared to a reference line that represents CSR: if K( h) [ pd2 then clustering is present (Bailey and Gatrell 1995, 90–95; Kaluzney et al. 1998, 162–163).

A bivariate- K (also called a cross K) function is used to test for independence between movement patterns. The bivariate- K answers whether the pattern of street segments belonging to one trajectory is related to the pattern of street segments in another trajectory

(Bailey and Gatrell 1995; Rowlingson and Diggle 1993). As described by Rowlingson and

Diggle (1993) and applied here, the bivariate- K function expresses the expected number of street segments of a particular trajectory (e.g., high decreasing) within a distance of an arbitrary point of a second type of street segment (e.g., high increasing), divided by the overall density of high increasing street segments. As with Ripley’s K, simulation is used to test whether two patterns are independent.

10 The output is a graph representing three

possible relationships: independence, attraction, and repulsion for the tested distance range

(400 foot bins from 0 to 2,800 feet). If the two patterns are independent of one another;

they are most likely the result of different processes (Bailey and Gatrell 1995). A finding of

spatial interaction between them can take two forms, attraction or repulsion. Since street

segments are stationary, attraction in this context refers to a tendency for street segments of

one trajectory to be found in closer proximity to street segments of another trajectory than

would be expected under independence (i.e., their patterns are similar). Repulsion refers to

a tendency for street segments of one particular trajectory to be found at longer distances

from another.

10 This is accomplished by using a series of random toroidal shifts on one set of points and comparing the

cross K-function of the shifted points with another fixed set (Rowlingson and Diggle 1993). A toroidal shift provides a simulation of potential outcomes under the assumption of independence by repeatedly and randomly shifting the set of locations for one type of street segment and calculating the cross K-function for that iteration. The outcomes are used to create test statistics in the form of an upper and lower envelope. One thousand iterations are used for each simulation. In order to better explore micro level relationships, the bivariate - K analysis examines the distribution of the trajectory pairs at distances up to 2,800 feet (using 400 foot bins which approximate one street block). This strategy also allows us to more closely inspect the relationship of the bivariate k statistic to the upper bound of the simulation envelope. The null hypothesis of the bivariate- K test is independence (i.e., the spatial pattern of one trajectory group is unrelated to the pattern of the other group being compared).

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Findings

We begin with a simple visualization of the distribution of all the temporal trajectories

across Seattle. Seattle is divided into three sections: northern (Fig. 2), middle (Fig. 3) and

southern (Fig. 4). Lower crime street segments are represented by thinner lines and the

higher crime street segments by thicker lines. For example, crime free groups are the thinnest and lightest grey lines. Low stable street segments are darker but still very thin. Street segments which are low increasing are symbolized using thin dark lines and those that are high increasing are thicker and darker in color.

At first glance the impression is one of large areas in which streets are the same color

and thickness broken up by linear patterns. Closer inspection reveals the variety in the

pattern. While there are large areas consisting of predominantly crime free and low stable groups (not surprising given their overwhelming numbers, 12,033 and 7,696 street seg-

ments, respectively), street segments from higher rate trajectory groups are interspersed

within those areas. Street segments that are thicker (designating a high rate group) are most

often arterial roads (i.e., roads which have higher speed limits and collect traffic from

residential streets). A closer examination of the pattern of temporal trajectories reveals

differences by section of Seattle.

In the northern part of the city (Fig. 2), we see most places are crime free or in another low crime trajectory. However, there are thicker/darker lines interspersed throughout.

There is a definite linear arrangement to the patterns related to connected streets with

similar temporal patterns. The area with the most concentrated heterogeneity is a

Fig. 2 Spatial distribution of temporal trajectories (northern Seattle)

J Quant Criminol (2010) 26:7–32 17

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commercial area near the University of Washington campus (solid gray area). In this one

small area (approximately 50 street segments) we see tremendous street segment by street

segment variation in high rate temporal trajectory groups; all high rate trajectory groups are

represented there. There is also a sharp transition at the commercial area’s edge into

predominantly crime free and low stable trajectory pattern street segments. Turning to the middle section of the city we immediately notice the influence of the

downtown area of Seattle on the western portion (Fig. 3). It is here we observe the greatest

magnitude of variation of the temporal trajectories from street to street and the greatest

concentration of streets with high crime. 11

A high degree of variability is also present in the

older residential sections east of downtown. Finally, in the southwestern and southeastern

sections of the city a slightly different pattern emerges (Fig. 4). Once again large areas of

predominantly low crime and crime free street segments are interspersed with the other trajectory types. However, the southern section seems to have the greatest number of linear

patterns of high rate street segments often following arterial roads.

While these maps do not provide quantitative substantiation for the significance of the

spatial associations revealed, they do offer a strong indication of heterogeneity as well as

homogeneity in crime patterns at the street segment level. Thus, it is not the case that we

can understand the action of crime by simply extrapolating from large area trends.

Something seems to be going on at the micro level that needs explanation.

Fig. 3 Spatial distribution of temporal trajectories (central)

11 Readers should note slight scale changes among maps 1, 2 and 3. These were necessary to provide

maximum enlargement of the three sections of Seattle.

18 J Quant Criminol (2010) 26:7–32

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Next, we use Ripley’s K to examine whether street segments of the same temporal trajectory are clustered in space.

12 The statistic reported from Ripley’s K in CrimeStat

(Levine 2005) is the L value. This is a rescaled Ripley’s K where CSR is represented by a horizontal zero line. Figure 5 shows the L( t) line for each of the trajectories; the higher the line, the greater the degree of clustering among the places which are members of the

trajectory group. The X-axis provides information as to the scales at which the clustering occurs. Since we are interested in comparing neighborhood level clustering with street

segment level clustering only distances up to about one-half mile (2,640 feet) are

discussed.

In general, we find the degree of clustering increases with the rate of crime exhibited by

the temporal trajectory (Fig. 5). More specifically, chronic high street segments have the greatest degree of local clustering. There is essentially no difference in the lines for the

other three higher trajectory groups until between 1.5 and 2 blocks (.12 and .15 miles)

where they begin to diverge. At this point, high increasing street segments are most likely to be near other one another followed by high decreasing and moderate stable groups. At the other end of the spectrum, among low rate street segments, it is the low increasing

street segments which are most likely to be found near one another followed by the low

Fig. 4 Spatial distribution of temporal trajectories (southern Seattle)

12 Ripley’s K also reveals whether the observed clustering is greater or less than would be expected under

an assumption of Complete Spatial Randomness (CSR). CSR is of limited use when examining human- related distributions such as crime because the opportunity for a crime to occur is constrained to accessible areas adjacent to streets. By calculating the Ripley’s K for the street network, we can provide a more realistic metric with which to compare patterns in the trajectory group member ship of street segments (see ‘Street segments’ line in Fig. 4).

J Quant Criminol (2010) 26:7–32 19

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decreasing street segments. The crime free and low stable street segments are the least clustered which reflects their ubiquitousness. All are more clustered than the reference line

representing the level of intrinsic clustering in the street network. In sum, street segments

of the same temporal trajectory group to be generally more clustered at distances of less

than a half-mile than would be expected as compared to either the intrinsic clustering in the

street network or under CSR. Only the degree of clustering varies by the temporal pattern

of crime. Street segments exhibiting high crime trajectory patterns show the strongest

clustering.

While Ripley’s K provides important information about the degree and scale of clus-

tering of street segments within the same temporal trajectory, we must use a variation on

Ripley’s K, the bivariate- K statistic, to answer our question about whether there are specific temporal trajectory pairs that tend to be physically proximal (e.g., low increasing streets near high increasing streets). We conduct a series of pairwise comparisons to evaluate the patterns of each group as compared to those of every other group (i.e., group 1

to group 2, group 1 to group 3 etc.). The results indicate whether the patterns of two groups

tend to systematically ‘hang together’ (attraction) or be found in different places (repul-

sion) or whether their locations are independent of one another. 13

We find no evidence for repulsion; a pair of temporal trajectory patterns which are

consistently far from one another (Fig. 6). We did find evidence for independence (i.e.,

members of two trajectory patterns which are neither consistently near to nor far from one

another) and for spatial interaction in the form of attraction. To simplify our discussion, we

discuss independence first and then attraction. The patterns of streets in the following

trajectory pairs are not related (independent) at some distances up to about a half-mile:

• Crime free as compared to moderate stable (at distances less than or equal to about 1,200 feet), high decreasing, high increasing, and chronic high.

• Low stable as compared to chronic high (at distances less than or equal to about 800 feet).

• Low decreasing as compared to high increasing (at distances less than or equal to about 1,200 feet) and chronic high (at distances less than or equal to 400 feet).

Fig. 5 Ripley’s K of all trajectories

13 These analyses produced 28 graphs. Space constraints do not allow the inclusion of the graphs in the

paper; however, they are available from the authors.

20 J Quant Criminol (2010) 26:7–32

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• Moderate stable as compared to crime free (at distances less than or equal to about 1,200 feet).

• High decreasing as compared to crime free. • High increasing as compared to crime free and low decreasing (at distances less than

or equal to about 1,200 feet).

• Chronic high as compared to crime free, low stable (at distances less than or equal to about 800 feet), and low decreasing (at distances less than or equal to about 400 feet).

These results indicate the process which is producing crime free trajectory patterns is different from the one producing the high crime trajectory patterns but not independent

from the process or processes producing low rate crime patterns. Crime free street seg- ments are the most independent overall; they are unrelated to the patterns three of the other

crime trajectories at all distances and to one other (moderate stable) up to about 1,200 feet. Low stable street segments are independent from chronic street segments at distances two blocks and under (about 800 feet) before showing evidence of attraction at larger distances.

Low decreasing street segments are independent from both high increasing and chronic high at short distances (1,200 feet and 400 feet, respectively). These results make intuitive sense since low crime places are typically very different from very high crime places.

Overall, the degree of independence between crime free, low stable, and low decreasing as compared to high crime temporal patterns suggests the existence of distinct processes

underlying the level of crime observed across places.

The predominant type of spatial interaction in these pairwise relationships at micro

distances is attraction. The members of the following trajectory pattern pairs exhibit

attraction at most distances up to one-half mile (i.e., they tend to ‘hang together’):

• Crime free with low stable, low decreasing, low increasing, and moderate stable (at distances greater than 1,200 feet).

2 – Low Stable

3 – Low Decreasing

4 – Low Increasing

5 – Moderate Stable

6 – High Decreasing

7 – High Increasing

8 – Chronic

1 – Crime Free <=

1,200

2 – Low Stable

<= 800 ft

3 – Low Decreasing <=

1,200 ft <= 400 ft

4 – Low Increasing

5 –Moderate Stable

6 – High Decreasing

7 – High Increasing

Attraction Independence

Fig. 6 Graphical representation of pairwise comparisons

J Quant Criminol (2010) 26:7–32 21

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• Low stable with low decreasing, low increasing, moderate stable, high decreasing, and high increasing and chronic high (after 800 feet).

• Low decreasing with low increasing, moderate stable, high decreasing, and high increasing (after 1,200 feet), and chronic high (after 400 feet).

• Low increasing with moderate stable, high decreasing, and high increasing. • Moderate stable with crime free (after 1,200 feet), low stable, low decreasing, low

increasing, high decreasing, high increasing, and chronic high. • High decreasing with low stable, low decreasing, low increasing, moderate stable,

high increasing, and chronic high. • High increasing with low stable, low decreasing (after 1,200 feet), low increasing,

moderate stable, high decreasing, and chronic high. • Chronic high with low stable (after 800 feet), low decreasing (after 400 feet), low

increasing, moderate stable, high decreasing, and high increasing.

The finding of attraction indicates common processes may be underlying these pairs of

temporal crime patterns. Turning first to the low rate groups, we find spatial attraction

between street segments with crime free, low stable, low decreasing, and low increasing trajectory patterns at all distances (Fig. 6). The attraction between moderate stable and crime free street segments is only present at distances of greater than 1,200 feet. Thus moderate stable street segments tend to be within a half mile of crime free street segments

but not within approximately three blocks of them.

Low stable trajectory patterns are attracted to all other trajectory patterns except chronic high (but only at distances of greater than 800 feet). Here again, we uncover a pattern of independence related to adjacent streets and those within two blocks but general attraction at

distances up to one-half mile. One explanation for this finding is that the processes under-

lying the realization of low stable places are similar to those underlying the places with both higher and lower crime trajectories. However, there may be place-specific differences that

occur at nearby streets which cause their temporal crime patterns to change. Alternatively, it

could indicate the role low stable places play as ‘jumping off points’ for changes in crime

rates. At the same time, the finding confirms the significant geographic dispersion of crime free and low stable places. They are found all across Seattle, not just in ‘good’ areas.

The spatial distribution of low rate street segments that are experiencing decreasing or

increasing temporal crime patterns is especially interesting. Low decreasing street seg- ments tend to be found near higher rate crime trajectory members such as moderate stable and high decreasing at all distances. However, similar to the finding just discussed for low stable street segments, they are only attracted to high increasing (at distances greater than 1,200 feet) and chronic high (at distances of greater than 400 feet) at distances greater than one block. On the other hand, low increasing street segments are also near moderate stable, high decreasing, high increasing, and chronic high. It may be the processes sustaining or increasing crime on streets nearby are also influencing the low increasing street segments.

We find much more heterogeneity among temporal trajectory groups representing high

rate crime places. The pattern of chronic high street segments is related to moderate stable, high decreasing, and high increasing places at all distances. High increasing street seg- ments tend to be found near low increasing, moderate stable, high decreasing, and chronic high at all distances. However, high increasing street segments are independent of low decreasing streets up to 1,200 feet. This spatial separation hints at the existence of sig- nificant differences between the processes underlying low decreasing and high increasing street segments. Streets which are part of high decreasing temporal patterns tend to found near street segments of both low and high rate trajectory patterns (i.e., low stable, low

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increasing, high increasing, and chronic high) at most distances. Thus, in general, streets with higher crime or changing temporal trajectories tend to be more frequently associated

with other streets which are also higher crime or undergoing a change in crime rates.

In sum, our descriptive and statistical analysis reveals temporal crime trajectory pattern

membership often varies from street segment to street segment. This is apparent in both the

descriptive maps and in our finding of widespread spatial attraction among different

temporal trajectory groups. Overall, these results are consistent with the spatial pattern we

saw on the descriptive maps showing proximal places can have very divergent temporal

crime trajectories. While certain areas of the city consist of predominantly crime free and low stable trajectory patterns, other areas are characterized by extreme spatial heteroge- neity with street to street variation in both the level and pattern of temporal crime pattern

membership. A set of any four nearby streets might include one crime free, one moderate stable, one high decreasing and one chronic street segment.

The outcomes of the pairwise comparisons are consistent with earlier analyses. In

general, high rate groups tend to attract other high rate groups. This may be a reflection of

the built environment. Place characteristics such as land use patterns, disorder, housing

quality etc. tend to be shared among nearby street segments. Thus it is not surprising that

high level and low level streets might generally attract. Our results clearly indicate inde-

pendence between the pattern of crime free streets and all high rate street segments

(although moderate stable is only up to three blocks away). Thus the processes which

create crime free streets are most likely very different from those which result in streets

with significant amounts of crime.

The finding of attraction between low stable and low decreasing and the high rate groups is more puzzling. It may be the geographic patterns of street segments in low stable and low decreasing temporal patterns are a reflection of their proximity to high crime places. Low stable street segments are found near both low and high crime street segments and thus are influenced by processes operating to produce both low crime and high crime

places. Similarly, low decreasing street segments may be ones which have experienced changes in their characteristics which run counter to the general high crime environment in

which they are situated and have resulted in a decreasing crime trajectory pattern. Together

these results demonstrate the micro level of analysis is capturing important variability in

trends; variability that would be masked at a more macro level.

At the same time, we do find two phenomena that demand we explain and understand

clustering at different scales. First, we find that there is a dependence of street segments

within the same temporal crime trajectories at distances up to one-half mile. Second, we

find evidence for significant attraction between specific types of temporal crime trajecto-

ries. This is especially true with low or no crime segments, where we observe large areas in

which these trajectories prevail. However, it is also true of chronic segments and other high

crime segments for example in the city center.

Discussion

But what do these findings mean to our understanding of the spatial pattern of temporal

crime trajectories and of the structure of crime concentration at places more generally?

What does the existence of varying degrees of heterogeneity in street to street temporal

crime trajectories indicate? How do we interpret our finding that street segments within a

single trajectory group are more likely to be found near one another? What can we learn

from the knowing certain pairs of temporal crime patterns tend to be found in the same

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general areas? Since these observations are not mutually exclusive, how do we reconcile

places where more than one of the above relationships exist? Absent data on the nature of

places we can of course only speculate on what might be driving the observed patterns.

One interpretation is that there are simply two sets of influences operating on crime in

the city. One set of influences may reflect the existence of ecological labels. Ecological

labels provide a shared perception of places that is generally homogeneous across larger

areas of a city (Brantingham and Brantingham 1991 [1981]). Assuming perceptions

influence actions, shared perceptions can lead to shared actions which in turn serve to

reinforce the existing crime situations. While these perceptions apply to an area they are

‘played out’ at the micro level of street segments. This cycle may explain the existence of

generally high crime rate areas as well as generally low crime rate areas and thus our

finding that streets with similar trajectories are generally clustered.

A second interpretation focuses on micro level influences of opportunity factors on

crime and the importance of local trends. Theories such as routine activity theory (Cohen

and Felson 1979; Felson 1986, 2002) and crime pattern theory (Brantingham and Bran-

tingham 1984, 1991 [1981], 1993) focus on the spatio-temporal characteristics of crime

events in an attempt to explain why some places have more crime than others (Eck 1995).

The urban backcloth provides a local baseline susceptibility to crime through a shared set

of crime-related characteristics (Brantingham and Brantingham 1991 [1981]; Jacobs 1961).

In this way, place characteristics also play a significant role in shaping the differential

distribution of potential targets and motivated offenders across space and time.

At the micro level, opportunity theories would predict variation in micro level crime if

there was significant variation in characteristics of places that facilitate the convergence of

the elements necessary for a crime to occur. However, variation in crime across larger

areas and even neighborhoods could also be consistent with the perspective. If there is no

significant variation in characteristics which affect the frequency of convergence, then we

would expect relatively homogenous and stable crime patterns over time. In this way,

opportunity theories can explain both heterogeneity and homogeneity in the geographic

pattern of temporal crime rates.

In the same vein, our finding of street to street variability in temporal crime trajectories

may simply be a reflection of the street to street variability in the urban environment.

Activity generators such as employment centers and retail stores attract both residents and

non-residents, law abiding and those open to criminal opportunity, to specific places and in

doing so produce local variation above and beyond the baseline for a place (Brantingham

and Brantingham 1995; Felson 1987; Kinney et al. 2009). Additionally, the influence of

activity generators/attractors is not confined to the street segment on which they are

located. As individuals travel to and from the activity/crime generator they become

familiar with the places along their route. Their awareness of opportunities is highest along

their route and at the places they visit; it declines with distance from each.

In this way, micro level patterns can also explain the pair-wise clustering of street

segments with different temporal crime patterns. While there is much street to street

variability in the urban backcloth, there is also general continuity. The general continuity

explains why low rate temporal crime trajectories tend to be found near other low rate

street segments. At the same time, the character of streets is influenced by the mixture of

land uses present and by unique characteristics of places that may influence the amount of

crime experienced.

Together these assumptions lead us to believe that crime attractors/generators have a

differential impact. Levels of traffic along blocks provide an indicator of the number of

people for whom that block is part of their activity space. The more activity spaces of

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which a block is a part, the greater the potential that an offender and a target will converge

without a capable guardian present and thus supply the necessary elements for a crime to

occur (Cohen and Felson 1979). These mechanisms are at work regardless of neighborhood

level influences. As we saw in our results, what varies by neighborhood is the degree to

which increased opportunity translates into increased crime (e.g., is the change from crime free to low stable or from moderate stable to chronic high). The exact form of the change may also be a function of variability in the temporal crime patterns of nearby places.

Given the above, our work would propose the importance of recognizing commonalities

among micro level places that stem from consistency in the built environment. However,

while our data suggest the importance of recognizing crime trends at micro places, they do

not necessarily reinforce the importance of community level influences. This is especially

the case in our finding of clustering of similar trajectories in certain areas. For example, the

clustering of chronic segments near one another may reflect important similarities in social and built environment characteristics. In this context it would be the street segment

characteristics that are important to understanding the concentration in crime at place.

Other researchers have advocated for starting small (Brantingham and Brantingham

2008; Groff et al. 2008; Oberwittler and Wikstrom 2008; Rengert and Lockwood 2008; van

Wilselm 2008). Even if some of the variability present at micro levels does work its way up

to more macro levels, we will still have wanted to know that. Otherwise, we will be unable

to separate observed macro level patterns which are being driven by variability at micro

levels from those that are not.

We think it also important to recognize that there may not be a simple dichotomy

between social disorganization and opportunity theories as they have often been linked to

units of geographic analysis. Since both social disorganization and opportunity variables

are related to the urban backcloth, it may be too simplistic a model to attribute area trends

to social disorganization factors and local trends to opportunity factors. The story of crime

at place may be more complex.

At a policy level, our research reinforces the importance of initiatives like ‘hot spots

policing which address specific streets within relatively small areas (Braga 2001; Sherman

and Weisburd 1995; Weisburd and Green 1995). If police become better at recognizing the

‘good streets’ in the bad areas, they can take a more holistic approach to addressing crime

problems. For example, they can more precisely target community building and law

enforcement operations to maximize efficiency and effectiveness. More broadly, they can

work with other city agencies to change the physical and social environment of problem

places (Johnson et al. 2008). Alterations to the built environment which improve sur-

veillability, control access, and increase the capacity for territoriality among legitimate

users can reduce crime (Lab 2007).

Before concluding we want to note some specific limitations of our work and make

suggestions for future study. While our findings break new ground in describing the spatio-

temporal relationships among street blocks, there are two limitations to our micro-level

examination of spatio-temporal crime patterns. Although our study examined a longer time

period than has been available to other scholars at the micro level of analysis, it remains

only a snapshot in the development of places. Accordingly, our analysis might have

underestimated some dynamic elements while overestimating others. Another limitation

involves the low numbers of incidents at street segments. This feature of street segment

level research necessitated we use total crime events rather than exploring specific types of

crime as is recommended in place-based research (Clarke 1983; Clarke and Felson 1993).

Future research may want to follow Braga et al. (2009) and isolate high crime street

segments for more in-depth study.

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This paper establishes the importance of looking at the micro level. Of course, our data

only allow us to identify the importance of temporal crime patterns. To advance the field

future work should collect street segment level characteristics which would allow the

identification of characteristics which are associated with particular temporal crime pat-

terns. Only by conducting research that uses street blocks as the unit of analysis and by

thoroughly describing their characteristics will we be able to shed additional light on these

questions. Future work should utilize prospective data collection to take advantage of the

more robust information systems available today. In addition, prospective data collection

allows for the inclusion of systemic social observation (SSO) (Sampson and Raudenbush

1999) and ethnographic components that are crucial to ‘adding flesh to the bones’ of

quantitative research. Neither of which are possible with retrospective studies.

Conclusions

While there is a growing body of evidence regarding the value of examining micro level

places, the predominant paradigms of place and crime focus on large area trends. Firmly

grounded in the ecological tradition, no one doubts the importance or salience of these

investigations to understanding crime and delinquency. However, the work herein rein-

forces the idea that we need to focus on small area trends as well.

Our systematic examination of the spatial patterns within temporal crime trajectory

groups allows us to enrich our understanding of the structure of concentration in crime

patterns at micro level places. In doing so our study reveals the significant geographic

variability in temporal trends from street segment to street segment which suggests

something is going on at the micro level that requires explanation. Since our study lacks

any information on the characteristics of places we are unable to explain why these patterns

are occurring or to conjecture about the underlying processes at work. We leave those

explorations to future work.

Acknowledgments This research was supported by grant 2005-IJ-CX-0006 from the National Institute of Justice (US Department of Justice). Points of view in this paper are those of the authors and do not necessarily represent those of the US Department of Justice. We would like to thank Dan Nagin for his thoughtful suggestions regarding trajectory analysis, Richard Heiberger for his assistance with R pro- gramming, and Breanne Cave and the anonymous reviewers whose comments were invaluable in strengthening the paper. We also want to express our gratitude for the cooperation of the Seattle Police Department, and especially to Lieutenant Ron Rasmussen for playing the crucial role of our main data contact and former Chief Gil Kerlikowske (now Director of the Office of National Drug Control Policy) for his interest in and support of our work.

Appendix 1: Technical Note of the Production of Developmental Trajectories for Street Segments

The trajectory modeling reported here was developed for a larger study of crime and place

in Seattle, WA (Weisburd et al. 2009a). The group-based trajectory model, first described

by Nagin and Land (1993) and further elaborated in Nagin (1999, 2005), is specifically

designed to identify clusters of individuals with similar developmental trajectories, and it

has been utilized extensively to study patterns of change in offending and aggression as

people age (see Nagin 1999). As such, we believe it is particularly well suited to our goal

of exploring the patterns of change in the Seattle data.

26 J Quant Criminol (2010) 26:7–32

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Formally, the model specifies that the population is comprised of a finite number of

groups of individuals who follow distinctive developmental trajectories. Each such group

is allowed to have its own offending trajectory (a map of offending rates throughout the

time period) described by a distinct set of parameters that are permitted to vary freely

across groups. This type of model has three key outputs: the parameters describing the

trajectory for each group, the estimated proportion of the population belonging to each

group, and the posterior probability of belonging to a given group for each individual in the

sample. The posterior probability, which is the probability of group membership after the

model is estimated, can be used to assign individuals to a group based on their highest

probability. 14

This approach is less efficient than linear growth models but allows for qualitatively

different patterns of behavior over time. There is broad agreement that delinquency and

crime are cases where this group-based trajectory approach might be justified, in large part

because not everyone participates in crime, and people appear to start and stop at very

different ages (Nagin 1999, 2005; Raudenbush 2001). Given that we have no strong

expectation about the basic pattern of change, the group-based trajectory approach appears

to be an excellent choice for identifying major patterns of change in our data set. 15

There are two software packages available that can estimate group-based trajectories:

Mplus, a proprietary software package, and Proc Traj, a special procedure for use in SAS,

made available at no cost by the National Consortium on Violence Research (for a detailed

discussion of Proc Traj, see Jones et al. 2001). 16

In using Proc Traj, we had three choices

when estimating trajectories of count data: parametric form (Poisson vs. normal vs. logit),

functional form of the trajectory over time (linear vs. quadratic vs. cubic), and number of

groups.

The Poisson distribution is a standard distribution used to estimate the frequency

distribution of offending that we would expect given a certain unobserved offending rate

(Lehoczky 1986; Maltz 1996; Osgood 2000). 17

We found that the quadratic was uni-

formly a better fit than the linear model, and that the cubic model did not improve the fit

over the quadratic in the case of a small number of groups. In choosing the number of

groups we relied upon the Bayesian Information Criteria (BIC) because conventional

14 The group-based trajectory is often identified with typological theories of offending such as Moffitt

(1993) because of its use of groups (see Nagin et al. 1995). But it is important to keep in mind that group assignments are made with error. In all likelihood, the groups only approximate a continuous distribution. The lack of homogeneity in the groups is the explicit trade off for the relaxation of the parametric assumptions about the random effects in the linear models (Bushway et al. 2003). For a different perspective on this issue, see Eggleston et al. (2004). 15

Those interested in a more detailed description of the group-based trajectory approach should see Nagin (1999) or (2005). 16

The procedure, with documentation, is available at www.ncovr.heinz.cmu.edu. 17

Proc Traj also provides the option of estimating a Zero Inflated Poisson (ZIP) model. The ZIP model builds on a Poisson by accommodating more non-offenders in any given period than predicted by the standard Poisson distribution. The zero-inflation parameter can be allowed to vary over time, but cannot be estimated separately for each group. It is sometimes called an intermittency parameter, since it allows places to have ‘‘temporary’’ spells of no offenses without recording a change in their overall rate of offending. In this context, the ZIP model’s differentiation between short-term and long-term change is problematic. The Poisson model, on the other hand, tracks movement in the rate of offending in one parameter, allowing all relatively long-term changes to be reflected in one place. We believe this trait of the Poisson model makes it the better model for modeling trends, especially over relatively short panels, even though the ZIP model provides a better fit according to the BIC criteria used for model selection. For a similar argument see Bushway et al. (2003).

J Quant Criminol (2010) 26:7–32 27

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likelihood ratio tests are not appropriate for defining whether the addition of a group

improves the explanatory power of the model (D’Unger et al. 1998). BIC = log( L) - .5 9 log( n) 9 ( k); where ‘‘ L’’ is the value of the model’s maximized likelihood estimates, ‘‘ n’’ is the sample size, and ‘‘ k’’ is the number of parameters estimated in a given model. Because more sophisticated models almost always improve the fit of a

given analysis, the BIC encourages a parsimonious solution by penalizing models that

increase the number of trajectories unless they substantially improve fit. In addition to

the BIC, trajectory analysis requires that researchers also consider posterior probabilities

of trajectory assignments, odds of correct classification, estimated group probabilities,

and whether meaningful groups are revealed (for a more detailed discussion, see Nagin

2005).

These models are highly complex, and researchers run the risk of arriving at a local

maximum, or peak in the likelihood function, which represents a sub-optimal solution. The

stability of the answer when providing multiple sets of starting values should be considered

in any model choice. In the final analysis, the utility of the groups is determined by their

ability to identify distinct trajectories, the number of units in each group, and their relative

homogeneity (Nagin 2005).

We began our modeling exercise by fitting the data to three trajectories. We then fit the

data to four trajectories and compared this fit with the three-group solution. When the four-

group model proved better than the three-group, we then estimated the five-group model

and compared it to the four-group solution. We continued adding groups, each time finding

an improved BIC, until we arrived at 24 groups. Models for 23 and 24 groups were not

stable and could not be replicated consistently. After reviewing the Bayesian Information

Criteria and the patterns observed in each solution, it was determined that a 22 group

model was the most optimal model for the crime data. We therefore chose the 22 group

model.

The validity of the model was also confirmed by conducting the posterior probability

analysis. The majorities of the within-group posterior probabilities in the model are above

.90, and the lowest posterior probability is .77. The lowest value of the odds of correction

classification (OCC) is 26.58. Nagin (2005) suggests that when average posterior proba-

bility is higher than .7 and OCC values are higher than 5, the group assignment represents a

high level of accuracy. Judging by these standards, the 22-group model performs satis-

factorily in classifying the various crime patterns into separate trajectories.

References

Aitken SC, Cutter SL, Foote KE, Sell JL (1989) Environmental perception and behavioral geography. In: Wilmott Gaile (ed) Geography in America. Columbus, Merrill, pp 218–238

Bailey TC, Gatrell AC (1995) Interactive spatial data analysis. Longman Group Limited, Essex Baller RD, Anselin L, Messner Steven F, Deane G, Hawkins DF (2001) Structural covariates of US county

homicide rates: incorporating spatial effects. Criminology 39:561–590 Baumer EP, Lauritsen JL, Rosenfeld R, Wright R (1998) The influence of crack cocaine on robbery,

burglary, and homicide rates: a cross-city, longitudinal analysis. J Res Crime Delinq 35:316–340 Boggs S (1965) Urban crime patterns. Am Sociol Rev 30:899–908 Braga AA (2001) The effects of hot spots policing on crime. Ann Am Acad Pol Soc Sci 578:104–125 Braga AA, Papachristos AV, Hureau D (2009) The concentration and stability of gun violence at micro

places in Boston, 1980–2008. J Quant Criminol. doi:10.1007/s10940-009-9082-x Brantingham PJ, Brantingham PL (1984) Patterns in crime. Macmillan, New York Brantingham PJ, Brantingham PL (1991) Environmental criminology. Waveland Press, Inc., Prospect

Heights (1981)

28 J Quant Criminol (2010) 26:7–32

123

Brantingham PL, Brantingham PJ (1993) Environment, routine, and situation: toward a pattern theory of crime. In: Clarke RV, Felson M (eds) Routine activity and rational choice, vol 5. Transaction Pub- lishers, New Brunswick, pp 259–294

Brantingham PL, Brantingham PJ (1995) Criminality of place: crime generators and crime attractors. Eur J Crim Pol Res 3:5–26

Brantingham PL, Brantingham PJ (1999) Theoretical model of crime hot spot generation. Stud Crime Crime Prev 8:7–26

Brantingham PL, Brantingham PJ (2008) Crime analysis at multiple scales of aggregation: a topological approach. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research. Springer, New York, pp 87–108

Brantingham PJ, Dyreson DA, Brantingham PL (1976) Crime seen through a cone of resolution. Am Behav Sci 20:261–273

Bursik RJJ, Grasmick HG (1993) Neighborhoods and crime: the dimensions of effective community control. Lexington Books, New York

Bursik RJJ, Webb J (1982) Community change and patterns of delinquency. Am J Soc 88:24–42 Bushway SD, Thornberry TP, Krohn MD (2003) Desistance as a developmental process: a comparison of

static and dynamic approaches. J Quant Criminol 19:129–153 Byrne JM, Sampson RJ (eds) (1986) Social ecology of crime. Springer, New York Chainey S, Tompson L, Uhlig S (2008) The utility of hotspot mapping for predicting spatial patterns of

crime. Secur J 21:4–28 Chakravorty S, Pelfrey WVJ (2000) Exploratory data analysis of crime patterns: preliminary findings from

the Bronx. In: Goldsmith V, McGuire P, Mollenkopf G,JH, Ross TA (eds) Analyzing crime patterns: frontiers of practice. Sage Publications, Thousand Oaks, pp 65–76

Chilton RJ (1964) Continuity in delinquency area research: a comparison of studies for Baltimore, Detroit, and Indianapolis. Am Sociology Rev 29:71–83

Clarke RV (1983) Situational crime prevention: its theoretical basis and practical scope. In: Tonry M, Morris N (eds) Crime and justice: an annual review of research, vol 14. University of Chicago Press, Chicago, pp 225–256

Clarke RV, Felson M (1993) Introduction: criminology, routine activity, and rational choice. In: Clarke RV, Felson M (eds) Routine activity and rational choice, vol 5. Transaction Publishers, New Brunswick, pp 1–14

Cohen LE, Felson M (1979) Social change and crime rate trends: a routine activity approach. Am Sociol Rev 44:588–608

Cohen J, Tita G (1999) Diffusion in homicide: exploring a general method for detecting spatial diffusion processes. J Quant Criminol 15:451–493

Cork D (1999) Examining space-time interaction in city-level homicide data: crack markets and the dif- fusion of guns among youth. J Quant Criminol 15:379–406

Crow W, Bull J (1975) Robbery deterrence: an applied behavioral science demonstration—final report. Western Behavioral Science Institute, La Jolla

D’Unger AV, Land KC, McCall PL, Nagin DS (1998) How many latent classes of delinquent/criminal careers? Results form mixed poisson regression analysis. Am J Sociol 103:1593–1630

Eck JE (1995) Examining routine activity theory: a review of two books. JQ 12:783–797 Eck JE, Weisburd D (1995) Crime places in crime theory. In: Eck JE, Weisburd D (eds) Crime and place.

Willow Tree Press, Monsey, NY, pp 1–33 Eck JE, Chainey S, Cameron JG, Leitner M, Wilson RE (2005) Mapping crime: understanding hotspots.

National Institute of Justice, Washington, DC Eggleston EP, Laub JH, Sampson RJ (2004) Methodological sensitivities to latent class analysis of longterm

criminal trajectories. J Quant Criminol 20:1–26 Felson M (1986) Predicting crime potential at any point on the city map. In: Figlio RM, Hakim S, Rengert

GF (eds) Metropolitan crime patterns. Criminal Justice Press, Monsey, pp 127–136 Felson M (1987) Routine activities and crime prevention in the developing metropolis. Criminology

25:911–931 Felson M (2002) Crime in everyday life, 3rd edn. Sage, Thousand Oaks Fotheringham AS, Brundson C, Charlton M (2000) Quantitative geography. Sage Publications, London Gold JR (1980) An introduction to behavioural geography. Oxford University Press, New York Golledge RG, Timmermans H (1990) Applications of behavioural research on spatial problems I: cognition.

Prog Hum Geogr 14:57–99 Griffiths E, Chavez JM (2004) Communities, street guns, and homicide trajectories in Chicago, 1980–1995:

merging methods for examining homicide trends across space and time. Criminology 42:941–978

J Quant Criminol (2010) 26:7–32 29

123

Groff ER, LaVigne NG (2001) Mapping an opportunity surface of residential burglary. J Res Crime Delinq 38:257–278

Groff ER, Weisburd D, Morris N (2008) Where the action is at places: examining spatio-temporal patterns of juvenile crime at places using trajectory analysis and GIS. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research. Springer, Berlin

Grubesic TH, Mack EA (2008) Spatio-temporal interaction of urban crime. J Quant Criminol 24:285–306 Guerry A-M (1833) Essai sur la Statisticque morale de la France. Crochard, Paris Hillier B (1999) The common language of space: a way of looking at the social, economic and environ-

mental functioning of cities on a common basis. Retrieved February 17, 2004, from http://www. spacesyntax.org/publications/commonlang.html

Hillier B (2004) Can streets be made safe? Urban Des Int 9:31–45 Jacobs J (1961) The death and life of great American cities. Vintage Books, New York Jefferis E (2004) Criminal places: a micro-level study of residential theft . Unpublished Dissertation,

University of Cincinnati, Cincinnati Jeffery CR (1971) Crime prevention through environmental design. Sage Publications, Beverly Hills Johnson SD, Lab SP, Bowers KJ (2008) Stable and fluid hotspots of crime: differentiation and identification.

Built Environ 34:32–45 Jones BL, Nagin DS, Roeder K (2001) A SAS procedure based on mixture models for estimating devel-

opmental trajectories. Sociol Methods Res 29:374–393 Kaluzney SP, Vega SC, Cardoso TP, Shelly AA (1998) S?SpatialStats: users manual for Windows and

UNIX. Insightful, Seattle, WA Kinney JB, Brantingham PL, Wuschke K, Kirk MG, Brantingham PJ (2009) Crime attractors, generators

and detractors: land use and urban crime opportunities. Built Environ 34:62–74 Kornhouser R (1978) Social sources of delinquency: an appraisal of analytic models. University of Chicago,

Chicago Kubrin CE, Herting JR (2003) Neighborhood correlates of homicide trends: an analysis using growth-curve

modeling. Sociol Quart 44:329–350 Lab SP (2007) Crime prevention: approaches, practices, evaluations. Anderson Publishing, Cincinnati Lehoczky J (1986) Random parameter stochastic process models of criminal careers. In: Blumstein A,

Cohen J, Roth JA, Visher CA (eds) Criminal careers and career criminals. National Academy of Sciences Press, Washington, DC

Levine N (2005) CrimeStat: a spatial statistics program for the analysis of crime incident locations, vol 3.0. Ned Levine & Associates, Houston, TX and National Institute of Justice, Washington, DC

Linnell D (1988) The geographic distribution of hot spots of robbery, rape, and auto theft in Minneapolis . Unpublished MA, University of Maryland, College Park

Loftin C, Hill RH (1974) Regional subculture and homicide: an examination of the Gastil-Hackney thesis. Am Sociol Rev 39:714–724

Maltz MD (1996) From Poisson to the present: applying operations research to problems of crime and justice. J Quant Criminol 12(1):3–61

Merry SE (1981) Defensible space undefended: social factors in crime control through environmental design. Urban Aff Q 16:397–422

Messner SF, Anselin L (2004) Spatial analysis of homicide with areal data. In: Goodchild MF, Janelle DG (eds) Spatially integrated social science. Oxford University Press, New York, pp 127–144

Moffitt TE (1993) Adolescence-limited and life-course persistent antisocial behavior: a developmental taxonomy. Psychol Rev 100:674–701

Morenoff JD, Sampson RJ (1997) Violent crime and the spatial dynamics of neighborhood transition: Chicago, 1970–1990. Soc Forces 76:31–64

Nagin DS (1999) Analyzing developmental trajectories: a semiparametric group-based approach. Psychol Methods 4:139–157

Nagin DS (2005) Group-based modeling of development over the life course. Harvard University Press, Cambridge

Nagin DS, Land KC (1993) Age, criminal careers, and population heterogeneity: specification and esti- mation of a nonparametric, mixed poisson model. Criminology 31:327–362

Nagin DS, Farrington DP, Moffitt TE (1995) Life-course trajectories of different types of offenders. Criminology 33:111–139

Newman O (1972) Defensible space: crime prevention through environmental design. Macmillan, New York

Oberwittler D, Wikstrom P-O (2008) Why small is better: advancing the study of the role of behavioral context in crime causation. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research, vol 35–60. Springer, New York

30 J Quant Criminol (2010) 26:7–32

123

Osgood DW (2000) Poisson-based regression analysis of aggregate crime rates. J Quant Criminol 16(1): 21–43

Pierce G, Spaar S, Briggs LR (1986) The character of police work: strategic and tactical implications. Center for Applied Social Research, Northeastern University, Boston

Potchak MC, McGloin JM, Zgoba KM (2002) A spatial analysis of criminal effort: auto theft in Newark, New Jersey. Crim Just Pol Rev 13:257–285

Quetelet AJ (1831[1984]) Research on the propensity for crime at different ages (trans: Test Sylvester SF). Anderson Publishing Co, Cincinnati

Ratcliffe JH (2004) Geocoding crime and a first estimate of a minimum acceptable hit rate. Int J Geogr Inf Syst 18:61–72

Raudenbush SW (2001) Comparing personal trajectories and drawing causal inferences from longitudinal data. Annu Rev Psychol 52:501–525

Reiss A J Jr, Tonry M (1986) Preface. In: Reiss A J Jr, Tonry M (eds) Communities and crime. University of Chicago Press, Chicago, pp 1–34

Rengert G, Lockwood B (2008) Geographical units of analysis and the analysis of crime. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research, vol 109–122. Springer, New York

Robinson JB, Lawton BA, Taylor RB, Perkins DD (2003) Multilevel longitudinal impacts of incivilities: fear of crime, expected safety, and block satisfaction. J Quant Criminol 19:237–274

Roman CG (2002) Schools as generators of crime: routine activities and the sociology of place . Unpub- lished Dissertation, American University, Washington DC

Roncek DW (2000) Schools and crime. In: Goldsmith V, McGuire P, Mollenkopf G,JH, Ross TA (eds) Analyzing crime patterns: frontiers of practice. Sage Publications, Thousand Oaks, pp 153–165

Rowlingson BS, Diggle PJ (1993) Splancs: spatial point pattern analysis code in S-plus. Comput Geosci 19:627–655

Sampson RJ, Raudenbush SW (1999) Systematic social observation of public spaces: a new look at disorder in urban neighborhood s. Am J Soc 105:603–651

Schuerman L, Kobrin S (1986) Community careers in crime. In: Reiss AJJ, Tonry M (eds) Communities and crime. University of Chicago, Chicago, pp 67–100

Sherman LW (1995) Hot spots of crime and criminal careers of places. In: Eck J, Weisburd DL (eds) Crime and place, vol 4. Willow Tree Press, Monsey

Sherman LW, Weisburd D (1995) General deterrent effects of police patrol in crime ‘hot spots’: a ran- domized, controlled trial. JQ 12:625–648

Sherman LW, Gartin P, Buerger ME (1989) Hot spots of predatory crime: routine activities and the criminology of place. Criminology 27:27–55

Skogan WG (1986) Fear of crime and neighborood change. In: Reiss AJJ, Tonry M (eds) Communities and crime. University of Chicago Press, Chicago, pp 203–230

Smith WR, Frazee SG, Davison EL (2000) Furthering the integration of routine activity and social disor- ganization theories: small units of analysis and the study of street robbery as a diffusion process. Criminology 38:489–523

Spring JV, Block CR (1988) Finding crime hot spots: experiments in the identification of high crime areas. Paper presented at the Midwest Sociological Society

Stark R (1987) Deviant places: a theory of the ecology of crime. Criminology 25:893–909 Taylor RB (1997) Social order and disorder of street blocks and neighborhoods: ecology, microecology, and

the systemic model of social disorganization. J Res Crime Delinq 34:113–155 Taylor RB (1998) Crime and small-scale places: what we know, what we can prevent, and what else we

need to know. In: Taylor RB, Bazemore G, Boland B, Clear TR, Corbett RPJ, Feinblatt J, Berman G, Sviridoff M, Stone C (eds) Crime and place: plenary papers of the 1997 conference on criminal justice research and evaluation, National Institute of Justice, Washington, DC, pp 1–22

Taylor RB (2001) Breaking away from broken Windows: baltimore neighborhoods and the nationwide fight against crime, grime, fear and decline. Westview Press, Boulder

Taylor RB, Gottfredson SD (1986) Enivronmental design, crime, and prevention: an examination of community dynamics. In: Reiss A J Jr, Tonry M (eds) Communities and crime. University of Chicago Press, Chicago, pp 387–416

Taylor RB, National Consortium on Violence Research (1999) A longitudinal look at the incivilities thesis: does disorder bring later crime and decline? Paper presented at the Eastern Sociological Association, Boston, MA

Timmermans H, Golledge RG (1990) Applications of behavioural research on spatial problems II: prefer- ence and choice. Prog Hum Geogr 14:311–354

J Quant Criminol (2010) 26:7–32 31

123

van Wilselm J (2008) Urban streets as micro contexts to commit violence. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research. Springer, Berlin, pp 199–216

Walmsley DJ, Lewis GJ (1993) People and environment: behavioral approaches in human geography. Longman Scientific & Technical, Essex

Weisburd D (2002) From criminals to criminal contexts: reorienting crime prevention. In: Waring E, Weisburd D (eds) Crime & social organization, vol 10. Transactions Publishers, New Brunswick, pp 197–216

Weisburd D, Braga AA (2002) Hot spots policing. In: Kury H, Fuchs O (eds) Crime prevention: new approaches. Mainz, Germany, Weisner Ring

Weisburd D, Green L (1994) Defining the drug market: the case of the Jersey City DMA system. In: MacKenzie DL, Uchida CD (eds) Drugs and crime: evaluating public policy initiatives. Sage, Newbury Park

Weisburd D, Green L (1995) Policing drug hot spots: the Jersey City drug market analysis experiment. JQ 12:711–735

Weisburd D, Mazerolle L (2000) Drug hot spots and crime. Police Quart 3:331–349 Weisburd D, McEwen T (1997) Introduction: crime mapping and crime prevention. In: Weisburd DL,

McEwen T (eds) Crime mapping and crime prevention: crime prevention studies, vol 8. Criminal Justice Press, Monsey, pp 1–26

Weisburd D, Maher L, Sherman LW (1992) Contrasting crime general and crime specific theory: the case of hot spots of crime. In: Adler F, Laufer WS (eds) Advances in criminological theory, vol 4. Transaction Press, New Brunswick, NJ, pp 45–70

Weisburd D, Bushway S, Lum C, Yang S-M (2004a) Trajectories of crime at places: a longitudinal study of street segments in the city of Seattle. Criminology 42:283–321

Weisburd D, Lum C, Yang S-M (2004b) The criminal careers of places: a longitudinal study. US Department of Justice, National Institute of Justice, Washington, DC

Weisburd D, Bruinsma G, Bernasco W (2008) Units of analysis in geographic criminology: historical development, critical issues and open questions. In: Weisburd D, Bernasco W, Bruinsma G (eds) Putting crime in its place: units of analysis in spatial crime research. Springer, New York, pp 3–31

Weisburd D, Groff E, Yang S-M (2009a) Explaining developmental crime trajectories at places: a study of ‘‘crime waves’’ and ‘‘crime drops’’ at micro units of geography. National Institute of Justice, Wash- ington DC (in progress)

Weisburd D, Morris N, Groff ER (2009b) Hot spots of juvenile crime: a longitudinal study of street segments in Seattle, Washington. J Quant Criminol 24:443–467

Weisburd D, Groff E, Yang S-M (under review) Understanding developmental crime trajectories at places: social disorganization and opportunity perspectives at micro units of geography. National Institute of Justice, Washington, DC

Wolfgang ME, Figlio RM, Sellin T (1972) Delinquency in a birth cohort. The University of Chicago Press, Chicago

32 J Quant Criminol (2010) 26:7–32

123

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