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Research Article | Open Access

Volume 2014 |Article ID 492461 | 8 pages | https://doi.org/10.1155/2014/492461

Analyses of Crime Patterns in NIBRS Data Based on a Novel Graph Theory Clustering Method: Virginia as a Case Study

Academic Editor: X. Meng
Received28 Jan 2014
Accepted25 Feb 2014
Published20 Mar 2014

Abstract

This paper suggests a novel clustering method for analyzing the National Incident-Based Reporting System (NIBRS) data, which include the determination of correlation of different crime types, the development of a likelihood index for crimes to occur in a jurisdiction, and the clustering of jurisdictions based on crime type. The method was tested by using the 2005 assault data from 121 jurisdictions in Virginia as a test case. The analyses of these data show that some different crime types are correlated and some different crime parameters are correlated with different crime types. The analyses also show that certain jurisdictions within Virginia share certain crime patterns. This information assists with constructing a pattern for a specific crime type and can be used to determine whether a jurisdiction may be more likely to see this type of crime occur in their area.

1. Introduction

The National Incident-Based Reporting System (NIBRS) is a crime reporting program for local, state, and federal law enforcement agencies that provides a wealth of incident level data for use in analysis. It is part of the Uniform Crime Reporting (UCR) Program which is administered by the FBI. The UCR Program provides a nationwide view of crime based on data submitted through state programs or directly to the national UCR Program and has been operational for around 70 years. The NIBRS was implemented in the late 1970s to meet law enforcement need for the 21st century. This vast system houses information on offenses, victims, offenders, property, and persons arrested, as well as the incident itself. The data of NIBRS are well structured and readily available for researchers and law enforcement agencies to assist with understanding the intricate nature of crime.

Akiyama and Nolan [1] outlined the structure of the NIBRS data set and provided methods for understanding and analyzing the data. Dunn and Zelenock [2] also describe and test procedures to facilitate the use of this vast system. Building on these initial works, many authors continue to investigate how this storehouse of data can be turned into useful information for researchers and law enforcement agencies. Much of the work employs descriptive statistics applied in various sophisticated ways to extract information from the files. For example, Thompson et al. [3] apply descriptive statistics to examine intimate partner violence and make connections between this crime and other crimes that occurred in the same incident. They were able to show a link between intimidation and more serious violent crimes and extract information about the relationships between the victim and the offender from the NIRBS data that helps to better understand this type of crime. Snyder [4] used logistic regression techniques to predict the arrest of juvenile robbery offenders. More recently, Addington and Rennison [5] used logistic regression models along with the NIBRS and the National Crime Victimization Survey (NCVS) data for predicting rape cooccurrence to provide a critical initial look at rapes that occur with other crimes.

For criminologists the NIBRS data holds the answers to many long-standing questions about crime, criminal offending, and crime victimization. However, gaining access to some of these answers has remained difficult because of the size and complexity of the data. Effective techniques, such as data mining and clustering, for criminal justice data are of increasing importance to both the research and law enforcement communities [6]. In recent years, clustering categorical data has gained more importance because it is one of the fundamental methods in data mining [7]. Clustering crime data, as with other categorical data, is unsupervised learning that aims at partitioning a data set into groups of similar items. The goal is to create clusters of data objects where the within-cluster similarity is maximized and the between-cluster similarity is minimized. Over the years, many clustering algorithms have been developed and tested. Some clustering techniques have developed specifically for use with categorical data. Abdu [8] presented three new clustering algorithms that were applied to the clustering of the NIBRS data. Two of his approaches combine spectral analysis and clustering techniques that are scalable to large data sets such as the NIBRS.

Clustering categorical data poses a challenge not encountered in clustering numerical data because the attribute categories are not ordered and defining a metric with which to measure the distance between data objects in a data set becomes a challenge. Many of the algorithms that have emerged for clustering categorical data rely on the occurrence/cooccurrence frequencies of attribute values in the data set to determine clusters of similar data objects. The basic goal is to choose a set of attribute categories that provide a summary of the data objects in a cluster. There are a wide range of clustering algorithms for categorical data, including K-modes [9], STIRR [10], CACTUS [11], ROCK [12], COOLCAT [13], LIMBO [14], and CLICKS [15]. A good summary can be found in [8]. The clustering algorithm used in this research is a mathematically well-defined model implemented in a polynomial time algorithm that guarantees an optimal solution [16].

Due to the lack of well-defined mathematical models and optimization goals, most existing graph theory clustering approaches could not guarantee a proper clustering result in general cases. For example, agglomerative hierarchical clustering methods could not produce proper clusters with larger sizes, while divisive hierarchical methods could not produce clusters with smaller sizes, and clusters with large difference in their sizes and k-core method may produce clusters with small edge-cuts, and so forth. Many papers and articles have mentioned these problems and frustration among users (e.g., see [1719]). Even the most popular commercial software, SAS, is unable to produce proper outputs for some simple data.

The purpose of this paper is to present a novel multidimensional clustering method for the NIBRS data. We firstly outlines a new measure, called the likelihood index, that helps examine quantitatively how likely a crime is to occur in a particular jurisdiction. This measure compares a vector that describes a jurisdiction with a vector that represents a crime type. Then according to the defined distance between these two vectors, we can determine how closely the jurisdiction aligns with that crime type. The data used in this study were obtained from the 2005 NIBRS which is stored at the National Archive for Criminal Justice Data at the University of Michigan. This work explores the following research questions. Do specific crimes exhibit certain quantifiable characteristics? Do different types of crimes share similar quantifiable characteristics? Do jurisdictions of a state cluster with respect to different crime types? What is the likelihood that if one type of crime is occurring in an area, other types of crime with similar quantifiable characteristic will also occur in that area?

The rest of the paper is organized as follows. In Section 2 we summarize the data unit of analysis and preparation. In Section 3 we introduce the methods to deal with the data matrix and take Virginia as a case study. Section 4 provides some additional results and Section 5 gives the conclusions of the research.

2. Data Unit of Analysis and Preparation

The data sets available in the NIBRS provide a wealth of incident level data about each reported crime. As for 2010, approximately 40 states contribute their data to the massive data set. The data and tools are made available by University of Michigan for use by law enforcement agencies and researchers. In order to devise a manageable set of data for preliminary testing of techniques and for preliminary data analysis, only the 2005 data on assaults were explored. From the 2005 assault data, 121 jurisdictions (counties or cities) in Virginia were selected for examination. These represent all jurisdictions within Virginia with populations greater than 10,000. There were 10,183 incidents reported in these 121 chosen jurisdictions.

For this study, 21 indexes from the NIBRS were chosen from the 246 available indexes. These 21 indexes were deemed important to provide the relevant characteristics of the victim(s), offender(s), and the circumstances of each incident. The selected particular indexes were listed in Table 1.


SegmentIndexNumber of subindexes

Victim indexesType of victim3
Victim age3
Victim sex2
Victim race5
Victim ethnicity3
Victim residence status3
Aggravated assault/homicide circumstances10
Type of injury6

Offender indexesOffender age3
Offender sex3
Offender race5

Additional indexesInjury2
Juvenile1
Violent crime1
Juvict1
Multiple victims1
Multiple offenders1
Multiple offenders and victims1
Multiple offenders and one victim1
One offender and multiple victims1
One offender and one victims1

In order to facilitate the selected analysis techniques, the data was expanded from one column, with many possible entries, to multi columns that contained zero or one. For example, the Offender Segment index contains the sex of the offender and has the possible entries of male, female, or unknown. This index column was split into three individual columns where an entry in the three columns of (1 0 0) means female, (0 1 0) means male, and (0 0 1) means unknown. This turns the column for sex of the offender to three columns. All created columns were binary (0/1) columns that were used to help classify the characteristics of the incident. From the expansion of the original 21 indexes, 57 binary columns were created. This led to the creation of a 121 × 57 Crime Data Matrix, where each row represents the th jurisdiction and each column represents the th parameter related to the incident (e.g., offender sex, victim resident status) or a crime type (e.g., hate crime, drug dealing). The Crime Data Matrix construction is pictured below: where is the entry of the th crime index from the th jurisdiction.

Normalization of the rows of the matrix was completed by dividing each row entry by the population of the jurisdiction. This gave a per person rate for each crime parameter or each crime type. Normalization of the columns was completed by averaging the columns and subtracting the average from each entry in the column. Then each entry in the column was divided by the vector length of the column. Equation (2) shows these normalization step-by-step operations.

3. Finding Patterns in the Data

This section explains several different analyses that were performed on the data in the matrix described above in order to attempt to answer the research questions listed in Section 1. These analyses include comparing the columns of the matrix to determine the correlation of crime parameters to crime types and to determine the correlation of different crime types. Also there were two analyses performed comparing the row vectors to develop the likelihood index and to cluster the jurisdictions by crime types.

3.1. Correlation of Crime Parameters and Crime Types (Comparing the Column Vectors)

The motivation for comparing the different crime parameters to crime types is to determine if there are some characteristics that can tell us about the likelihood of a crime type to occur in a certain jurisdiction. Each crime type may have factors that contribute to a specific crime appearing in a certain place. An overall increase in crime in an area may or may not correlate to an increase in any one particular type of crime, say hate crime, in that area. However, there may be individual parameters whose increase may indicate an increase in a particular type of crime. For example, if juvenile offenders are up in a certain area, this may indicate that hate crimes will also be up in that area. Also crime type vectors were compared against each other in a similar way. For example, crimes like juvenile gang were compared against hate crime to see if these crime types also have a correlation.

In order to determine the relationship between the th parameter (th column of the Crime Data Matrix) and any other column (another parameter or another crime type), the correlation coefficient (cosine of the angle between the two column vectors over the norm of the vectors) was calculated. Each of the columns of the crime data matrix forms a vector in a 121-dimensional space and the vectors can be geometrically compared, with the correlation between two parameters represented by the cosine of the angle in this space. For example, the column for the juvenile offender could be compared against the column for hate crime or the column for hate crime can be compared to the column for gang-related crime. These comparisons are made by calculating the angle between these two columns to determine if there is any relationship and how strong that relationship may be. This method can assist in determining whether two columns vary directly, inversely, or separately.

For this comparison of two column vectors, the variation in two vectors must be transformed to eliminate the effects of mean differences. Once the mean deviation is determined, then the correlation can be determined by the cosine of the angle between the vectors. As an example, let be the column for juvenile offender and let be the column for the hate crime indicator. To find the correlation coefficient of this two columns calculate the following: The sign of the answer is ignored, since either a strong positive relationship (close to 1 meaning that the two angles are in the same direction and close to one another) or a strong negative relationship (close to −1 meaning that the two angles are in opposite directions, but nearly opposite one another) indicates that the vectors appear to be related in some way.

In doing this comparison, with one vector fixed and comparing it to all other vectors and itself, we form a row vector of size 57. Consider the example that compares the hate crime vector with all other vectors. We construct another vector that we refer to as the Hate Crime Character Vector of size 57, which contains all the cosine (the correlation coefficient of the th parameter with respect to hate crime). A new vector is formed from all these correlation coefficients (), where each of the 57 parameters has a correlation coefficient vector associated with it. Now consider the corresponding components in the Hate Crime Character Vector. Table 2 tells us how hate crimes are related to aggravated assault crimes. Among those crimes, juvenile gang (0.4058) has higher correlation with hate crime, while the others, for example, argument (0.1325) and drug dealing (0.0265) have relatively lower correlation.


Crime typeHate crime correlation coefficient ( )

Argument0.1325
Assault on law enforcement officers0.0371
Drug dealing0.0265
Gangland (organized crime involvement)0.1565
Juvenile gang0.4058
Lovers’ quarrel
Other felony involved
Other circumstances0.0961
Unknown circumstance0.1356

Table 3 also shows similar evidence that for victim’s age, the age group less than 18 is also more correlated to hate crime than the other two age groups. Many such comparisons can be observed using these correlation coefficient vectors.


Age of victimHate crime correlation coefficient ( )

Age < 18 0.4944
18 ≤ Age ≤ 600.0540
Age > 600.0474

3.2. Analyses of Jurisdictions (Comparing the Row Vectors)

By using the correlation coefficients, a 57-dimensional vector, two data analyses can be performed, each of which indicates the likelihood of a particular crime for each jurisdiction. The first one is a numerical index, called the likelihood index assigned to each of the 112 jurisdictions. The second one is a clustering analysis of all jurisdictions. Jurisdictions with similar recorded crime patterns (adjusted corresponding to correlation coefficients) form clusters.

Let be the likelihood index vector where is the likelihood index between jurisdiction and a particular crime (e.g., hate crime) or crime parameter (e.g., juvenile offender). For this comparison of two row vectors, the variation in two vectors must be transformed to eliminate the effects of mean differences. Once the mean deviation is determined, then the correlation can be determined by the cosine of the angle between the vectors. As an example, let be the row for the Norfolk jurisdiction and let be the row for the hate crime indicator. To find the correlation coefficient of these two rows calculate the following: Table 4 gives the top 30 jurisdictions of Virginia with respect to the hate crime likelihood index and also provides the data on the clustering of the jurisdictions discussed in the next section.


Name ClusterHate crime likelihood index

1NEWPORT NEWSB0.867
2NORFOLKB0.837
3CHESAPEAKEB0.810
4GREENSVILLEB0.762
5PORTSMOUTHB0.759
6RICHMONDB0.716
7WYTHEB0.681
8ALEXANDRIAB0.678
9BRISTOLB0.667
10NEW KENTB0.618
11FAUQUIERB0.531
12ROANOKEA0.439
13RICHMONDA0.386
14WILLIAMSBURGA0.377
15VIRGINIA BEACHB0.318
16CHARLOTTESVILLEA0.312
17PETERSBURGA0.280
18SPOTSYLVANIAA0.226
19HOPEWELLA0.205
20RUSSELLA0.121
21CLARKEA0.121
22WINCHESTERA0.118
23SUFFOLKA0.111
24MARTINSVILLEA0.075
25STAUNTONA0.073
26GALAXA0.070
27SHENANDOAHA0.063
28CAROLINEA0.044
29SURRYA
30DANVILLEA

3.3. Using the Correlation Coefficients to Cluster the Jurisdictions

To begin the clustering, a weighted complete graph with 121 vertices is formed. The weight on each edge is the correlation coefficient between the jurisdictions. The novel graph theory clustering method we proposed in [20] is used to find all the dense (highly weighted) subgraphs of the complete graph. A distinguished feature of this method, nonbinary hierarchical tree, clearly highlights meaningful clusters which significantly reduces further manual efforts for cluster selections. The results of clustering the jurisdictions with respect to hate crimes are also displayed in Figure 1 and Table 4.

It can be seen that of the top 30 jurisdictions with respect to hate crimes, 12 of the top 15 are in Cluster A and the others are in Cluster B. The remaining 91 jurisdictions also fall into Cluster B. The 12 higher hate crime rate counties (cities) are showed in Figure 1 (in red and blue).

4. Additional Results

Similar analyses where performed with respect to other crime types: drug-dealing, juvenile gang, and gangland (organized crime involvement). The results are summarized in Tables 510.


Drug dealing correlation coefficient

Type of victim
 individual0.5789
 Business0.1721
 Society/public0.1729
Age of victim
 Age < 180.2933
 Age ≥ 600.3872
 18 ≤ age < 600.5993
Sex of victim
 Male0.5774
 Female0.4989
Race of victim
 White0.3906
 Black0.5028
 Asia/Pacific Islander0.0764
 Unknown0.0003
 American Indian0.0393
Ethnicity of victim
 Hispanic origin0.0688
 Not of Hispanic origin0.6009
 Unknown−0.0616
Resident status of victim
 Nonresident0.2631
 Resident0.5740
 Unknown0.1290


Offender age
 Age < 180.3514
 Age 600.3762
 18 age < 600.5678
Offender sex
 Male0.5331
 Female0.5685
 Unknown0.1972
Offender race
 White0.3145
 Black0.5133
 American Indian/Alaskan native−0.0479
 Unknown0.1948
 Asian/Pacific Islander0.113


Other crime types

Argument0.4854
Assault on law enforcement officer(s)0.3117
Gangland (organized crime involvement)0.3456
Juvenile gang0.0706
Lovers’ quarrel0.2967
Other felony involved0.2955
Hate crime0.0265


Name ClusterDrug dealing likelihood index

1CHARLOTTESVILLEB0.9094
2NEWPORT NEWSB0.9012
3CHESAPEAKEB0.8851
4PETERSBURGB0.8637
5RICHMONDB0.8365
6PORTSMOUTHA0.8281
7HOPEWELLA0.7917
8ROANOKEB0.791
9SUFFOLKB0.7839
10NORFOLKB0.7756
11BRISTOLB0.7365
12DANVILLEA0.707
13GREENSVILLEA0.7032
14GALAXA0.6881
15CAROLINEA0.6852
16SUSSEXA0.6396
17WINCHESTERA0.6327
18FREDERICKSBURGB0.6031
19CLARKEA0.6021
20FRANKLINA0.5808
21RICHMONDA0.5667
22LYNCHBURGA0.5395
23MECKLENBURGA0.5185
24RADFORDA0.5133
25GOOCHLANDA0.5114
26MANASSASA0.494
27HENRYB0.4844
28NORTONA0.4456
29WISEB0.4381
30WILLIAMSBURGA0.395


Name ClusterGangland likelihood index

1NEWPORT NEWSB0.9217
2ROANOKEB0.8786
3CHESAPEAKEB0.8713
4PETERSBURGB0.8421
5NORFOLKB0.8129
6RICHMONDB0.7953
7LYNCHBURGB0.6965
8FREDERICKSBURGB0.689
9BRISTOLB0.6787
10ALEXANDRIAB0.6569
11NORTHAMPTONB0.632
12LOUDOUNB0.4939
13CHARLOTTESVILLEA0.4152
14STAFFORDB0.3853
15PORTSMOUTHA0.3075
16HOPEWELLA0.2724
17GALAXA0.2121
18CAROLINEA0.1372
19SUFFOLKA0.1077
20GREENSVILLEA0.0817
21CLARKEA0.0689
22DANVILLEA0.0519
23HENRYA0.0518
24RICHMONDA0.0074
25WINCHESTERA
26FRANKLINA
27SUSSEXA
28MANASSASA
29WILLIAMSBURGA
30GOOCHLANDA


Name ClusterJuvenile gang likelihood index

1NORFOLKB0.8424
2ROANOKEB0.8387
3RICHMONDB0.7987
4PORTSMOUTHB0.7977
5GREENSVILLEB0.7575
6CHESAPEAKEB0.7211
7NEWPORT NEWSB0.7112
8WILLIAMSBURGB0.6639
9WYTHEB0.5719
10ALEXANDRIAB0.5597
11LYNCHBURGB0.5452
12MARTINSVILLEB0.5355
13PULASKIB0.4884
14HAMPTONB0.4848
15SPOTSYLVANIAB0.451
16POWHATANB0.442
17PETERSBURGA0.4207
18CHARLOTTESVILLEA0.4064
19HOPEWELLA0.3519
20RICHMONDA0.208
21GALAXA0.2025
22BRISTOLA0.2012
23SUFFOLKA0.1925
24CAROLINEA0.121
25CLARKEA0.0939
26WINCHESTERA0.0893
27DANVILLEA0.0834
28SUSSEXA0.051
29CAMPBELLA0.0119
30FRANKLINA

Table 6 summarizes the drug dealing vector comparison with the offender parameter vectors.

Tables 5 and 6 summarize the comparison of the drug dealing vector with other crime parameter vectors. Each of the other crime parameters related to the victim and the offender was compared to the drug dealing vector. A high value for this correlation coefficient implies that there is a correlation between this crime parameter and drug dealing. Again, the motivation for comparing the different crime parameters to crime types is an attempt to determine if there are some characteristics that can tell us about the likelihood of a crime type to occur in a certain jurisdiction. For example, the correlation between the drug dealing vector and individual victim is 0.5789, which is much higher than the correlation between drug dealing and victim type business (0.1721) or victim type society/public (0.1729). This would seem to imply that for jurisdictions where the individual victim crimes are evident they may also have the likelihood for drug dealing related crimes. The tables show the correlation coefficient values: the correlation values in bold show a higher correlation with the drug dealing crime type than the other parameters in that category.

Table 7 summarizes the drug dealing vector comparison with the other crime types. The highest correlation is with argument.

Each of the other crime parameters related to the victim and the offender was compared to the gangland (organized crime involvement) vector. There were no significant relationships to report from this comparison.

Table 8 gives the top 30 jurisdictions of Virginia with respect to the drug dealing likelihood index and also shows how the jurisdictions are clustered. Given the 121 Virginia jurisdictions being considered, of the top 30 with respect to the likelihood index, twelve are clustered together. Only three others jurisdictions from Cluster B appear outside the top 30; these jurisdictions are Smyth (35), Tazewell (37), and Pittsylvania (50). Each of the other crime parameters related to the victim and the offender was compared to the gangland (organized crime involvement) vector. There were no significant relationships to report from this comparison.

Table 9 gives the top 30 jurisdictions of Virginia with respect to the gangland (organized crime involvement) likelihood index and also shows how the jurisdictions are clustered. Given the 121 Virginia jurisdictions being considered, of the top 30 with respect to the likelihood index, thirteen are clustered together. Only one other jurisdiction from cluster B appears outside the top 30; this one jurisdiction is Fairfax County PD (31). Each of the other crime parameters related to the victim and the offender was compared to the juvenile gang vector. There were no significant relationships to report from this comparison.

Table 10 gives the top 30 jurisdictions of Virginia with respect to the juvenile gang likelihood index and also shows how the jurisdictions are clustered. Given the 121 Virginia jurisdictions being considered, of the top 30 with respect to the likelihood index, sixteen are clustered together. No other jurisdiction from Cluster B appears outside the top 30.

5. Conclusion

The NIBRS provides a wealth of incident level data for use in analysis. The methods investigated in this research yielded promising preliminary results. The methods were applied only to the assault data from 2005 but can easily be extended to other crime types and to other years to validate these results and also provide longitudinal investigation.

The comparison between the crime type vector and the individual parameters vectors helped in two cases (hate crimes and drug dealing) to determine which factors was more related to those crimes. The different types of analyses that were conducted on the jurisdictions helped to validate one another. The likelihood index looked at whether a certain crime pattern existed in that jurisdiction, while the clustering method sought to cluster all the jurisdictions based on the crime patterns of that jurisdiction. This information could be useful to assist law enforcement agencies or policy makers in determining which jurisdictions share common challenges that could possibly be addressed through cooperation and sharing resources between jurisdictions.

The next steps would be to utilize this same approach for data from other states or perhaps a larger region to examine if the same information is observed from the analyses. It will be interesting to see if Virginia data and other states have the same patterns or if different patterns emerge. Further research and refinement of these methods should yield tools that would provide researchers, law enforcement agencies, and government officials with a means to find patterns of different crime types and possibly identify jurisdictions that may be likely to experience that type of crime.

Conflict of Interests

Peixin Zhao, Marjorie Darrah, Jim Nolan, and Cun-Quan Zhang certify that there is no actual or potential conflict of interests in relation to this paper.

Acknowledgments

First author was partially supported by the China Postdoctoral Science Foundation Funded Project (2011M501149), the Humanity and Social Science Foundation of Ministry of Education of China (12YJCZH303), the Special Fund Project for Postdoctoral Innovation of Shandong Province (201103061), the Informationization Research Project of Shandong Province (2013EI153), the National Statistical Science Project (2013LZ38), and Independent Innovation Foundation of Shandong University (IIFSDU) (IFW12109).

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Copyright © 2014 Peixin Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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