Analyzing Space–Time Crime Incidents Using

Eigenvector Spatial Filtering: An Application to

Vehicle Burglary

Yongwan ChunSchool of Economic, Political and Policy Sciences, University of Texas at Dallas, Richardson, TX,USA

Poisson models generally are utilized in analyzing spatial patterns of crime count data.When spatial autocorrelation is present, these models are extended to account for it. Amongvarious methods, eigenvector spatial filtering (ESF) furnishes an efficient means of analy-sis. However, because space–time crime data have temporal components as well as spatialcomponents, Poisson models need to be further adjusted to reflect the two types of compo-nents simultaneously. This article discusses how the ESF method can be utilized to modelspace–time crime data, extending the generalized linear mixed model specification for it.This approach is illustrated with an application to space–time vehicle burglary incidents inthe city of Plano, Texas, during 2004–2009.

Introduction

Aggregated count data in small areas are frequently used in studies analyzing spatial patterns ofcrime. Although various spatial units, such as neighborhoods, cities, and counties, also areutilized, individual events are aggregated into census units, such as census block groups andcensus tracks, because socioeconomic variables are available for these units in most cases. Forcounts of crime incidents, which are seldom expected to follow a normal distribution, generalizedlinear models (GLMs) are frequently utilized, specifically, Poisson regression and negativebinomial (NB) regression (Osgood 2000; Berk and MacDonald 2008). However, the presence ofspatial autocorrelation in count data violates the independence assumption, and, consequently,standard Poisson and NB regression models may yield biased, inefficient, and inconsistentparameter estimates. Hence, a proper model specification is required to accommodate spatialautocorrelation (Griffith and Haining 2006; Haining, Law, and Griffith 2009).

Some well-known methods exist for incorporating spatial autocorrelation into a Poissonmodel specification. Besag (1974) specifies the auto-Poisson model to account for spatial auto-correlation in count data utilizing the Hammersley–Clifford theorem. However, the auto-Poisson

Correspondence: Yongwan Chun, School of Economic, Political and Policy Sciences, University of Texas atDallas, 800 W. Campbell Rd., GR31, Richardson, TX 75080-3021e-mail: ywchun@utdallas.edu

Submitted: August 26, 2013. Revised version accepted: December 10, 2013.

Geographical Analysis (2014) 46, 165–184

doi: 10.1111/gean.12034© 2014 The Ohio State University

165

model has integrability conditions that ensure it is able to describe only spatial competitionbetween neighboring sites; that is, negative spatial autocorrelation. Kaiser and Cressie (1997)propose a Winsorized auto-Poisson model to overcome the integrability condition so that it canaccommodate positive spatial autocorrelation. Alternatively, Augustin, McNicol, and Marriott(2006) utilize a truncated Poisson distribution to avoid this restriction of the auto-Poisson model.Lambert, Brown, and Florax (2010) extend count data modeling to a spatial autoregressivespecification. Bayesian approaches also have been popularly utilized to model the number ofcrime incidents, capturing spatial autocorrelation with spatially structured random effects (e.g.,Law et al. 2006; Matthews et al. 2010). Often, a conditional autoregressive specification isutilized in a Bayesian framework (Besag, York, and Mollié 1991). The eigenvector spatialfiltering (ESF) approach furnishes an alternative way to incorporate spatial autocorrelation inPoisson regression (Griffith 2002; Chun 2008). Griffith (2006) provides an empirical comparisonbetween the ESF and the Winsorized auto-Poisson methods. Unlike Poisson cases, a NBmodel specification to incorporate spatial autocorrelation is considerably underinvestigated.An auto-NB model has the same limitation: that it can account for only negative spatial auto-correlation. Although the aforementioned methods to account for spatial autocorrelation inPoisson regression potentially can be extended to a NB case, such an extension rarely is foundin the literature.

The purpose of this article is to model space–time crime incidents, extending the ESFmethod to a generalized linear mixed model (GLMM) context. Modeling spatial autocorrelationin count data is often performed in a GLMM framework (e.g., Mohebbi, Wolfe, and Jolley 2011)because it frequently requires researchers to account for temporal effects in individual spatialunits as well as cross-sectional spatial autocorrelation. A GLMM specification is popular forcapturing latent or hidden effects with random effects (e.g., D’Unger et al. 1998). Random effectscan be defined to reflect different structural effects such as temporal, spatial, or unstructuredrandom phenomena. A complex GLMM can include multiple random components. However,estimation of a complex GLMM remains difficult in practice in a non-Bayesian setting. Thisdrawback may provide one reason for a lack of space–time modeling of crime counts in theliterature. ESF combined with GLMM furnishes an efficient and effective way to incorporatespatial and temporal effects simultaneously (Chun and Griffith 2011; Patuelli et al. 2011). In thismodel specification, the cross-sectional spatial component (or spatial autocorrelation) is capturedby eigenvectors that are introduced as independent variables, whereas temporal effects arecaptured with random effects in a GLMM specification. Griffith (2013) also employs an ESFmethod in a GLMM specification, but his research focuses on imputations for missing valuesutilizing spatial and temporal structure.

ESF in GLMM

ESF is a nonparametric statistical method to model spatial autocorrelation in the context of linearand generalized linear regression. It utilizes eigenfunctions of a transformed spatial weightmatrix to account for spatial autocorrelation in a georeferenced random variable. Specifically, theESF method focuses on linearly decomposing spatial autocorrelation in a georeferenced variablewith uncorrelated and orthogonal eigenvectors. In linear regression, the eigenvectors can be usedto capture spatial autocorrelation in residuals, and, hence, estimation of a standard linear regres-sion model does not suffer from a violation of the independence assumption caused by spatialautocorrelation.

Geographical Analysis

166

Given a geographical landscape with n spatial units, eigenvectors that can be extracted fromthe decomposition of MCM (where M = I – 11′/n, C is an n-by-n binary spatial weight matrix,I is an identity matrix, 1 is a vector of ones, and ′ is the matrix transpose operator) exhibit distinctorthogonal and uncorrelated spatial patterns. That is, when these eigenvectors are portrayed in thegeographical landscape from which C is generated, their maps show a certain level of spatialautocorrelation, and their Moran’s I values are a function of their corresponding eigenvalues(Tiefelsdorf and Boots 1995). These eigenvectors can function as proxies for missing variables ina model specification and, accordingly, can be used to identify and isolate spatial autocorrelationin the unexplained variation of a dependent variable (see Griffith 2003 for details). An ESF modelspecification for Poisson or NB regression describing cross-sectional data (e.g., Griffith 2002)can be specified as

Exp[ ] ( ),Y = +−g 1 X Eb g (1)

where Y is a response variable assumed to follow a Poisson or a NB distribution, X is a designmatrix with p covariates and a vector of ones (the intercept term), E is a set of eigenvectors, β,and γ are vectors of parameters to be estimated, g(·) is a link function, which is the naturallogarithm in most Poisson cases, and Exp[·] denotes the calculus of expectations operator.Identification of E can be done in a stepwise manner to maximize model fit or to minimize spatialautocorrelation at each selection step (Tiefelsdorf and Griffith 2007). However, because fewstudies address the problem of measuring spatial autocorrelation in GLM residuals, includingthose for Poisson or NB regression, minimizing spatial autocorrelation in the residuals of a GLMstill remains more challenging than maximizing a model fit based on some well-known measuresuch as the Akaike information criterion (AIC).

A GLMM framework is widely used, particularly when repeated measurements are availablefor the same observational units or measurements (Breslow and Clayton 1993). Random effectsare used to capture underlying latent effects and/or hidden effects among repeated observationsfor each individual measurement unit. A Poisson or NB-GLMM with random effects specified interms of its intercept and repeated measures at each location constituting a time series can bewritten as follows:

Exp[ ] ( ), , , , , , ,y g b i n t Tit it i= ′ + = =−1 1 1x b … … (2)

where yit is a measure for observational unit i at time t following a Poisson or NB distribution,′xit is a row vector of covariates for unit i at time t, and bi is a random effects term for unit i that

is common for all t. A GLMM specification allows for different sources of variability in a meanresponse through its random effects, which can be specified in various ways, including two verycommon structures: temporal and spatial effects. Temporal effects are widely utilized for alongitudinal or panel data analysis (e.g., Zhang 2004), and spatial effects are often modeled withgeostatistical approaches (e.g., Diggle, Tawn, and Moyeed 1998; Yoo, Hoagland, and Cao 2013).In many cases, the random effects term, bi, can be specified as having zero mean and anautocorrelation structure, such as temporal or spatial correlation. Although generally, these twotypes of effects are simultaneously embedded in a space–time data set, empirical data analysessimultaneously including them in a non-Bayesian setting are rare because of complexities in theestimation of such models. When independent variables do not vary over t (e.g., elevation),equation (2) further simplifies to

Yongwan Chun Analyzing Space–Time Crime Incidents

167

Exp[ ] ( ), , , , , , .y g b i n t Tit i i= ′ + = =−1 1 1x b … … (3)

An ESF specification coupled with a GLMM furnishes a flexible way to incorporate these twoeffects simultaneously. An ESF GLMM model for space–time data (e.g., Chun and Griffith 2011)can be specified as

Exp[ ] ( ), , , , , , ,y g b i n t Tit i i i= ′ + ′ + = =−1 1 1x b gE … … (4)

where ′Ei is a row vector of elements from the matrix of eigenvectors (E) corresponding to uniti. In this model specification, the linear combination of eigenvectors constitutes a spatial filter thatcaptures spatial autocorrelation across n spatial units so that the random effects terms do notcontain spatial autocorrelation. This specification allows the random effects term to capture onlypure temporal effects in each unit in a space–time data set. An ESF GLMM can be estimated intwo steps: identification of the eigenvectors and then estimation of the GLMM with the eigen-vectors as independent variables. Identification of relevant eigenvectors can be achieved in astepwise manner similar to cross-sectional cases, after eigenvectors are stacked to match the totalnumber of observations in a space–time data set. That is, n eigenvectors are concatenated Ttimes to match the total number of observations (n × T). A stacked eigenvector can be written as

E

EE

E

stacked =

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

�. A common model fit measure, such as the AIC, can be used with a preset threshold

criterion. This selection procedure is based on an assumption that spatial and temporal compo-nents are separable. Griffith (2011) presents various ESF-based frameworks for space–timemodeling.

This ESF model specification coupled with GLMM furnishes advantages in modelingspace–time counts. First, the ESF specification provides a flexible way to account for positivespatial autocorrelation. Especially, the auto-model specification for Poisson and NB randomvariables can account for only negative spatial autocorrelation. Furthermore, an alternativemethod to accommodate positive spatial autocorrelation in a NB model specification currently isunavailable, unlike a Poisson specification that has a Winsorized and a truncated version. Second,this ESF specification allows the use of conventional statistical functions that are readily availablein statistical packages. A sophisticated spatial model often needs a complicated estimationtechnique that may not be readily available in general statistical packages and, hence, requiresa customized function or program. In contrast, the ESF method can be easily estimated witha conventional stepwise regression procedure.

Background literature: analyzing crime in space

In the literature, spatial aspects of crime have been well recognized and have been described withvarious spatial data analytical models. Even simple mapping with a geographic informationsystem (GIS) furnishes a useful tool to reveal “place matter” of crime incidents (Chainey andRatcliffe 2005). Researchers also are interested in detecting spatial clusters or hot spots of crime(Sherman, Gartin, and Buerger 1989; Grubesic 2006; Johnson et al. 2007) and in explainingcrime patterns with socioeconomic status and/or physical environments using spatial regressionmodels (Morenoff, Sampson, and Raudenbush 2001; Malczewski and Poetz 2005; Haining andLaw 2007; Congdon 2013). Interests about crime in a spatial context are further extended to

Geographical Analysis

168

forecasting crime rates for different spatial scales and resolutions (Cohen, Gorr, andOlligschlaeger 2007), and effective distance from well-known crime-related places such as barsserving alcohol (Ratcliffe 2012). Also, software packages for crime mapping and modeling exist(e.g., Levine 2005).

Research about spatial patterns of crime, especially burglary, has been mostly based on twotheoretical frameworks. The first is routine activity theory, which explains crime in terms ofopportunity (Cohen and Felson 1979). Three factors essential for crime in this framework aremotivated offenders, suitable targets, and absence of capable guardians; these factors should bepresent simultaneously for a crime to occur. That is, a crime occurs where a motivated offendercan find a suitable target with an absence of a capable guardian. Routine activity theory alsosuggests that crimes tend to occur close to the locations where offenders live (Ratcliffe 2003;Bernasco and Nieuwbeerta 2005). The other theoretical framework is social disorganizationtheory, which explains crime in terms of neighborhood characteristics (Bursik and Grasmick1993; Sampson 1993). This theory contends that disadvantaged communities tend to have highcrime rates; “disadvantaged communities” are those with economic deprivation, residentialinstability (or higher mobility), and social heterogeneity (socially mixed neighborhood). Studiesconstantly find that the economic status of a neighborhood is strongly correlated with its crimerate (e.g., Bursik 1988; South and Messner 2000; Ludwig, Duncan, and Hirschfield 2001; Cahilland Mulligan 2003), although residential instability (e.g., Ceccato, Haining, and Signoretta 2004)and social heterogeneity (e.g., Block 1979; Peeples and Loeber 1994) still display a significantrelationship.

Modeling space–time crime data has drawn much attention more recently. Space–time crimeclusters have been investigated with exploratory analysis tools such as three-dimensionalmapping and space–time kernel density (Nakaya and Yano 2010), and have been further extendedto a formal statistical test for space–time clustering (e.g., Johnson et al. 2007; Grubesic and Mack2008; Kim and O’Kelly 2008). Also, space–time crime data have been analyzed with regression-type modeling techniques. Although spatial panel models have been analyzed in the context ofthe frequentist statistical paradigm using, for example, linear regression and ordinary linearsquares estimation (e.g., Ye and Wu 2011), the Bayesian space–time modeling paradigm also hasbeen frequently used (Yu et al. 2008; Law, Quick, and Chan 2014), especially with non-Gaussianmodel (e.g., Law and Haining 2004). However, space–time crime modeling is relatively under-investigated compared with other thematic domains such as health studies (e.g., Kim and Oleson2007; Abellan, Richardson, and Best 2008; Lawson 2009).

An application: vehicle burglary in Plano, Texas

This article summarizes an analysis of vehicle burglary incidents in the city of Plano, Texas. Thiscity is located in the northern part of the Dallas–Fort Worth metropolitan area, has approximately260,000 inhabitants according to the 2010 Census of Population and is one of the most affluentcities in the United States.1 The burglary incidents that occurred in a 6-year span, from 2004 to2009, were provided along with their x, y coordinates by the Plano Police Department under theTexas Open Records Act.2 During this 6-year span, 17,549 vehicle burglary incidents occurred;for analysis purpose, these incidents are tabulated into 159 census block groups according to theiryear of occurrence. That is, each of the 159 block groups (i = 1, . . . , 159) has six observations(T = 1, . . . ,6). The small value of T (i.e., 6) leaves a small number of degrees of freedom toestimate parameters, especially in a GLMM model specification. Fig. 1 portrays the number of

Yongwan Chun Analyzing Space–Time Crime Incidents

169

vehicle burglary incidents aggregated over the 6 years for the census block groups in the city. Itshows that clusters of census block groups with a high number of vehicle burglaries in the easternand northwestern parts of this city. The central areas of this city have relatively few vehicleburglary incidents.

Sixteen covariates are included as surrogates for neighborhood characteristics of the studyarea. These variables were selected based on routine activity theory and social disorganizationtheory, which are briefly reviewed in the previous section. Six of these variables are employed toreflect economic status of the census block groups: income per capita, median house value, highschool or higher graduation rates, poverty rates, unemployment rates, and the rate of householdswith no more than one vehicle. Empirical studies reveal that low-economic-status areas tend tohave higher crime rates (e.g., Conley and Topa 2002; Cahill and Mulligan 2003). Because peoplewho are at home may play the role of guardians, two variables reflecting this tendency also areincluded: the rates of people working at home and the rates of households with young children.Three variables are included to reflect social instability (e.g., Crutchfield, Geerken, and Gove2006): home ownership rates, residential mobility rates, and house vacancy rates. Further, malepopulation rates and young adult rates are included as demographic factors. As for built envi-ronment, the proportions of land use dedicated to parks, whether a railroad traverses a censusblock group, and distance to a major highway are included. Specifically, distance to a majorhighway is widely discussed in the literature as an important factor for burglars in the context ofjourney-to-crime (Rengert 2004; Levine and Block 2011) as well as journey-after-crime (Lu2003). Additionally, various functional models for distance decay effects have been investigatedby scholars (Rossmo 2000; Kent, Leitner, and Curtis 2006). These socioeconomic variableswere obtained from the 5-year American Community Survey (ACS) estimates (2005–2009).GIS data sets were obtained from the North Central Texas Council of Government (http://www.nctcog.org/).

The space–time counts of vehicle burglaries were analyzed with three different Poissonregression model specifications and, similarly, three different NB model specifications using thelogarithm of total population as an offset variable. First, pooled Poisson regression was utilized,ignoring the characteristics of repeated measures over time. That is, observations are treated as across-sectional data set without considering the effects of individual units. Then, this model was

Figure 1. The spatial pattern of vehicle burglary in the city of Plano, Texas, during 2004–2009.

Geographical Analysis

170

extended to two Poisson GLMM specifications with unstructured random effects (P-GLMM-UN)and random effects with a first-order (i.e., lag 1) temporal autoregressive structure (P-GLMM-AR1), respectively. These three model specifications also were analyzed with NB regression:pooled NB, NB-GLMM-UN, and NB-GLMM-AR1. Further, these three Poisson and three NBmodel specifications were estimated with the ESF method to account for spatial autocorrelation,as described by equation (4).

Empirical results

This section presents the analysis results of the vehicle burglary counts in Plano, Texas, using theaforementioned Poisson and NB model specifications. First, the results of the standard Poissonand NB models are presented, and then the results of the ESF Poisson and NB models follow laterin this section.

Results of standard Poisson and NB regressionTable 1 summarizes the estimation results for the three Poisson models without accounting forspatial autocorrelation: the pooled Poisson, the P-GLMM-UN, and the P-GLMM-AR1. Thesemodels were estimated with the residual pseudo-likelihood estimation technique. Three notice-able issues exist here. First, the significance levels of the estimates in the pooled Poissonregression model are different from their counterparts for the two Poisson GLMM models. Forthe pooled Poisson model, 12 variables are significant with the 5% level, but only two variablesare significant at the same level for the two Poisson GLMMs: home ownership rates and distanceto the closest highway. The standard errors of the pooled Poisson models are adjusted with theestimated overdispersion parameter (13.70), whose theoretical value is one for a Poisson distri-bution. This high level of overdispersion may indicate that the pooled model has a modelspecification problem or a substantial level of spatial autocorrelation (Haining, Law, and Griffith2009). Second, the pseudo-AIC values of the Poisson GLMMs are much smaller than the AICvalue of the pooled Poisson model, which indicates that the Poisson GLMMs perform better thanthe pooled model. That is, the Poisson GLMMs improve model performance by accounting forwithin-unit variations with random effects. Further, a comparison of the two Poisson GLMMsreveals that the P-GLMM-AR1 model has a smaller pseudo-AIC and better model behavior. Theestimated autoregressive (AR) parameter is statistically significant at the 5% level; its 95%confidence interval is (0.0794, 0.2262). Third, these three models exhibit a significant level ofspatial autocorrelation. The random effects of the P-GLMM-UN and P-GLMM-AR1 containsignificant positive spatial autocorrelation; their Moran’s I z-scores are 4.7855 and 4.7081,respectively. This outcome indicates that unexplained spatial autocorrelation in the models isembedded in their random effects. Therefore, these model specifications need to be extended toaccount properly for spatial autocorrelation. These z-scores were calculated without consideringestimation effects. That is, independent variables were not incorporated in calculations of theexpected value and variance of Moran’s I because, unlike for residuals in linear regression, thestatistical distribution theory of Moran’s I for random effects in unknown. This remains a topicof future research. Meanwhile, yearly residual spatial autocorrelation for the pooled model has amean z-score for Moran’s I values of 3.0953, indicating a high level of positive spatial autocor-relation in the pooled model.

Table 2 reports the results of the three NB models: the pooled NB, the NB-GLMM-UN,and the NB-GLMM-AR1. Overall, these results are compatible with the results of their

Yongwan Chun Analyzing Space–Time Crime Incidents

171

corresponding Poisson models. First, the small pseudo-AIC values of the NB-GLMMs confirmthat the two NB-GLMMs with random effects terms are significantly improved vis-à-vis thepooled NB model. Furthermore, when the two NB-GLMMs are compared with each other, theNB-GLMM-AR1 is preferred because it has a smaller pseudo-AIC value and its AR parameter issignificant; its 95% confidence interval is (0.0667, 0.2537). Second, these three NB models havea significant level of positive spatial autocorrelation. The z-scores of Moran’s I values for therandom effects of the NB-GLMMs are 4.7819 and 4.7475, respectively. For the pooled NB

Table 1 The Results of Standard Poisson Models: Pooled Poisson, Poisson GLMM withUnstructured Random Effects, and Poisson GLMM with Temporal AR with Lag 1

Variables Pooled Poisson PoissonGLMM-UN

PoissonGLMM-AR1¶

Coeff. Std. error† VIF‡ Coeff. Std. error Coeff. Std. error

Intercept −4.3932 1.1833*** — −4.9447 2.1414* −4.8005 2.1444*Income −0.1812 0.117 3.5342 0.0689 0.2097 0.0422 0.2103House value −0.0192 0.0123 1.6322 −0.0245 0.0267 −0.0253 0.0266Education 0.1302 0.376 3.9943 −0.9358 0.7221 −0.869 0.7217Poverty −0.0835 0.5571 2.5661 −0.8855 1.0206 −0.8438 1.0229Unemployment −1.9219 0.6289** 1.2973 −1.5653 1.0639 −1.5005 1.0687Vehicle (0 or 1) 0.8192 0.3422* 6.0909 0.9361 0.5936 0.9248 0.5969Working home 1.0980 0.3236*** 1.3050 0.1610 0.5931 0.1663 0.5942Family with young

children−0.7008 0.2197** 1.2576 −0.6694 0.4053 −0.6503 0.4062

Home ownership 1.0334 0.2447*** 7.1777 0.9578 0.4542* 0.9702 0.4549*Residential mobility 0.8661 0.3783* 2.0098 0.3727 0.7662 0.4226 0.7616Vacant house 2.4969 0.5636*** 1.5788 1.5813 1.065 1.5954 1.0662Male 2.5732 0.5492*** 1.5242 1.2548 0.9356 1.3981 0.9381Young adult −2.5621 0.7376*** 1.3092 −2.2506 1.3522 −2.3565 1.3543Park −0.8594 0.3254** 1.1103 −0.6207 0.5944 −0.6118 0.597Railroad 0.1984 0.0887* 1.5041 0.1329 0.2065 0.1516 0.2054Distance to highway −0.4636 0.0514*** 1.5683 −0.3818 0.0895*** −0.3879 0.09***AIC or pseudo-AIC 12,976.00 2,116.40 1,540.57Z (Moran’s I) of

random effects (RE)3.0953

(SD = 1.6221)§

4.7855(P-value = 0.0000)

4.7081(P-value = 0.0000)

RE mean — 0.0000 0.0000RE standard deviation — 0.6228 0.5722RE Shapiro–Wilk test — 0.9818

(P-value = 0.0342)0.9815

(P-value = 0.0315)

Significance codes: ***0.001, **0.01, *0.05.†The P-values are calculated based on a quasi-Poisson with 13.70 for its overdispersionparameters.‡A maximum variance inflation factor (VIF) value in excess of 10 generally is considered as anindication of multicollinearity.§Mean and standard deviation of Moran’s I among residuals for each year.¶The estimated AR parameter is 0.1728 with a 95% confidence interval of (0.0794, 0.2662).

Geographical Analysis

172

model, the yearly residual spatial autocorrelation has a mean z-score for Moran’s I values of2.4263. Third, the NB-GLMMs have the same statistical significances for the independentvariables at the 5% level as the Poisson GLMMs. That is, only two variables are significant at the5% level: home ownership rates and distance to the closest highway.

The distributions of the random effects in the Poisson and NB-GLMM models are reportedin Tables 1 and 2. The random effects in the four GLMMs are distributed around 0.0000. That is,their means are 0.0000. Their standard deviations are between 0.5722 and 0.6228. The Shapiro–Wilk normality test results show that the random effects in the four GLMMs follow a normal

Table 2 The Results of Standard NB Models: Pooled NB, NB-GLMM with UnstructuredRandom Effects, and NB-GLMM with Temporal AR with Lag 1

Variables Pooled NB NB-GLMM-UN NB-GLMM-AR1§

Coeff. Std. error VIF‡ Coeff. Std. error Coeff. Std. error

Intercept −3.7918 0.9553*** — −4.9431 2.1414* −4.8646 2.1434*Income −0.0810 0.0936 2.9182 0.0687 0.2097 0.0555 0.2101House value −0.0375 0.0116** 1.7911 −0.0245 0.0267 −0.0249 0.0267Education −0.4473 0.3199 3.1961 −0.9352 0.7221 −0.9141 0.7221Poverty −1.1047 0.4540* 2.1888 −0.8859 1.0205 −0.8594 1.0220Unemployment −0.7109 0.4773 1.3009 −1.5636 1.0639 −1.5468 1.0665Vehicle (0 or 1) 0.6757 0.2665* 5.3282 0.5936 0.5936 0.9356 0.5954Working home −0.1609 0.2652 1.2033 0.1602 0.5931 0.1560 0.5938Family with young

children−0.4102 0.1809* 1.1646 −0.6691 0.4052 −0.6578 0.4058

Home ownership 0.8230 0.2027*** 6.5245 0.9576 0.4542* 0.9694 0.4547*Residential mobility 0.5290 0.3391 2.0436 0.3727 0.7662 0.4051 0.7643Vacant house 1.3169 0.4730** 1.6426 1.5807 1.0650 1.5846 1.0659Male 2.2965 0.4257*** 1.4110 1.2567 0.9357 1.3242 0.9370Young adult −3.9072 0.6022*** 1.3475 −2.2533 1.3522 −2.2967 1.3535Park −0.2212 0.2627 1.1389 −0.6202 0.5943 −0.6155 0.5958Railroad 0.4921 0.0895*** 1.3445 0.1334 0.2065 0.1412 0.2061Distance to highway −0.3992 0.0398*** 1.5577 −0.3818 0.0895*** −0.3859 0.0898***AIC or pseudo-AIC 7,283.31 1,502.26 1,494.65Z (Moran’s I) of

random effects(RE)

2.4263(SD = 1.3458)†

4.7819(P-value = 0.0000)

4.7475(P-value = 0.0000)

RE mean — 0.0000 0.0000RE standard deviation — 0.5993 0.5803RE Shapiro–Wilk test — 0.9817

(P-value = 0.0335)0.9820

(P-value = 0.0367)

Significance codes: ***0.001, **0.01, *0.05.†Mean and standard deviation of Moran’s I among residuals for each year.‡A maximum VIF value in excess of 10 generally is considered as an indication ofmulticollinearity.§The estimated AR parameter is 0.1602, with a 95% confidence interval of (0.0667, 0.2537).

Yongwan Chun Analyzing Space–Time Crime Incidents

173

distribution at the 1% level but not at the 5% level: their P-values are 0.0342, 0.0315, 0.0335, and0.0367.

In sum, these results suggest that the space–time counts of vehicle burglary have a significantlevel of within-unit variation that enables the Poisson and NB-GLMMs with random effects toexplain variations better in the data set. Specifically, these count data have a significant temporalstructure (here, an AR one with lag 1). However, a significant level of positive spatial autocor-relation is present in all three of the models and needs to be properly accounted for.

Results of ESF Poisson and NB regressionTable 3 summarizes the results of three spatial filter Poisson models described in equation (4).The standard errors of the ESF pooled model also are adjusted with the estimated dispersionparameter (10.74). Three points about the results are noticeable. First, the three ESF Poissonmodels do not display a significant level of residual spatial autocorrelation. The mean of theresidual Moran’s I z-scores for each year for the ESF pooled model is −0.1172, and the randomeffects in the two ESF Poisson GLMMs also have insignificant Moran’s I values; the Moran’s Iz-scores for them are −1.2121 and −1.3522, respectively. Fig. 2 portrays the patterns of theestimated random effects for the standard Poisson GLMMs and ESF Poisson GLMMs. Therandom effects of the standard Poisson GLMMs show a positive spatial autocorrelation pattern.Specifically, a noticeable cluster of positive values can be observed along a north-south swath onthe east side of Plano. Also, a cluster of negative random effects can be observed in the northernpart of the city. In contrast, the random effects of the ESF Poisson GLMMs do not showsubstantial spatial clusters of positive or negative values. This can indicate that the linearcombinations of eigenvectors included in the ESF Poisson GLMMs successfully account forspatial autocorrelation, which compromises the standard Poisson models. The smaller estimatefor the overdispersion parameter of the ESF pooled model also indicates that spatial autocorre-lation is well explained by the spatial eigenvectors (Haining, Law, and Griffith 2009).

Second, the ESF Poisson models have smaller (pseudo-) AIC values than their counterpartstandard Poisson models. This outcome indicates that accounting for spatial autocorrelation withthe ESF approach improves the Poisson model descriptions and behavior. The Fig. 3 scatterplotsof observed versus fitted counts for the Poisson models also show this model improvement,although it is more noticeable for the pooled models, which is also confirmed with the larger AICdecrease for the pooled models. A comparison of the six Poisson models reveals that the ESFP-GLMM-AR1 model with the smallest AIC value is preferred over the other five models. Thatis, the model specification in which spatial and temporal correlation structures are incorporatedsimultaneously produces the best model performance by the six specifications. These scatterplotsalso reveal that the empirical data set has a strong skewed pattern: more observations have smallrather than large values.

Third, the ESF models render different statistical decisions for the covariates vis-à-vis theircounterpart standard models. Specifically, two variables—house vacancy rates and housevalues—are significant at the 5% level in the ESF Poisson GLMMs but not in the standardPoisson GLMMs. Similar changes also occur between the standard and ESF pooled models. Thisempirical example furnishes another illustration of the impact of unexplained spatial autocorre-lation on statistical decisions. A comparison of the three ESF Poisson models reveals that the twoESF Poisson GLMMs have the same four variables with a 5% level of statistical significance,and, in contrast, the ESF Poisson pooled model has more variables with a 5% level of statisticalsignificance. This pattern is similar to the standard Poisson models. That is, the consideration of

Geographical Analysis

174

within-unit variation triggers statistical decision changes for covariates. However, the differentspecifications for the random effects (i.e., unstructured and an AR1 temporal structure) do notlead to different statistical inferences for all of the covariates at the 5% level, although theP-GLMM-AR1 and the ESF P-GLMM-AR1 have a smaller pseudo-AIC value with a significanttemporal AR estimate at the 5% than their counterpart standard models.

Table 3 The Results of ESF Poisson Models: Pooled Poisson, Poisson GLMM withUnstructured Random Effects, and Poisson GLMM with Temporal AR with Lag 1

Variables ESF pooled Poisson ESF PoissonGLMM-UN

ESF PoissonGLMM-AR1¶

Coeff. Std. error† VIF‡ Coeff. Std. error Coeff. Std. error

Intercept −6.4950 1.1931*** −5.9436 2.1133** −5.9745 2.1154**Income −0.0537 0.1208 4.8860 0.0286 0.2085 0.0215 0.2088House value −0.0517 0.0123*** 2.0320 −0.0550 0.0243* −0.0559 0.0242*Education 1.3663 0.3743*** 4.8892 0.7075 0.6924 0.7341 0.6889Poverty 0.2450 0.5528 3.1707 −0.5483 0.9093 −0.4859 0.9129Unemployment −0.8492 0.6381 1.7103 −0.1982 1.0374 −0.1595 1.0430Vehicle (0 or 1) 0.7350 0.3232* 7.3771 0.7842 0.5301 0.7800 0.5343Working home 1.1029 0.3265*** 1.5815 0.5557 0.5400 0.5537 0.5420Family with young

children−0.4121 0.2124 1.5096 −0.3856 0.3683 −0.3681 0.3688

Home ownership 1.1354 0.2458*** 9.2871 1.0899 0.4129** 1.0974 0.4147**Residential mobility 0.3403 0.3511 2.1550 −0.2723 0.6684 −0.2175 0.6623Vacant house 2.9693 0.5361*** 1.8417 2.0601 0.9517* 2.0657 0.9540*Male 2.2485 0.5321*** 1.9864 1.3784 0.8260 1.5177 0.8315Young adult −2.1817 0.7505** 1.6655 −2.0519 1.2309 −2.1691 1.2305Park −1.1773 0.3545*** 1.8663 −0.2915 0.5677 −0.3243 0.5740Railroad −0.1394 0.0954 2.2231 −0.0867 0.1937 −0.0628 0.1926Distance to highway −0.5031 0.0545*** 2.4592 −0.5430 0.0966*** −0.5378 0.0966***AIC or pseudo-AIC 10,010.00 2,072.89 1,502.09Z (Moran’s I) of

random effects(RE)

−0.1172(SD = 0.2682)§

−1.2121(P-value = 0.2255)

−1.3522(P-value = 0.1763)

RE mean — 0.0000 0.0000RE standard deviation — 0.4859 0.4294RE Shapiro–Wilk test — 0.9227

(P-value = 0.0000)0.9220

(P-value = 0.0000)

Significance codes: ***0.001, **0.01, *0.05†The P-values are calculated based on a quasi-Poisson with 10.74 for its overdispersionparameter.‡A maximum variance inflation factor (VIF) value in excess of 10 generally is considered as anindication of multicollinearity.§Mean and standard deviation of Moran’s I among residuals for each year.¶The estimated AR parameter is 0.1743, with a 95% confidence interval of (0.0811, 0.2675)

Yongwan Chun Analyzing Space–Time Crime Incidents

175

Fig

ure

2.T

hepa

ttern

sof

rand

omef

fect

sfo

rth

est

anda

rdan

dE

SFPo

isso

nG

LM

Ms.

Geographical Analysis

176

Fig

ure

3.Sc

atte

rplo

tsof

obse

rved

vers

usfit

ted

coun

tsfo

rth

est

anda

rdan

dE

SFPo

isso

nm

odel

s.

Yongwan Chun Analyzing Space–Time Crime Incidents

177

The four significant variables for the ESF P-GLMM-AR1 are median house value, homeownership rate, vacant house rate, and distance to major highway. These estimates show thatvehicle burglary tends to occur more often in locations with a lower house value (–0.0559) anda higher house vacancy rate (2.0657). These two variables are significant at the 5% level. Theyare consistent with social disorganization theory (Sampson 1983; Bursik and Grasmick 1993),which postulates that economic deprivation is positively associated with crime. The positiveassociation with house vacancy rates also can be explained in terms of a high social instability.Distance to highway is significant at the 0.1% level and has a negative sign (–0.5378). This meansthat census block groups that have better access to a highway tend to experience more vehicleburglaries during the 6-year period. This result can be explained in the context of journey-to-crime and/or journey-after-crime, which posits that transportation is a significant factor forvehicle burglaries. The final significant variable is home ownership rates, which has a positivesign (1.0974). This result is different from findings in the literature. Home ownership ratesgenerally are expected to be negatively associated with crime (e.g., Glaeser and Sacerdote 1999)because they frequently reflect a high level of social stability and social connection. One possiblereason for this particular negative relationship is the popularity of gated multifamily housing inPlano, which might have a negative impact on crime (e.g., Helsley and Strange 1999), but thismismatch merits a further formal investigation.

Table 4 reports the results of the three ESF NB models. These results are fairly compatiblewith the results of the ESF Poisson models. First, the positive spatial autocorrelation is wellexplained in the ESF NB models. The yearly residuals of the ESF pooled NB model do not havea significant level of spatial autocorrelation. Also, the random effects of the NB-GLMMs do nothave significant positive spatial autocorrelation but rather exhibit significant negative spatialautocorrelation. This outcome may indicate an overcorrection for spatial autocorrelation. Second,the ESF NB-GLMM-AR1 with the smallest (pseudo-) AIC value is preferred to the other ESF NBmodels. Third, the ESF NB and Poisson GLMMs have the same statistical significances for theindependent variables at the 5% level. That is, the four significant variables are median housevalue, home ownership rate, vacant house rate, and distance to highway.

Fig. 4 portrays the patterns of the estimated random effects for the standard NB-GLMMs andESF NB-GLMMs. The random effects maps for the standard NB-GLMMs are very similar tothose of the standard Poisson GLMMs. These maps visually suggest that positive spatial auto-correlation is embedded in the random effects of the standard NB-GLMMs. In contrast, therandom effects maps of the ESF NB-GLMMs do not have a prominent cluster of positive ornegative values. This outcome may suggest that the eigenvectors in these models successfullyaccount for spatial autocorrelation.

Tables 3 and 4 report the distributions of the random effects in the ESF Poisson and ESFNB-GLMM models. The means of the random effects in the four ESF GLMMs are 0.0000. Thatis, they are the same as those of the four standard GLMMs. Their standard deviations (which arebetween 0.4276 and 0.4859) are smaller than those of the four standard GLMMs. This outcomemay suggest that part of the variability of the random effects in the standard GLMMs is explainedby the eigenvector spatial filter. However, the Shapiro–Wilk normality test results for the fourESF GLMMs show that these random effects do not follow a normal distribution (their P-valuesare 0.0000). This non-normality of the random effects might be attributable to the small numberof degrees of freedom resulting from T = 6. The ESF GLMMs have a smaller number of degreesof freedom than the standard GLMMs because of the inclusion of eigenvectors. This smalldegrees of freedom issue may need further investigation.

Geographical Analysis

178

Summary and conclusions

This article discusses modeling space–time crime incidents using the ESF method coupled withGLMMs. This proposed method allows simultaneous incorporation of two correlation structuresin a model specification: a spatial correlation structure across observational spatial units and atemporal correlation structure within individual observational units. Because a set of selectedeigenvectors isolates spatial autocorrelation, random effects in a standard GLMM specificationdo not contain a substantial level of spatial autocorrelation. That is, this specification leads to alinear decomposition of spatially and temporally structured random effects.

Table 4 The Results of ESF NB Models: Pooled NB, NB-GLMM with Unstructured RandomEffects, and NB-GLMM with Temporal AR with Lag 1

Variables ESF pooled NB ESFNB-GLMM-UN

ESFNB-GLMM-AR1§

Coeff. Std. error VIF‡ Coeff. Std. error Coeff. Std. error

Intercept −4.3259 1.0047*** — −4.8072 2.2420* −4.8902 2.2523*Income −0.1042 0.1001 4.6956 −0.0575 0.2222 −0.0530 0.2234House value −0.0689 0.0109*** 2.2165 −0.0529 0.0249* −0.0533 0.0249*Education 0.9942 0.3210** 4.5630 0.9045 0.7183 0.9041 0.7194Poverty −1.3880 0.4238** 2.6580 −1.0986 0.9345 −1.0582 0.9414Unemployment 0.1539 0.4709 1.7517 −0.0162 1.0402 0.0121 1.0463Vehicle (0 or 1) 0.8293 0.2445*** 6.2277 0.7996 0.5343 0.8082 0.5390Working home −0.0387 0.2405 1.3731 0.3329 0.5318 0.3144 0.5348Family with young

children−0.1362 0.1683 1.4126 −0.3163 0.3745 −0.3125 0.3764

Home ownership 1.1405 0.1975*** 8.6610 1.0067 0.4306* 1.0115 0.4343*Residential mobility −0.2204 0.3166 2.4844 −0.4302 0.7068 −0.3821 0.7074Vacant house 1.6673 0.4315*** 1.8734 1.9083 0.9560* 1.9084 0.9615*Male 1.9864 0.3932*** 1.6793 1.3896 0.8476 1.4376 0.8542Young adult −3.1904 0.5714*** 1.6828 −2.3935 1.2594 −2.4393 1.2657Park 0.6977 0.2592** 1.5462 0.0983 0.5709 0.0894 0.5758Railroad 0.0147 0.0856 1.7392 −0.1655 0.1949 −0.1574 0.1952Distance to highway −0.8511 0.0622*** 5.3407 −0.7590 0.1392*** −0.7544 0.1399***AIC or pseudo-AIC 6,982.69 1,410.27 1,405.27Z (Moran’s I) of

random effects(RE)

−0.4416(SD = 1.2170)†

−2.3001(P-value = 0.0214)

−2.3249(P-value = 0.0201)

RE mean — 0.0000 0.0000RE standard deviation — 0.4461 0.4276RE Shapiro–Wilk test — 0.9466

(P-value = 0.0000)0.9458

(P-value = 0.0000)

Significance codes: ***0.001, **0.01, *0.05.†Mean and standard deviation of Moran’s I among residuals for each year.‡A maximum VIF value in excess of 10 generally is considered as an indication of multicol-linearity.§The estimated AR parameter is 0.1570, with a 95% confidence interval of (0.0641, 0.2499).

Yongwan Chun Analyzing Space–Time Crime Incidents

179

Fig

ure

4.T

hepa

ttern

sof

rand

omef

fect

sfo

rth

est

anda

rdan

dE

SFN

B-G

LM

Ms.

Geographical Analysis

180

The analysis of vehicle burglary in the city of Plano, Texas, illustrates that the proposedmethod provides an effective way to model space–time crime data. Specifically, a set of spatialeigenvectors successfully accounts for spatial autocorrelation, and, accordingly, random effectsalone then can reflect pure unstructured or temporal within-unit variation. These different modelspecifications lead to different statistical inferences about parameter estimates for covariates.This modeling exercise furnishes additional empirical evidence for the need for proper modelspecifications to describe space–time crime data. A comparison of the six standards and ESFPoisson models reveals that the ESF P-GLMM-AR1, which contains both spatial and temporalcomponents, furnishes the best model specification with the smallest pseudo-AIC value. The ESFP-GLMM-AR1 and the ESF P-GLMM-UN do not differ much in terms of significance levels ofparameter estimates for covariates. The six NB model specifications produce similar results. Thatis, the ESF NB-GLMM-AR1 with spatial and temporal random effects simultaneously in itsmodel specification is preferred to the other models. Based on the results of the ESF P-GLMM-AR1 and the ESF NB-GLMM-AR1, vehicle burglary in Plano during the 2005–2009 period ispositively associated with house vacancy rates and negatively associated with median housevalue and distance to highway. Vehicle burglary also exhibits a significant positive associationwith home ownership rate, a relationship meriting further investigation.

Studies show that Poisson regression is an appropriate method to analyze crime count data,which are not likely to follow a Gaussian distribution (e.g., Osgood 2000). However, few studiesundertake space–time crime data modeling using Poisson regression, which is in stark contrast tothe type of space–time studies undertaken in other disciplines such as health studies. This ESFspecification coupled with a GLMM provides an effective method for space–time crime datamodeling. However, the specification employed in this article currently has a notable limitationbecause it is based on an assumption that there is no interaction between the temporal and spatialcomponents. Subsequent research needs to be evaluated for a myriad of different space–timestructures.

Acknowledgements

I thank the editor as well as the referees for their valuable suggestions and comments.

Notes

1 According to the 2007–2011 American Community Survey (ACS), the medium household income inPlano is $82,901, which is greater than the U.S. medium income of $52,762.

2 Information about the Texas Open Records Act can be found at http://www.plano.gov/index.aspx?nid=765.

References

Abellan, J., S. Richardson, and N. Best. (2008). “Use of Space–Time Models to Investigate the Stabilityof Patterns of Disease.” Environmental Health Perspective 116(8), 1111–19.

Augustin, N. H., J. McNicol, and C. Marriott. (2006). “Using the Truncated Auto-Poisson Model forSpatial Correlated Counts of Vegetation.” Journal of Agricultural, Biological, and EnvironmentalStatistics 11(1), 1–23.

Berk, R., and J. M. MacDonald. (2008). “Overdispersion and Poisson Regression.” Journal of Quantita-tive Criminology 24(3), 269–84.

Bernasco, W., and P. Nieuwbeerta. (2005). “How Do Residential Burglars Select Target Areas?” BritishJournal of Criminology 44, 296–315.

Yongwan Chun Analyzing Space–Time Crime Incidents

181

Besag, J. (1974). “Spatial Interaction and the Statistical Analysis of Lattice Systems.” Journal of theRoyal Statistical Society, Series B 36(2), 192–236.

Besag, J., J. York, and A. Mollié. (1991). “Bayesian Image Restoration with Two Application in SpatialStatistics.” Annals of the Institute of Statistical Mathematics 43(1), 1–20.

Block, R. (1979). “Community, Environment, and Violent Crime.” Criminology 17, 46–57.Breslow, N. E., and D. G. Clayton. (1993). “Approximate Inference in Generalized Linear Mixed

Models.” Journal of the American Statistical Association 88(421), 9–25.Bursik, R. (1988). “Social Disorganization and Theory of Crime and Delinquency: Problems and Pros-

pects.” Criminology 26(4), 519–51.Bursik, R., and H. Grasmick. (1993). Neighborhoods and Crime. San Francisco: Lexington Books.Cahill, M., and G. Mulligan. (2003). “The Determinants of Crime in Tucson, Arizona.” Urban Geogra-

phy 24(7), 582–610.Ceccato, V., R. Haining, and P. Signoretta. (2004). “Exploring Offence Statistics in Stockholm City

Using Spatial Analysis Tools.” Annals of the Association of American Geographers 92(1), 29–51.Chainey, S., and J. Ratcliffe. (2005). GIS and Crime Mapping. Chichester, U.K.: Wiley.Chun, Y. (2008). “Modeling Network Autocorrelation within Migration Flows by Eigenvector Spatial

Filtering.” Journal of Geographical Systems 10(4), 317–44.Chun, Y., and D. A. Griffith. (2011). “Modeling Network Autocorrelation in Space–Time Migration Flow

Data: An Eigenvector Spatial Filtering Approach.” Annals of the Association of American Geogra-phers 101(3), 523–36.

Cohen, J., W. L. Gorr, and A. M. Olligschlaeger. (2007). “Leading Indicators and Spatial Interactions: ACrime-Forecasting Model for Proactive Police Deployment.” Geographical Analysis 39(1), 105–27.

Cohen, L. E., and M. Felson. (1979). “Social Change and Crime Rate Trends: A Routine ActivityApproach.” American Sociological Review 44, 588–608.

Congdon, P. D. (2013). “A Model for Spatially Varying Crime Rates in English Districts: The Effects ofSocial Capital, Fragmentation, Deprivation and Urbanicity.” International Journal of Criminologyand Sociology 2, 138–52.

Conley, T. G., and G. Topa. (2002). “Socio-Economic Distance and Spatial Patterns in Unemployment.”Journal of Applied Economics 17(4), 303–27.

Crutchfield, R., M. R. Geerken, and W. R. Gove. (2006). “Crime Rate and Social Integration: TheImpact of Metropolitan Mobility.” Criminology 20, 467–78.

D’Unger, A., K. C. Land, P. L. McCall, and D. S. Nagin. (1998). “How Many Latent Classes ofDelinquent/Criminal Careers? Results from Mixed Poisson Regression Analyses.” American Journalof Sociology 103(6), 1593–630.

Diggle, P., J. Tawn, and R. Moyeed. (1998). “Model-Based Geostatistics.” Applied Statistics 47(3), 299–350.

Glaeser, E., and B. Sacerdote. (1999). “Why Is There More Crime in Cities? “ Journal of PoliticalEconomy 107, 225–58.

Griffith, D. A. (2002). “A Spatial Filtering Specification for the Auto-Poisson Model.” Statistics andProbability Letters 58(3), 245–51.

Griffith, D. A. (2003). Spatial Autocorrelation and Spatial Filtering: Gaining Understanding throughTheory and Scientific Visualization. Berlin: Springer.

Griffith, D. A. (2006). “Assessing Spatial Dependence in Count Data: Winsorized and Spatial FilterSpecification Alternatives to the Auto-Poisson Model.” Geographical Analysis 38, 160–79.

Griffith, D. A. (2011). “Space, Time, and Space–Time Eigenvector Filter Specifications That Account forAutocorrelation.” Estadística española 53(177), 211–37.

Griffith, D. A. (2013). “Estimating Missing Data Values for Georeferenced Poisson Counts.” Geographi-cal Analysis 45, 259–84.

Griffith, D. A., and R. Haining. (2006). “Beyond Mule Kicks: The Poisson Distribution in GeographicalAnalysis.” Geographical Analysis 38, 123–39.

Grubesic, T. H. (2006). “On the Application of Fuzzy Clustering for Crime Hot Spot Detection.” Journalof Quantitative Criminology 22(1), 77–105.

Grubesic, T. H., and E. A. Mack. (2008). “Spatio-Time Interaction of Urban Crime.” Journal of Quanti-tative Criminology 22(1), 77–105.

Geographical Analysis

182

Haining, R., and J. Law. (2007). “Combining Police Perceptions with Police Records of Serious CrimeAreas: A Modeling Approach.” Journal of Royal Statistical Society Series A 170, 1019–34.

Haining, R., J. Law, and D. A. Griffith. (2009). “Modelling Small Area Counts in the Presence ofOverdispersion and Spatial Autocorrelation.” Computational Statistics and Data Analysis 53,2923–37.

Helsley, R. W., and W. C. Strange. (1999). “Gated Communities and the Economic Geography ofCrime.” Journal of Urban Economics 46(1), 80–105.

Johnson, S. D., W. Bernasco, K. J. Bowers, H. Elffers, J. Ratcliffe, G. Rengert, and M. Townsley.(2007). “Space–Time Patterns of Risk: A Cross National Assessment of Residential Burglary Vic-timization.” Journal of Quantitative Criminology 23, 201–19.

Kaiser, M. S., and N. Cressie. (1997). “Modeling Poisson Variables with Positive Spatial Dependence.”Statistics and Probability Letters 35(4), 423–32.

Kent, J., M. Leitner, and A. Curtis. (2006). “Evaluating the Usefulness of Functional Distance MeasuresWhen Calibrating Journey-to-Crime Distance Decay Functions.” Computers, Environment andUrban Systems 30, 181–200.

Kim, H., and J. J. Oleson. (2007). “A Bayesian Dynamic Spatio-Temporal Interaction Model: An Appli-cation to Prostate Cancer Incidence.” Geographical Analysis 40(1), 77–96.

Kim, Y., and M. E. O’Kelly. (2008). “A Bootstrap Based Space–Time Surveillance Model with AnApplication to Crime Occurrences.” Journal of Geographical Systems 10(2), 1–25.

Lambert, D. M., J. P. Brown, and R. J. G. M. Florax. (2010). “A Two-Step Estimator for A Spatial LagModel of Counts: Theory, Small Sample Performance and An Application.” Regional Science andUrban Economics 40, 241–52.

Law, J., and R. Haining. (2004). “A Bayesian Approach to Modeling Binary Data: The Case of HighIntensity Crime Areas.” Geographical Analysis 36(3), 197–216.

Law, J., R. Haining, R. Maheswaran, and T. Pearson. (2006). “Analyzing the Relationship betweenSmoking and Coronary Heart Disease at the Small Area Level.” Geographical Analysis 38,140–59.

Law, J., M. Quick, and P. Chan. (2014). “Bayesian Spatio-Temporal Modeling for Analysing Local Pat-terns of Crime over Time at the Small-Area Level.” Journal of Quantitative Criminologydoi:10.1007/s10940-013-9194-1. Available at http://link.springer.com/article/10.1007/s10940-013-9194-1 (accessed February 20, 2014).

Lawson, A. B. (2009). Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology.New York: CRC Press.

Levine, N. (2005). “Crime Mapping and the Crimestat Program.” Geographical Analysis 38(1), 41–56.Levine, N., and B. Block. (2011). “Bayesian Journey-to-Crime Estimation: An Improvement in Geo-

graphic Profiling Methodology.” Professional Geographer 62(2), 213–29.Lu, Y. (2003). “Getting Away with the Stolen Vehicle: An Investigation of Journey-after-Crime.” Profes-

sional Geographer 55(4), 422–33.Ludwig, J., G. Duncan, and P. Hirschfield. (2001). “Urban Poverty and Juvenile Crime: Evidence from A

Randomized Housing-Mobility Experiment.” Quarterly Journal of Economics 116(2), 655–79.Malczewski, J., and A. Poetz. (2005). “Residential Burglaries and Neighborhood Socioeconomic Context

in London, Ontario: Global and Local Regression Analysis.” Professional Geographer 57(4), 516–29.Matthews, S. A., T. C. Yang, K. L. Hayslett, and R. B. Ruback. (2010). “Built Environment and Property

Crime in Seattle, 1998–2000: A Bayesian Analysis.” Environment and Planning A 42(6), 1403–20.Mohebbi, M., R. Wolfe, and D. Jolley. (2011). “A Poisson Regression Approach for Modeling Spatial

Autocorrelation between Geographically Referenced Observations.” BMC Medical Research Meth-odology 11, 133.

Morenoff, J. D., R. J. Sampson, and S. W. Raudenbush. (2001). “Neighborhood Inequality, CollectiveEfficacy, and the Spatial Dynamic of Urban Violence.” Criminology 39, 517–58.

Nakaya, T., and K. Yano. (2010). “Visualising Crime Clusters in A Space–Time Cube: An ExploratoryData-Analysis Approach Using Space–Time Kernel Density Estimation and Scan Statistics.” Trans-actions in GIS 14(3), 233–39.

Osgood, D. W. (2000). “Poisson-Based Regression Analysis of Aggregate Crime Rates.” Journal ofQuantitative Criminology 16(1), 21–43.

Yongwan Chun Analyzing Space–Time Crime Incidents

183

Patuelli, R., D. A. Griffith, M. Tiefelsdorf, and P. Nijkamp. (2011). “Spatial Filtering and EigenvectorStability: Space–Time Models for German Unemployment Data.” International Regional ScienceReview 34(2), 253–80.

Peeples, F., and R. Loeber. (1994). “Do Individual Factors and Neighborhood Context Explain EthnicDifferences in Juvenile Delinquency? “ Journal of Quantitative Criminology 10, 141–57.

Ratcliffe, J. H. (2003). “Suburb Boundaries and Residential Burglars.” Trends and Issues in Crime andCriminal Justice, no. 246, Australian Institute of Criminology, Canberra.

Ratcliffe, J. H. (2012). “The Spatial Extent of Criminogenic Places: A Changepoint Regression of Vio-lence around Bars.” Geographical Analysis 44(4), 302–20.

Rengert, G. F. (2004). “The Journey to Crime.” In Punishment, Places and Perpetrators: Developmentsin Criminology and Criminal Justice Research, 169–81, edited by G. Bruinsma, H. Elffers and J. deKeijser. Portland: Willan Publishing.

Rossmo, K. D. (2000). Geographic Profiling. Boca Raton: CRC Press.Sampson, R. (1983). “Structural Density and Criminal Victimization.” Criminology 21, 276–93.Sampson, R. J. (1993). “The Community Context of Violent Crime.” In Sociology and the Public

Agenda, 267–74, edited by W. J. Wilson. Newbury Park: Sage Publications.Sherman, L., P. Gartin, and M. Buerger. (1989). “Hot Spots of Predatory Crime: Routine Activities and

the Criminology of Place.” Criminology 27, 27–55.South, S., and S. Messner. (2000). “Crime and Demography: Multiple Linkages, Reciprocal Relations.”

Annual Review of Sociology 26, 83–106.Tiefelsdorf, M., and B. Boots. (1995). “The Exact Distribution of Moran’s I.” Environment and Planning

A 27(6), 985–99.Tiefelsdorf, M., and D. A. Griffith. (2007). “Semiparametric Filtering of Spatial Autocorrelation: The

Eigenvector Approach.” Environment and Planning A 39, 1193–221.Ye, X., and L. Wu. (2011). “Analyzing the Dynamics of Homicide Patterns in Chicago: ESDA and

Spatial Panel Approaches.” Applied Geography 31(2), 800–77.Yoo, E.-H., B. W. Hoagland, and G. Cao. (2013). “Spatial Distribution of Trees and Landscapes of the

Past: A Mixed Spatially Correlated Multinomial Logit Model Approach for the Analysis of thePublic Sand Survey Data.” Geographical Analysis 45(4), 420–41.

Yu, Q., R. Scribner, B. Carlin, K. Theall, N. Simonsen, B. Ghosh-Dastidar, D. Cohen, and K. Mason.(2008). “Multilevel Spatio-Time Dual Changepoint Models for Relating Alcohol Outlet Destructionand Changes in Neighborhood Rates of Assaultive Violence.” Geospatial Health 2(2), 161–272.

Zhang, D. (2004). “Generalized Linear Mixed Models with Varying Coefficients for Longitudinal Data.”Biometrics 60(1), 8–15.

Geographical Analysis

184

Copyright of Geographical Analysis is the property of Wiley-Blackwell and its content maynot be copied or emailed to multiple sites or posted to a listserv without the copyright holder'sexpress written permission. However, users may print, download, or email articles forindividual use.