Benchmark - Recommendation for Crime Reduction

Computers, Environment and Urban Systems 39 (2013) 93–106

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Com puters, Environ ment and Urban System s

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Understanding the spatial distribution of crime based on its related variables using geospatial discriminative patterns

0198-9715/$ - see front matter � 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compenvurbsys.2013.01.008

⇑ Corresponding author. E-mail addresses: ding@cs.umb.edu (W. Ding), chenp@uhd.edu (P. Chen).

Dawei Wang a, Wei Ding a,⇑, Henry Lo a, Melissa Morabito b, Ping Chen e, Josue Salazar c, Tomasz Stepinski d a Department of Computer Science, University of Massachusetts Boston, United States b Department of Criminology and Criminal Justice, University of Massachusetts Lowell, United States c Department of Computer Science, Rice University, United States d Department of Geography, University of Cincinnati, United States e Computer and Mathematical Sciences Department, University of Houston-Downtown, Texas, United States

a r t i c l e i n f o

Article history: Received 23 April 2012 Received in revised form 25 January 2013 Accepted 31 January 2013 Available online 9 April 2013

Keywords: Crime related variable Geospatial Discriminative Pattern Hotspot Optimization Tool Footprint

a b s t r a c t

Crime tends to clust er geographi cally. This has led to the wide usage of hotspot analysis to identify and visualize crime. Accurately identified crime hotspots can greatly benefit the public by creating accurate threat visualizations, more efficiently allocating police resources, and predicting crime. Yet existing map- ping methods usually identify hotspots without considering the underlying correlates of crime. In this study, we introduce a spatial data mining framework to study crime hotspots through their related vari- ables. We use Geospatial Discriminative Patterns (GDPatterns) to capture the significant difference between two classes (hotspots and normal areas) in a geo-spatial dataset. Utilizing GDPatterns, we develop a novel model—Hotspot Optimization Tool (HOT)—to improve the identification of crime hotspots. Finally, based on a similarity measure, we group GDPattern clusters and visualize the distribution and characteristics of crime related variables. We evaluate our approach using a real world dataset collected from a northeast city in the United States.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Crime is understood to be related to the interactio n of victims and offenders, and to the strength of guardianship (Cornish & Clarke, 1986 ). In practice, these concepts can be measure d using a variety of socio-economi c and crime opportunity variables, such as population density, economic investment, and arrest rate.

Geographical studies reveal that crime is often concentr ated in clusters, which in the literature are called hotspots. Hotspot map- ping techniques for crimes draw continuo us attention from researchers and public safety agencies. This is because accurately identified and clearly visualized crime hotspots , and understanding their relation to underlyin g crime related variables, can signifi- cantly benefit crime analysis and police practices by providing a solid basis for threat visualization, police resource allocation, and crime predictio n.

Existing hotspot mapping methods can be essentially divided into three main categories: point mapping, choropleth mapping, and kernel density estimation (KDE) (Eck, Chainey, Cameron, Leit- ner, & Wilson, 2005; Williamson , McGuire, Ross, Mollenkopf , & Goldsmith, 2001; Boba, 2005 ). Usually, these methods aggregat e

the density of a target crime, which results in a net loss of informa- tion (Van Patten, McKeldin -Coner, & Cox, 2009 ). For example, in chorople th mapping, incident-level data is first aggregated into arbitrary administrat ive or political boundary areas. During this step, spatial details within and across the thematic areas are lost. Second, when hotspots are generate d based on aggregat ed data, there is a further decline of precision in the resulting map. Because traditional methods mainly rely on target crime density, particular areas with relatively less crime may be left out of hotspots, even though crime related variables indicate they are under similar risks as those hotspots .

A reasonable way to reduce this accuracy and precision loss in chorople th mapping is to use more related information in the map- ping process. Crime related variables can be aggregated and used along with target crime data in the hotspot identification process. Informati on carried by these variables can provide clues on whether the relatively high crime rate in a certain area happens by chance. Compared to traditional methods, the utilization of re- lated informat ion in hotspot mapping can reduce information loss during analysis.

Addition ally, such an approach can benefit further analysis on the characteristics of crime related variables. Instead of just evaluating crime by itself, recent studies also integrate crime related data into a unified framework that assists the analysis and exploration of crime hotspots (Maciejews ki et al., 2010 ). Using

94 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

related variables in hotspot mapping can additionally benefit such visualization and analyzation processes by providing an intuitive linkage between target crime and its related data.

In this paper, we present a framework that uses spatial data mining concepts to map hotspots and investigate the relationship between socio-eco nomic and criminal variables. Recently , spatial data mining has emerged as an active research area in studies of spatial relationshi ps that try to answer the questions like ‘‘why’’ and ‘‘where’’ (Ester, Kriegel, & Sander, 1997; Mu, Ding, Mor- abito, & Tao, 2011 ). It has been proven to be very powerful in iden- tifying the linkage between target objects and its related factors. The components of our method are shown in Fig. 1. In particular, we:

� Introduce a spatial data mining concept, Geospatial Discrimina - tive Patterns (GDPatterns), to study the relationship between target crime hotspots and their underlying related variables. � Introduce a model, Hotspot Optimization Tool (HOT), to identify

crime hotspots through their related variables.

Fig. 1. The framework of our methods. With the help of GDPatterns, criminal hotspot m are clustered and visualized for domain scientists.

� Use a similarity based method to cluster the crime related vari- ables that contribute to hotspots into groups. � Visualize the locations of those clusters in a rational way to

assist domain scientists in further analysis, using the footprint s of GDPatterns .

Utilizing the proposed framework, a case study is conducte d using a 6-year crime dataset from a city in northeast United States. We compare our mapping tool with a widely used hotspot evaluat- ing technique,the G�i statistics (Getis & Ord, 2010 ), and demon- strate the potential in assisting crime analysis using related variable clusters.

The rest of the paper is organized as follows. Section 2 discusses related work. Section 3 introduces the data represen- tation and formal definition of the research problems . HOT mod el and the implementation of the similarity mea sure are also presented in this section. Section 4 evaluates the proposed framework in a real-world case study. We conclude the paper in Section 5.

aps are generated using HOT. By applying a similarity measure method, GDPatterns

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2. Related work

In this section we briefly present some literatures related to criminology , spatial data mining, and hotspot mapping techniqu es. Additionally , we give a brief introduction to a choropleth mapping application—the Hotspot Analysis (HSA) tool implemented by Esri ArcGIS (ESRI, 2011 ).

Occurrence of crime has been linked to a number of different variables. Classic criminology theories, such as Routine Activities Theory (Cohen & Felson, 1979 ), conclude that three concepts con- tribute to crime: accessible and attractive targets, a pool of moti- vated offenders, and lack of guardianship (Brantingham & Brantingham, 1984; Cornish & Clarke, 1986 ). The concept of ‘‘disor- der’’ (Skogan, 1992 ) explains why adjacent areas of crime hotspots are at higher risk. The probability of arrest or the social penalties for committ ing crime may be lower in crime hotspots than in other neighborho ods, which leads to the ‘‘contagion’’ of criminal activity in crime hotspots (Ludwig, Duncan, & Hirschfield, 2001; Sah, 1991; Sampson, Raudenbush, & Earls, 1997 ). Recent work done by Short, Bertozzi, and Brantingham (2010) also discusses how an area is af- fected by the activity scope of offenders. Criminology theories ex- plain why crime is clustered in particular areas, and why certain victims are selected. They also help in deciding which variables are related to a certain type of crime.

Spa tial dat a min ing (Est er et al., 199 7) is a kno wle dge dis cov ery tech niq ue for ‘‘e xtra ctio n of imp lici t kno wle dge , spa tial relat ion s, or oth er pat tern s not expl ici tly sto red in spat ial dat abas es’’ (Kop ersk i & Han, 199 5). It has been pro ven to be very pow erfu l and efficien t for stu dyi ng com preh ens ive rela tion ship s in lar ge dat abas es (Mil ler & Han, 2009 ; Est er et al., 1997 ; Qia n, He, Chie w, & He, 2012 ). The GDP atte rn is an app lica tion of inte gra ting spat ial ass ocia tion rul es (Agr awal et al., 199 4; Kop ersk i & Han , 199 5) wit h emerg ing pat tern s (Don g & Li, 199 9; Her rera , Car mon a, Gonz ález, & del Jesu s, 201 1; Yu, Ding , Sim ovi ci, & Wu, 201 2). App lica tion s usin g ass oci atio n rul es have been dev elope d to expl ore the spa tial and tem por al rel atio n- ship s among obj ects usin g cens us data (Mal erb a, Espo sito , Lisi , & Ap- pice , 2002 ). In the wor k of Men nis (2006) and Menn is and Liu (2005), asso ciat ion rule mini ng tec hni ques hav e been use d to exp lore the non- lin ear rela tion shi ps amo ng soc ioe conom ic- vege tati on var i- abl es. In the wor k of Lin (1998) the auth ors pre sen t a simil ari ty mea- sure method for summ ari zin g larg e numb er of emerg ing pat tern s. Ding , Step insk i, and Sal aza r (2009) ado pts the rel ativ e risk rati o as the mea sure of pat tern eme rgen ce and use s spati al dat a mini ng tech niq ues in inv est iga ting veget atio n rem ote sen sin g dat aset s. In our wor k GDPatt ern s are use d as a tool to dis cover the sta tica lly sig- nificant dif fere nce betw een targ et crim e hots pot s and nor mal are as spat iall y, wit h res pec t to the und erl ying rel ated var iab les.

The Spatial and Temporal Analysis of Crime (STAC) program (Bates, 1987 ) is one of the earliest and widely used hotspot map- ping applications. Based on point mapping, STAC uses ‘‘standard deviational ellipses’’ to display crime hotspots on a map and does not pre-define any spatial boundaries. But some studies (Eck et al., 2005 ) show that STAC may be misleading because hotspots do not naturally follow the shape of ellipses. Another popular hot- spot representat ion method is choropleth mapping, in which boundary areas (geographic boundaries like census blocks or uni- form grids) are used as the basic mapping elements (Hirschfield, 2001). Unlike point mapping, choropleth mapping uses aggregate data, which removes spatial details within the thematic areas. Also, identified hotspots are restricted to the shape of these areas. The method of Kernel Density Estimation (KDE) (Wand & Jones, 1995) aggregates point data inside a user-spe cified search radius and generate s a continuous surface representi ng the density of points. It overcomes the limitatio n of geometric shapes but still lacks statistical robustness that can be validated in the produced

map. Reviews and comparative studies for the three methods have been done in the works of Chainey, Tompson, and Uhlig (2008), in which authors introduce a ‘‘prediction accuracy index’’ to evaluate the accuracy of the different methods in the content of predicting where crime may occur.

Esri ArcGIS (ESRI, 2011 ) is the most widely used Geographic Informati on System (GIS) and its newest component, ArcMap 10.1, includes a Hotspot Analysis (HSA) toolbox, which implements the G�i statistics (Getis & Ord, 2010 ) and provides users the ability to analyze the hotspots existed in the input spatial dataset (usually a polygon map with interested attributes). In particular, HSA calcu- late the G�i statistics and outputs z-scores and p-values for each spatial area (polygon) that tell the statistically significance of the polygon as a hotspot. To be a statistically significant hotspot, a polygon will have a high value of the target attribute and be sur- rounded by other polygons with high values as well. The local sum of the attribute values for a polygon and its neighbors are compare d proportionally to the sum of attribute values of all poly- gons. When the local sum is very different from the expected local sum (very high z-score), and that difference is too large to be the result of random chance (very small p-value), the polygon is con- sidered as a hotspot.

3. Methodology

The key insight behind our methods is identifyin g hotspots by searching , utilizing, and presenting patterns in geographic space. By preprocessing the crime related data sets into a transacti on- based geospatia l dataset, we develop a model, called HOT, to map crime hotspots through the related variables. Then we introduce a similarity method to summari ze the identified GDPatterns into clusters. Based on these clusters, a relevant report of crime hotspots and related variables is visually presented for domain experts.

3.1. Problem formulation and data representatio n

To discover GDPatterns from a target crime’s related variables, we firstly build a transacti on-based geospatial database, which we refer to as the database or simply D. A widely used method for representing spatial distribution of entities in D is through grid mapping (Harries, 1999; Janeja & Palanisamy, 2012 ). Both target crime and related variables in the original spatial dataset can be plotted onto grid maps with the same dimensions . The cell value in the grid is assigned to be the number of incidents falling into it. An illustrative example of D is shown on the top right of Fig. 1. Addition ally, instead of using the original values directly, the way to fairly represent all the variables in one pattern is to cat- egorize them and change the original values into categories. Stan- dard tools (Nguyen & Nguyen, 1998 ) such as the Jenks Optimization for Natural Breaks Classification (or Nature Breaks; Jenks, 1967 ), a method that is based on natural groupings inherited in data, can be used in the categorization process.

Definition 1 (Database object ). A object in D is a tuple of the form: {x, y, V1, V2, . . . , Vn, C}, where x, y indicate the object’s spatial coordi- nates, V1, V2, . . . , Vn are the values of the related variables, and C is the class label of target crime.

Using C, objects in D can be labeled into different classes. For example, we say C is 0 if the area is not a hotspot (or normal area) and 1 if the area is a hotspot. Then the geospatial database can be divided into two parts: Dh (hotspots) if C = 1, or Dn (normal area) if C = 0. Disregarding the location informat ion (x, y) and the class la- bel C, each object in D can be viewed as a transaction of n variable values. For example, in Table 1, T1, T2, T3, and T4 are transacti ons with three variable values.

Table 1 Examples of transactions, patterns and patterns’ supports. In the examples AR, POP and IC stand for arrest rate, population density and average income, respectively. Pattern X3 is not a closed pattern because X1, its immediate superset, has exactly the same support. X1 is a closed frequent pattern if we set the minimum support threshold q = 70%.

Transactions T1: {AR = high, POP = low, IC = low} T2: {AR = high, POP = low, IC = high} T3: {AR = high, POP = low, IC = medium} T4: {AR = medium, POP = low, IC = medium}

Patterns Support

X1: {AR = high, POP = low} supðX1Þ¼ 34 ¼ 75%ðT 1; T 2; T 3Þ X2: {AR = high, IC = high} supðX2Þ¼ 14 ¼ 25%ðT 2Þ X3: {AR = high} supðX3Þ¼ 34 ¼ 75%ðT 1; T 2; T 3Þ

96 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

3.2. Geospatial Discriminative Patterns (GDPatterns)

The GDPatterns we are looking for should meet two require- ments: (1) to significantly represent the situation or condition s of related variables in objects in database D; (2) to significantly dis- tinguish hotspots Dh from normal areas Dn. GDPatterns are built upon closed frequent patterns. Here we give a brief introduction of relevant concepts.

Definition 2 (Pattern). Given a set of related variables, a pattern is a set of values for a subset of those related variables.

For example, Table 1 gives an example of a database that has 3 related variables AR, POP, and IC, which can take the values of low, medium, or high. In the examples AR, POP and IC stand for arrest rate, population density and average income, respectively . A com- bination of these variables and values constitutes a pattern; e.g., X1: {AR = high, POP = low}, or X3: {AR = high}.

Definition 3 (Support and support count (Agrawal et al., 1994)). A pattern is said to be supported by a transacti on when it is a sub set of the transaction. The support count of a pattern X is the number of times X appears in a database D.

supportcountDðXÞ¼ jfT 2 DjX # Tgj ð1Þ

where T represe nts transactions in D. The support of a pattern X is calculated as the support count of X

divided by the total number of transactions in the database D.

supportDðXÞ¼ supportcountDðXÞ

jDj ð2Þ

For exampl e, in Table 1 pattern X1 = {AR = high, POP = low} is sup- ported by transactions T1,T2 and T3, then the support count of X1 is 3 for the database . Since there are totally 4 transac tions in this database , the support of X1 is 3/4 = 0.75.

Definition 4 (Closed pattern (Pasquier, Bastide, Taouil, & Lakhal, 1999)). A pattern is closed if none of its supersets has exactly the same support.

For example, in Table 1 X1 is a closed pattern and X3 is not, be- cause its immediate superset X1 has exactly the same support.

Note that if we consider only closed frequent patterns, we can deduce the support of non-clos ed frequent patterns from their cor- respondent closed patterns. To see why this is true, note that the supports of patterns exhibit a property called downwa rd closure:

If X � X0; then supportDðXÞ P supportDðX 0Þ

Thus, if X is closed, and X0 is not, then supportD(X) = supportD(X0). The benefit of considering only closed patterns is a reduction in

the set of considered patterns without losing informat ion. In Table 1 both X3 and X1 are supported by T1, T2 and T3. In other words, both X3 and X1 carry information about the characteri stics of these transactions . But X1 carries more information ({AR = high, - POP = low}) than X3 ({AR = high}) does and the informat ion carried by X3 ({AR = high}) is fully represented by X1. There is no informa- tion loss if we only consider X1 in further analysis.

Definition 5 (Closed frequent pattern (Pasquier et al., 1999)). A closed pattern whose support is above a user-defined threshold is considered as a closed frequent pattern.

Definition 6 (Growth ratio ). Let set {Dh, Dn} be an exhaustive par- tition of D. The growth ratio d of a pattern X is the ratio of X0s sup- port in one partition Dh to its support in the other partition Dn.

d ¼ supportDhðXÞ supportDnðXÞ

ð3Þ

Definition 7 (Geospatia l Discriminative Pattern (GDPattern)). A closed frequent pattern X whose growth ratio exceeds a user- defined threshold is considered a GDPattern.

With a rational growth ratio threshold, the GDPatterns mined from D carries information that is significantly different between a subset and the remainder in D. For example, if the growth ratio is greater than 20, thus a closed frequent pattern will be considered as a GDPattern when the pattern is 20 times more frequent in hot- spots than in normal areas. In other words, this pattern will have a more than 95% (19/20) chance of being found in hotspots. So the locations out of which such a pattern is mined are more than 95% (or ‘‘significantly’’) likely to be a hotspot.

Definition 8 (Footprint). The footprint of a GDPattern X is the objects that support X in database D. It is the set of cells in the grid map whose correspondi ng objects support X.

For example, in Fig. 2 a GDPattern: {Comme rcial Burglary- ‘‘low’’, Street Robbery-‘ ‘Average’’, Motor-Vehicle Larceny-‘‘Aver - age’’} is selected from the case study (Section 4) and the hollow squares with slash lines are footprints of this GDPattern. These areas (the footprint s) have similar characterist ics of the related variables (low in commercial burglary rate and average in street robbery and motor-vehic le larceny rate). The utilizing of footprint provides a way to measure the spatial distribution of the corre- sponding patterns in studied area.

3.3. Hotspot Optimization Tool

GDPatterns are capable of digging out the meaningful informa- tion underlying the spatial distribut ion of target crime hotspots . Utilizing the informative GDPatterns, here we develop a model, Hotspot Optimization Tool (HOT), to emphasize the identification of hotspots by optimizin g user-specified hotspot boundaries. The practicality of HOT is based on two concepts: firstly, a hot- spot can be considered as the source of disorder of its adjacent blocks, which means the adjacent areas have the possibility of being affected by crimes happening in hotspots. Also, from the point of view of spatial correlations (Bailey & Gatrell, 1995 ), adja- cent areas of a hotspot are more likely to fall into the active range of the same criminals. Therefore these cells can be considered as potential hotspots, especially those with a relatively high crime density. Secondly, according to the definition, GDPatterns are much more frequent in hotspots than in normal areas. Normal areas located in the footprint s of GDPatterns are more likely to be hotspots because in these areas the values of related variables are the same.

Fig. 2. A example map of GDPattern footprints. By selecting Residential Burglary (RB) data as the target crime, nine other variables are used as related variables from the experiment dataset and GDPatterns are mined with a growth ratio larger than twenty (d P 20). The hollow squares with slash lines are footprints of one example GDPattern (Commercial Burglary-‘‘low’’, Street Robbery-‘‘Average’’, Motor-Vehicle Larceny-‘‘Average’’) whose grawth ratio is 67.0. The red area are RB hotspots defined by a user-specific threshold. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106 97

In summary , by initializing hotspots of a target crime with a user-specified threshold, HOT considers a normal location as a hot- spot if (1) it is adjacent to current hotspots; (2) its crime rate is rel- atively high compared to the user-spe cified hotspot threshold; and (3) it is inside the footprints of GDPatterns mined out of current hotspots. The detailed process of HOT is showed in Algorithm 1.

Algorithm 1. The Hotspot Optimiza tion Tool.

This algorithm takes as input a geospatial dataset D, a hotspot threshold h, a hotspot candidate threshold h0, a support threshold q of closed frequent pattern, a growth ratio threshold d, and re- turns a new set of hotspots Dh, a set of GDPatterns G, and their foot- prints w. It does the following:

� Identify areas with a relatively high crime density (Dh0 , areas with high target crime density that are close to the density in hotspots , line 2). � Mine GDPatterns based on current hotspot boundari es and

draw the footprint s of GDPatterns (lines 6 and 7). � Generate candidate cells (lines 8–12): cells whose correspond-

ing objects belong to Dh0 and adjacent to some cell whose corre- sponding objects belong to Dh. � Test the hypothesis for candidate cells (line 14): a candidate cell

is inside the footprint s of GDPatterns (w). � If the hypothes is is true, the boundaries of the hotspot are mod-

ified by changing the current cell into a hotspot cell (moving its correspondi ng object from Dh0 to Dh) (line 15). � Iterate until all hypothesis tests are false (lines 3 and 19).

When hotspot boundari es are changed, a new set of GDPatterns will be generated based on the modified hotspots , followed by the change of footprints. If in the current loop the set of GDPatterns is the same as the former loop, it means there are no new footprints and there will be no ‘‘true’’ from the hypothesis test (lines 4–10 in Algorithm 1). The HOT will stop and a new optimized hotspot map is generate d.

3.4. Crime related variables demonstration

Hotspots of target crime extracted using GDPatterns carry a wealth of informat ion. But the GDPattern mining process usually results in an explosive number of possible patterns (Han et al.,

98 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

2000). It is desirable to organize these GDPatterns in a meaningful way in order to make the information usable to domain analysts. Here we present a pattern summarization method that can cluster GDPatterns into small groups which have similar structures.

Given two patterns X and Y that are mined out of m variables, the function to calculate similarity between X and Y is

s0ðX; YÞ¼ Pm

i¼1sðXi; Y iÞ m

ð4Þ

where s0(X, Y) is the similarity betwee n pattern X and Y; s(Xi, Yi) is the similarity between the ith variable s of X and Y; m is the number of variable s in each pattern. For example, s(Xi, Yi) = 1 if Xi and Yi are in the same category and 0 if they are not. We calculate the similar- ities between every variable and take the mean of the m similar ities as the overall similarity between the pattern s.

The categories of the crime related variables can be presented using ordinal numbers . For example, the categories of population density can be presented using ordinal numbers: 1 (‘‘low’’), 2 (‘‘medium’’) and 3 (‘‘high’’). The similarity between two ordinal values of the ith variable s(Xi, Yi) can be measured by the ratio be- tween the amount of informat ion needed to state the commonality between Xi and Yi, and the information needed to fully describe both Xi and Yi. In practice when we calculate the similarity be- tween patterns X and Y, the ith variable does not always exist in both patterns (Fig. 3). There are three cases according to the pres- ence of Xi and Yi.

Case 1: Both Xi and Yi are in the pattern:

sðXi; Y iÞ¼ 2 � log PðXi _ Z1 _ Z2 � � �_ Zk _ Y iÞ

log PðXiÞþ log PðY iÞ ð5Þ

where P() is the probability calculated using the known distrib ution of the values of ith variable in D and Z1, Z2, . . . , Zk is the ordinal inter- vals delimited by Xi and Yi. For example, in Fig. 3 the ordina l interv al between the first variable XAand YA is Z1 = 2.

Case 2: Either Xi or Yi is absent (here we use the case that Xi is absent):

sð�; Y iÞ¼ Xn k¼1

PXðZkÞsðZk; Y iÞ ð6Þ

where n is the amount of different values that the ith variable has, PX(Zk) is the probability of the ith variable having value Zk in all transactions that support pattern X. PX(Zk) = 0 if Zk does not exist in the footprin t of X at all and

Pn k¼1PXðZkÞ¼ 1. The similarity is a

weighte d average betwee n Yi and all ordinal values of the ith vari- able presented in the footprin t of pattern X. Example is shown in Fig. 3 case 2.

Case 3: Neither Xi or Yi is present:

sð�;�Þ¼ Xn l¼1

Xn k¼1

PXðZlÞPY sðZkÞsðZl; ZkÞ ð7Þ

Fig. 3. An illustrative example showing the similarit

In this case the probability of all ordinal values (Z1, Z2, . . . , Zn) of the ith variable in patterns X and Y are checked and a weighted average pairwise comparisons is calculated (case 3 in Fig. 3).

Using the similarity measureme nts, we can build a N � N dis- tance matrix of GDPatterns using distance ¼ 1similarity, where N is the number of GDPatterns . Standard clustering tools such as Hier- archical Agglomerati ve Clustering (HAC), which treat each GDPat- tern as a singleton cluster at the outset and then successivel y merge (or agglomerate) pairs of clusters according to their distance until all clusters have been merged into a single cluster that con- tains all GDPatterns , can be used to group the closest GDPatterns into clusters.

These clusters serve as compositi ons of crime related variables and carry rich information not only about relationships between variables, but also about their spatial distributions. Locations exhibiting certain socio-economi c and crime-related characteris- tics tShat are significantly related with target crime hotspots can be drawn using the clusters’ footprints. In Section 4 we present a case study to show how these GDPattern clusters can assist do- main experts in criminal studies.

4. Case study

Utilizing the proposed framework, a case study is conducte d with real world data from a northeastern city in the United States. We firstly describe the data preproces sing in Section 4.1. Secondly, with the purpose of comparison study, crime hotspot maps are drawn in Section 4.2 using HOT, HSA, and user-specified thresh- olds, respectively . Kappa Index (Cohen et al., 1960; Rossiter, 2004) and cell statistics are used to compare the results and the pros and cons of HOT are discussed. Finally, we cluster the GDPat- terns using the similarity method (Section 3.4) and discuss the potential s of utilizing GDPattern clusters in demonstrat ing the characteri stics of crime related variables in Section 4.3.

4.1. Data preprocessin g

The data in the case study includes reported crimes and associ- ated variables in a northeastern city in the United States from 2004 to 2009. The size of study area is 130.1 km2 and the approximate population is 600,000. As one of the most frequently reported and resource -demanding crimes in the studied city (according to the city’s police department report), residenti al burglary (RB) is selected as the target crime (Fig. 4). In addition to RB, total of eight social/criminal features (Table 2) are selected in this study as related variables with the help of a domain expert. Among those are:

� Commerc ial burglary (CB), street robbery (SR), and motor vehicle larceny (MV). These indicate the level of activity of related crimes, and also reflect the strength of guardianship in the area.

y measure approach between patterns X and Y.

Table 2 Crime related variables for the case study.

Variables Number of incidents (2005–2009)

Residential burglary (RB) 18,321 Street robbery (SR) 12,020 Commercial burglary (CB) 4438 Motor-vehicle larceny (MV) 29,685 Arrest (AR) 254,309 Foreclosed houses (FC) 11,671 Population (POP) – Number of houses units (HUs) – Distance to colleges (DCs) –

D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106 99

� Arrests (AR). This helps indicate the size of the pool of offenders. � Foreclosed homes (FC). A vacant house has a higher risk of being

broken into than an inhabited one, and is also a sign of lack of guardianship . � Population (POP) and housing density (HU). A hotspot of RB may

simply be a location of high housing density because such areas have a potential higher RB rate than areas with fewer houses. � Distance to colleges (DC). The studied city is heavily populated

by college students , which makes many properties easy targets for burglars during semester breaks. DC is calculated as the dis- tance to the geographical center of a university or college.

Fig. 4. Residential burglary rates in the studied city. Top is the grid density map of RB. On the bottom it is a graph showing the frequency of cell values.

100 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

The original criminal dataset comes as vector maps (points and polygons). We firstly convert all the variable data into grid maps (Fig. 4). The grid cell size selected is 100 m � 100 m, which results in a number of 12,984 cells in the study area. There are two con- cepts to consider when choosing the cell size. Firstly, the cell size (10,000 m2) is approximat ely half the size of average city block size (19,873 m2) in the studied city. According to domain experts, this will be a good representat ive of reality and helpful in police prac- tice. Secondly , at this size, the number of cells covering the study area is the same order of magnitude as the number of RB incidents (Table 2), which minimizes the loss of spatial information during aggregation . On the other hand, HSA needs to be conducted using polygon maps instead of rasters. The raster of RB is converted into a fishnet map with the same dimensio n as the mask. Each polygon in the fishnet map has an attribute of ‘‘RB Counts’’ indicating the amount of RB incidents in the area. In order to facilitate the discus- sion, we call the polygons in the fishnet map cells as well.

Since the related variables come from very different sources, the range of their values varies. As with most criminal activities, the counts of cells with same values in each grid map follow a power-law distribution (Cook, Ormerod, & Cooper, 2004 ) (Fig. 4). Using Nature Breaks (Jenks, 1967 ), every variable is divided into six categories: 0 – ‘‘empty’’, 1 – ‘‘lowest’’, 2 – ‘‘low’’, 3 – ‘‘average ’’, 4 – ‘‘high’’, and 5 – ‘‘highest’’. Using the Nature Break method the categories’ breaks are identified with best grouping of similar val- ues, and the differences between categories are maximized.

4.2. Hotspot mapping

An initial threshold of RB hotspots is needed to set the initial classes before utilizing HOT. From the study of (Short et al., 2010), a house is at relatively higher risk if a burglary happened nearby within the past 4 months. Therefore if three or more bur- glaries happened in the block in one year, the area is likely to be a burglary hotspot. Because the time span of our data is 6 years, we set an area (cell) to be a hotspot if there are eighteen or more burglary incidents (h P 18). We use the threshold of 9 RB incidents (18 > h0 P 9), to define the ‘‘potential hot’’ areas ðDh0Þ. The growth ratio for GDPatterns is set at more than twenty (d > 20), which in- sures an at least 95% confidence level (1:20) that these GDPatterns will reveal the difference between hotspots and normal areas. To test the tolerance of HOT, four different support thresholds (q = 0.001, 0.005, 0.01, 0.02) are used in the experiments.

For comparison, hotspot maps generate d by hard thresholds and the HSA method are presented. Three maps using the hard thresholds are generated. Two of them are just using the thresh- olds of h P 18 and h P 9. The third one is generated using an initial threshold of h P 18 and then locating cells with RB rate h P 9 that are also adjacent to the h P 18 cells.

HSA takes the fishnet map (Section 4.1) as input and calculates a G�i (Formula (8)) statistic for each polygon in the map. The G

� i statis-

tic is considered as the z-score of the polygon. Then a p-value, the probability distribution of the z-scores, is calculated for each poly- gon. In summary , a polygon with a high z-score and a p-value less or equal to 0.05 is considered as having a high enough attribute value to be statistically significant, and thus is considered as a hotspot.

X � ¼ Pn

j¼1 xj n

S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn j¼1 x

2 j

n �ðX

� Þ2

s

G�i ¼ Pn

j¼1 wi;j xj � X � Pn

j¼1 wi;j

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Pn

j¼1 w2

i;j � Pn

j¼1 wi;j

� �2� � n�1

vuut

ð8Þ

where xj is the value of the attribute (amount of incident s) for spa- tial polygon j, wi,j is the spatial weight between polygon i and j (In the case study we use inverse distances as the spatial weights (Deane, Beck, & Tolnay, 1998; Ratcliffe & Taniguchi , 2008; Tita & Greenbau m, 2009 ) and Euclidean Distance as the distance method.), n is the total number of polygon s.

We name the maps generate d using hard thresholds h P 18 and h P 9 HT18 and HT9, respectivel y. The map generate d using h P 18 cells and their adjacent cells with h P 9 is called HT18_9. The HOT produced maps using the support thresholds q = 0.001, q = 0.005, q = 0.01, q = 0.02 are called HOT001, HOT005, HOT01, and HOT02, respectively . The map generate d by HSA is named the HSA map. All these maps are shown in Fig. 5.

The standard Kappa Index k (Formula (9)) (Cohen et al., 1960; Rossiter, 2004 ) is used to compare the difference between hotspot maps (Table 3). The value of k is between �1 and 1, and two maps are considered more similar when the k between them is larger (closer to 1).

k ¼ p0 � pc 1 � pc

ð9Þ

where p0 is the proportio n of cells that classified into the same class (agreed) by both maps. pc is the proportio n of units for which the agreem ent is expected by chance.

From Fig. 5 and Table 3 we can tell that even using different support thresholds, the final HOT hotspot maps are very close to each other (the Kappa indices between them are all larger than 0.94). Although different support thresholds will result in different set of closed frequent patterns, by setting a relatively high growth ratio value, only the most significant patterns are selected as GDPatterns that contribute to hotspot mapping.

The HOT maps and the HT18_9 map are similar to each other (average Kappa Index 0.86) because they all contain the h P 18 cells. On the other hand, there are totally 344 (different hotspot cells between HT18 and HT18_9, Table 4) cells that having RB rate h P 9 and adjacent to the h P 18 cells and around 69.4% of them are considered as hotspots by HOT (calculated by dividing the average value of different hotspot cells between the HOT maps and the HT18 map by 344, Table 4). The differenc e between them (HOT maps and the HT18_9 map) can be considered as the infor- mation gained using HOT.

A land cover map of the studied city is drawn (Fig. 6) with the purpose of evaluating the precision of our hotspot maps. In Table 4 we calculated the cell statistics for each map. The percentages of RB hotspot cells that are actually located in residential areas can be seen as the precisions of the maps (Column 3, Table 4).

All the hotspot maps we generated are based on grid choropleth mapping. There is an intrinsic defect when using grid choropleth mapping for hotspot identification. By converting points represent- ing crime incidents into cells with crime counts, spatial details within and across the cells boundaries can be lost. In the case study, this limitation is reflected by the fact that cells in non-resi- dential areas (Fig. 6) are classified as hotspots of residential bur- glary (RB) in all the hotspot maps. For example, after the aggregat ion process a certain cell may contain 20% non-residenti al areas, like roads or parks, and 80% of residential areas. If during the hotspot analysis process the cell is classified as a residential bur- glary (RB) hotspot, then the precision of this hotspot is 80%.

The hotspot maps using the user-specified thresholds (HT18, HT9 and HT18_9) can be considered as benchmarks for the case study. In other words, using the current grid map (cell size 100 m � 100 m), the precision for describing residenti al areas in the studied city is around 85% (percentage of hotspot cells locating in residential area in the hard threshold hotspot maps; Table 4). HSA does not achieve this precision. Because during the hotspot analysis (the G�i statistic calculation) process, all the cells are only

Fig. 5. RB hotspot maps of the studied city. HT18 and HT9 are generated by the thresholds of h P 18 and h P 9, respectively. HSA is the hotspot map generated by the Hotspot Analysis tool in Esri ArcGIS. HOT001, HOT005, HOT01, HOT02 are the HOT generated hotspot maps with the support thresholds equal to 0.001, 0.005, 0.01, and 0.02, respectively. In the map of HT18_9, cells with RB rate h P 18 and cells with RB rate h P 9 that are also adjacent to the h P 18 cells are considered as hotspots.

Table 3 Comparison results of the hotspot maps. The number in front of the brackets is the amount of cells that being classified as hotspots in both maps. The number inside the brackets is the Kappa Index betwe en the two maps.

HT18 HT9 HSA HOT001 HOT005 HOT01 HOT02 HT18_9

HT18 301(1.00) HT9 301(0.38) 1245(1.00) HSA 262(0.39) 668(0.74) 1094(1.0) HOT001 301(0.69) 561(0.61) 456(0.61) 561(1.0) HOT005 301(0.73) 523(0.58) 428(0.58) 509(0.95) 523(1.0) HOT01 301(0.69) 567(0.62) 457(0.61) 546(0.98) 511(0.95) 567(1.0) HOT02 301(0.74) 508(0.57) 416(0.57) 504(0.95) 487(0.96) 507(0.94) 508(1.0) HT18_9 301(0.63) 645(0.67) 523(0.66) 496(0.87) 475(0.85) 501(0.88) 466(0.84) 645(1.0)

D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106 101

considered as areas with or without RB rates. There is not enough information for HSA to tell if a cell contains 80%, or only 20% resi- dential areas. This results in a further precision lost (82%). The HOT model outperform s HSA under current setting of paramete rs be- cause not only the target crime rate, but also the related variables have been taking into account in HOT. By using the informative GDPatterns, only the areas with similar background (or similar characterist ics of related variables) as the hard threshold hotspots

are considered . The use of GDPatterns ensures that the precision of the HOT hotspot maps (86% in average, Table 4) will consist with the original inputs.

To give an intuitive view of HOT’s performanc e, two of the hot- spot maps, HT18 and HOT001 (Fig. 5) are projected with satellite images of the studied city and a figure of sample site is extracted (Fig. 7). Using an initial threshold (h P 18) the red cells are classi- fied into hotspots and cells in same blocks (in the color of blue)

Table 4 Cell statistic of the hotspot maps. The number in front of the brackets is the amoun t of cells located in the corresponding area. The number inside the brackets shows the percentage.

Total hotspot cells

Cells in residential areas

Cells in non-residential areas

HT18 301 257(85.4%) 44(14.6%) HT9 1245 1056(84.8%) 189(15.2%) HSA 1094 901(82.4%) 192(17.6%) HOT001 561 484(86.3%) 77(13.7%) HOT005 523 451(86.2%) 72(13.8%) HOT01 567 488(86.1%) 79(13.9%) HOT02 508 435(85.6%) 73(14.4%) HT18_9 645 548(85.0%) 97(15.0%)

102 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

have been left out. Understandabl y, houses in the same block are at similar risk of being broken into. Our optimizati on method suc- cessfully captures these cells. Other than a chorople th mapping tool, the HOT performs a dasymetric mapping by modifying the hotspot boundari es rationally. Also, locations covered by natural land, parking lots, roads, and highways are identified and are clas- sified out of hotspots using our method (Fig. 7).

4.3. Demonstratin g crime related variables

One thousand five hundred GDPatterns in the experiment satis- fying a support threshold of 0.001 are selected for further analysis. These GDPatterns (H-GDPatterns) are sorted by growth ratios from

Fig. 6. A land cover map showing the r

high to low. All 1500 patterns have a growth ratio greater than 50 (d > 50). For comparison, a set of GDPatterns (N-GDPatterns) based on normal areas are also mined using HOT. Specifically, we set cells with h P 18 as Dn, cells with 18 > h0 P 9 into Dh0 and other cells into Dh (h < 9). In order to facilitate the compara tive analysis, 1500 top N-GDPatter ns are selected after running HOT. The growth ratios of these N-GDPat terns are all larger than 30 (d > 30).

Using the similarity method discussed in Section 3.4, the dis- tance between each pair of GDPatterns is calculated. We use the cluster heat map tool (Wilkinso n & Friendly, 2009 ) to visualize the clusters in sorted distance matrices (Fig. 8). In sorted distance matrices, the value of aij represents the distance between GDPat- tern i and GDPattern j, where GDPattern j is the ji � jjth closest to GDPattern i by distance. The heat maps use different colors to represent the different values in the sorted distance matrices.

After locating all the clusters, the footprint s of these clusters are drawn (Fig. 9), which demonst rate the spatial distribution of GDPatterns . Moreover, we use pie-chart to explore the structure of GDPatterns in the same clusters (Fig. 10), in which the values of variables are shown using different colors.

A lot of information can be revealed from these figures. For example, when we look at the H-GDPattern clusters in the studied city,

� High residential burglary (RB) rates are associated with high population density only in areas with few foreclosures (FC), commerc ial burglaries (CB), motor-ve hicle larcenies (MV),

esidential areas in the studied city.

Fig. 7. An example of re-projected hotspots with satellite images. The blue cells are hotspots defined using a threshold of h P 18 (HT18 in Fig. 5). Both the blue and red cells belong to the hotspots identified using HOT with a support threshold of 0.001 (HOT001 in Fig. 5). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Heat maps for distance matrices of GDPatterns. On the left side a heap map based on distance matrix of H-GDPatterns is drawn by using the color ramp from blue to red representing distances between H-GDPatterns from small to great. GDPattern clusters that identified using HAC (Section 3.4) are marked with white frames. On the right is the heat map for the distance matrix of N-GDPatterns with color ramp from black to white representing distances from small to great and GDPattern clusters are marked with blue frames. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106 103

street robberies (SR), and very low arrest rates (AR) (Cluster 1). These areas also have high residenti al density (HU) and are close to universities or colleges (DC). Such locations are shown in the footprint map of H-GDPattern Cluster 1 in Fig. 9. � High residential burglary (RB) rates are associate d with very

low foreclosure rate (FC) in most instances (Cluster 1–7). The only locations with many residenti al burglaries (RB) and a moderate number of foreclosu res (FC) are shown in Fig. 9,

H-GDPattern Cluster 8. These areas are usually far from univer- sities or colleges, have average population and house density, and low to moderate arrest (AR), commercial burglary (CB), motor-ve hicle larceny (MV) and street robbery (SR) rates (Clus- ter 8). � Areas with high residential burglary rates and not close to any

colleges or universities (low in DC) can be mainly considered in two categories (Clusters 4 and 7 in Fig. 10). One of them is

Fig. 9. Footprint maps of GDPatterns’ clusters. Areas inside blue circle are where most colleges located in the studied city and the green circle indicate the centre park of the city. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

104 D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106

Fig. 10. Pie-charts of GDPatterns’ clusters. The values of each related variable are shown in different colors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D. Wang et al. / Computers, Environment and Urban Systems 39 (2013) 93–106 105

characterized by high residenti al density (HU), as well as low motor-vehic le larcenies (MV) and street robberies (SR) rates (Cluster 4). The other has low residential density and average MV and SR rates (Cluster 4). The locations of the two categories are shown in H-GDPattern Cluster 4 and H-GDPattern Cluster 7 of Fig. 9, respectively.

The information revealed by our approach has been verified by domain scientists . For example:

� Offenders are known to focus on neighborho ods with large pro- portions of college students living in off-campus residences (the blue circles in Fig. 9 show areas where most colleges located), (Fig. 10, H-GDPattern Cluster 1 in which the value of DC is high). � Where college students are less significantly represented,

offenders take a different approach, and the FC rates become a more important indicator of RB offenders (Fig. 10, H-GDPatter n Cluster 8 in which the value of FC is relatively high). This also explains why high RB is associated with low FC in most areas of the city. � The footprint map of N-GDPatter n Cluster 7 (green circle in

Fig. 9) covers mostly non-residenti al areas like parks, because these areas have similar condition s and no RB incidents.

The case study and the comparison experime nts have shown the potential of using crime related variables in hotspot mapping. Our method helps maintain the mapping precision during the hot- spots representat ion process and also provides a comprehensive way for further analysis.

5. Conclusion

In this paper, we present a spatial data mining framework to study the spatial distribution of crimes through their related variables. To the best of our knowledge, it is the first attempt to use related variables in crime hot spot mapping. Spatial data min- ing is often said to ‘‘let the data speak for themselves’’. But the data cannot tell stories unless appropriate questions are formulated and asked, and appropriate methods are needed to solicit the answers from the data. In the framework we address an iterative and induc- tive learning process to study the spatial properties of crime. Experiment results show that our HOT model outperforms HSA in precisely identifying crime hotspots. Additionally , by using a similarity measure method, we demonstrate the characterist ics

of target crime’s related variables using GDPattern clusters and footprint maps, which help explainin g the varying of crime over space and deliver the knowled ge in a quantitat ive, as well as com- prehensive and systematic manner.

Acknowled gement

The work was partially funded by the National Institute of Jus- tice (No. 2009-DE-BX-K21 9).

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