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Jun 15, 2002 - analysis and cluster correspondence analysis) to the spatial dependence ... of geospatial lifelines, a special class of spatiotemporal data .... sesses the relationship between two point sets through evaluating the degree of mix-.
Yongmei Lu Jean-Claude Thill

Assessing the Cluster Correspondence between Paired Point Locations

Some complex geographic events are associated with multiple point locations. Such events include, but are not limited to, those describing linkages between and among places. The term multi-location event is used in the paper to refer to these geographical phenomena. Through formalization of the multi-location event problem, this paper situates the analysis of multi-location events within the broad context of point pattern analysis techniques. Two alternative approaches (vector autocorrelation analysis and cluster correspondence analysis) to the spatial dependence of pairedlocation events (i.e., two-location events) are explored, with a discussion of their appropriateness to general multi-location event problems. The research proposes a framework of cluster correspondence analysisfor the detection of local non-stationarities in the spatial process generating multi-location events. A new algorithmfor local analysis of cluster correspondence is proposed. It is implemented on a large-scale dataset of vehicle thejl and recovey location pairs in Buffalo, New York.

1. INTRODUCTION

Many spatial phenomena can be conceptualized as elemental events of zero dimension and can be represented as a single set of points for geographical investigation. The well-established techniques of spatial statistics (Diggle 1983; Upton and Fingleton 1985; Boots and Getis 1988; Bailey and Gatrell 1995; Fotheringham, Brunsdon, and Charlton 2000) provide methodological support for the analysis of such point patterns. When multiple sets of zero-dimensional events interact with each other during certain geographical processes (e.g., the coexistence and interdependence of different species of trees), multivariate spatial point analysis techniques exist to represent and analyze the distribution of multiple point sets (Diggle 1983).By extending the spatial statistical analysis of single point sets to that of multiple point sets, these techniques enable a broader set of research questions to be addressed, Yongmei Lu is an assistant professor in the Department of Geography, and the Texas Center for Geographic Information Science, Southwest Texas State University. E-mail: [email protected] Thill is a professor in the Department of Geography, University at Buffalo, The State University of New York, and a research scientistfor the National Center for Geographic Information and Analysis (NCGIA). Email: [email protected]. Geographical Analysis, Vol. 35, No. 4 (October 2003) The Ohio State University Submitted:June 15,2002. Revised version accepted: March 19,2003.

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such as whether the pattern in the occurrences of one type of event is related to that of another. Besides simple geographical processes that link each event to a single location, some geographical events exhibit the added complexity of association with multiple point locations. Such events include, but are not limited to, those describing some linkage between and among places. The linking can be based on physical connections, functional interactions, or any other process relating one place to another. Such is the case of geospatial lifelines, a special class of spatiotemporal data identified and defined by Mark and Egenhofer (1998) as “continuous set[s] of positions occupied by an object in geographic space over some time period.” Migration and journey-to-work associating origins with destinations, information exchange connecting places through telecommunication,and crime events developing spatial relationships between criminals’ base locations and the corresponding offense sites are but three examples of such complex geographical phenomena. To provide a complete description of these complex phenomena, related analysis will have to take into consideration the association of multiple locations through a single geographical event. In this paper, we use the term multi-location events to refer to such complex geographical phenomena. The most important aspect for multi-location events is that a certain point from one location set is associated with a specific point from another location set through the construction of a specific event. For example, any work trip links a specific residential place with a specific workplace. Because of the structure embedded in multi-location events, techniques of multivariate spatial pattern analysis cannot effectively describe their spatial pattern. Multivariate spatial pattern analysis considers the points belonging to distinct location sets to be generated by independent processes. These methods assume away any special relationship between elements from one point set and elements from another set. Hence, they fail to capture the full extent of spatial associations embedded in multi-location events and are ill suited to detect global or local spatial association properties of such events. Therefore, there is a true need to expand the family of point pattern analysis techniques to include new methodologies that can describe the spatial patterns of multi-location events (see Table 1). In this paper, we argue for a new class of spatial statistics for the exploratory analysis of multi-location events at global and local scales, and we present a conceptual

TABLE 1 Cluster Correspondence Analysis in the Family of Point Pattern Analysis Single Point Pattern Analysis (Spatial Autocorrelation) Multivariate Spatial Pattern Analysis (Extended Single Point Spatial Autocorrelation) Multivariate

Intra-pair

Point

Multi-location

Pattern

Event

Analysis

Analysis (Spatial Correspondence Analysis)

Direction Analysis

Analysis

Directional Autocorrelation Inter-pair

I

Distance Autocorrelation

1

292 / Geographical Analysis framework for this purpose. Two approaches are discussed. Spatial autocorrelation of multi-location events can first be tested by examining the spatial patterns of the onedimensional objects (vectors) linking point locations (vector autocorrelation). Alternatively, it can be revealed by the pattern of persistence of distance and direction relationships along the path of multi-location events. In particular, that points representing one end of vectors are clustered and that other point locations linked through the same vectors would also be clustered is an indication that the corresponding multi-location events are positively spatially correlated and that the vectors representing them form clusters. We refer to this complex pattern of cluster dependence within linked clusters as cluster correspondence.We argue for the approach of cluster correspondence over vector autocorrelation analysis because of the relative immaturity of the latter. After transposing the problem of multi-location events into that of a series of single location sets, we propose to tackle cluster correspondence analysis by extending well-developed univariate point pattern analysis techniques. The second section of the paper formalizes the spatial pattern detection problem of multi-location events and situates the analysis of multi-location events within the broader context of point pattern analysis. It also identifies cluster correspondence analysis as a suitable approach to this effect. The remainder of the paper focuses on the detection of local non-stationarity in multi-location spatial processes. Section 3 explains the requirements and technical tasks involved in evaluating the correspondence of clusters of paired point locations. This section also proposes a new algorithm for the local measurement of cluster correspondencewhen the distributions of paired locations are not random, while its implementationon a large scale dataset of vehicle theft and recovery location pairs is presented in section 4.The last section of the paper summarizes findings and discusses future research directions. 2. FROM SPATIAL AUTOCORRELATION TO CLUSTER CORRESPONDENCE

2.1. Formalizing the Multi-location Events Problem Let us consider a spatial process s that generates N events in space R . Each of the N events, n, comprises a set of M ordered locations that defines its spatial configuration, and can be referred to as a multi-location event, as defined previously. The ordered location set that defines the spatial configuration of multi-location event n in N is denoted by P,: pn =

{(xnj, y n j )

Ij

E

11, 2, ...> M I )

(1)

wherej denotes thejth location among the M ordered locations defining event n and ( x ~ ynj) , is the coordinate pair of thejth location for event n. The location set comprising the mth location of all N multi-location events can be denoted by Qm: Qm

= { ( x i m , yim)

I i E 11, 2, ...) NII

(2)

where m denotes the mth location among the ordered locations defining the spatial configuration of a multi-location event, i denotes multi-location event i from N , and (xi?,,rim) is the coordinate pair of the mth location for multi-location event i in N . P,, describes the spatial layout of event n in R while Qmdescribes the spatial distribution of all the mth locations in R. Put another way, P,, denotes a set of point locations defining single multi-location events, while Q,, denotes a set of point locations performing a certain function for the development of all the multi-location events. Location sets P, and Qm capture the variations of geographical events produced through some geographical process. Several special cases are quite familiar to spatial

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analysts. When m = 1in formula (I), the set of (P,) captures the location distribution of single location events, the analysis of which has been supported by the well-developed spatial statistical techniques for point pattern analysis (Diggle 1983; Upton and Fingleton 1985; Bailey and Gatrell 1995; Fotheringharn, Brunsdon, and Charlton 2000). When n = 1in (2), (Q,,) collapses to describe the spatial interaction among locations defining a single multi-location event and can lend support from the matured family of spatial interaction models and analysis (e.g., Fotheringham 1981; Fotheringham and O’Kelly 1989). In a more complex situation, when m = 1in (1)and when more than one geographical process generates more than one set of geographical events, we have multiple sets of single point locations, (P,), (P,,), {P,b,t],. . . It then becomes a typical multivariate spatial statistics problem, and the analysis boils down to the interaction among the different types of single point location events. Therefore, as portrayed in Table 1,there is a close relationship between traditional simple point pattern spatial statistics and the newly introduced multi-location pattern analysis. They belong to the same family of point pattern analysis notwithstanding the fact that new techniques needs to be developed to effectively handle the complexity of the multi-location event problem.

2.2. The Paired-Locution Events Problem: De3nition and Properties When m = 2 in ( l ) , multi-location events become paired-location events. The paired-location events problem is a special case of the multi-location events problem. Paired-location event pattern analysis has a broad domain of applications. Journey-towork, inter-state migration, global trade flows, and flows of capital pursuing maximum profit are common cases of paired-location events. In the remainder of this study, we focus our discussion on the spatial analysis of paired-location events and illustrate our proposed approach for multi-location events spatial pattern analysis by applying it to paired-location events analysis. The proposed methods for paired-location events analysis can be extended to higher orders of multi-location events rather easily. Assume that a spatial process s generates N paired-location events. Two point sets Qm and Qmrare created during this process. Every element from Qm is called afrompoint and is paired with one, and only one, element from Qln,,which is called a topoint. Both the from-point and the to-point are also termed endpoints as they define the ends of the vector linlung the paired point locations (see Figure Ib). The spatial relationship existing between the paired endpoints of all N pairs is termed spatial correspondence. The analysis of spatial correspondence between paired point locations enables new research questions to be addressed. Do point locations of one type form clusters when their corresponding points of another type cluster together? In other words, do the points indicated by dots in Figure l b cluster if their corresponding points indicated by crosses cluster? If they indeed do, is the degree of clustering of points strengthened or weakened through pairing? Are there spatial variations in the clustering of paired locations across the study regon? The stable and non-random correspondence of points from distinct sets is a key property that differentiates spatial correspondence analysis of point patterns from the traditional spatial pattern analysis of multiple point sets (Table 1).Although multivariate point pattern analysis (Diggle 1983)is used to describe patterns of several distinguishable point sets, it is fundamentally an extension of single point pattern techniques to multiple point distributions. Traditional spatial association analysis assesses the relationship between two point sets through evaluating the degree of mixture of points from different point sets (Sorensen 1974; Bailey and Gatrell 1995). A few other studies follow radically different principles. For example, Lenz (1979) advocates the use of information theoretic measures to track temporal change in spatial point patterns. In all these approaches, however, no consideration is given to the possible pairing of points belonging to the two sets whose spatial relationships are being

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A

B

FIG.1. Illustration of the Difference between Traditional Spatial Association Analysis of Point Sets (as shown in A) and Spatial Correspondence Analysis of Paired Points (as shown in B)

explored (see Figure la). The analysis of spatial correspondence of point pairs aims at assessing the spatial relationships between points forming a pair, and between and among vectors linhng anchor points of each pair (see Figure Ib). There is an explicit one-to-one relationship between points from different point sets in spatial correspondence analysis. There are two primary perspectives on the analysis of spatial correspondence of point pairs-the spatial relationship between point locations within a pair (intru-pair analysis) and the spatial relationship between and among vectors linking the paired locations (inter-pair analysis) (see Table 1).Both perspectives account for the one-toone relationship between points from different point sets, but they observe the spatial patterns of paired-location events from different perspectives. Their scope and limitation are discussed in detail in the following section. In this context, cluster correspondence analysis is proposed as an effective approach to the inter-pair analysis of paired-location events. 2.3. Analyzing Spatial Correspondencefor Paired-Location Events

Intra-pair analysis is the first perspective of spatial correspondence of paired-location events (see Table 1).Its focus is on the spatial relationships between point locations within a pair. It entails two complementary aspects; the distance between paired points and the direction from one element of the pair to the other (see Figure Ib). Some studies have described the distances between paired locations statistically without searching for theoretical explanation (e.g., Barton, David, and Fix 1963; Lu 2001). More importantly, spatial interaction models (Fotheringham and O’Kelly 1989) have a long tradition of explaining the distance between location pairs that are functionally related through flows in terms of the push and pull effects between the corresponding origins and destinations. Various approaches have been proposed by Fotheringham and Webber (1980), Griffith and Jones (1980), Fotheringham (1981), and others to incorporate the effect of the spatial arrangement among interacting locations on flow patterns. See also Roy and Thill(2004) on this matter. A few studies have focused on the directional relationship between paired points. The term orientation, rather than direction, is sometimes used to describe the directional relationships from dispersed origins to relatively concentrated destinations. For example, Rengert and Wasilchick (1985) found that residential burglars’journeyto-crime tends to be oriented towards their work places or recreational places. Other studies in the field analyze direction relationship between paired points by measuring the absolute directional deviations from north (e.g., Lu 2001).

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Originally proposed to compare geographical phenomena using a bi-dimensional regression technique, Tobler’s (1994) method can be applied to investigate the intrapair spatial relationship of paired locations through a function that models the coordinates of one end point based on that of the other. Tobler’s method aims at identifying a function f that satisfies

w =f(2) minimizing-

c

l N (W - W)’

(3)

k=l

where W and Z represent corresponding sets of points defining the spatial configurations of two geographical phenomena under comparison. When W and Z are used to represent Q,, and Q,,L. as defined in (2),Tobler’s method can explain both distance and directional relationships in paired points using mathematical models. The second perspective on spatial correspondence differs from the first one by its emphasis on inter-pair similarity relationships (see Table 1).Let us conceptualize pairs of linked point locations as vectors defined by both distance and direction measurements. This perspective is then about the spatial patterns between vectors. In the vector field constituted by a series of vectors, a vector cluster refers to the spatial pattern of a certain group of vectors located close to one another and showing similar direction and distance measures. Vector clustering is the spatial pattern of a number of positively autocorrelated vectors. On the contrary, negative vector autocorrelation exists when vectors start from close locations but go to totally different directions and distances or when vectors end at close locations but come from totally dlfferent directions and/or distances. Vector spatial autocorrelation statistics reveal the spatial pattern of vectors and therefore can be expected to provide methodological support for examining the inter-pair spatial correspondence of paired locations. The scarcity of research on spatial correspondence of paired locations sterns from the lack of formalization of the multi-location events problem in the field of geography, coupled with the immaturity of vector spatial autocorrelation analysis techniques. Most commonly used spatial statistics treat the spatial patterns of vectors by separating its direction property from its distance property. Yet, directional spatial autocorrelation techniques (e.g., Costanzo, Halperin, and Gale 1986; Oden and Sokal 1986; Rosenberg 2000) by themselves cannot provide full support for the spatial correspondence analysis of vectors because these techniques do not control vector length and treat vector direction as an attribute of each vector’s starting point. Similarly, distance spatial autocorrelation analysis only tests whether vectors starting from nearby locations are of similar length without considering whether vectors sharing similar length point in the same direction. Vector spatial autocorrelation in fact is a “combination” of vector direction autocorrelation and distance autocorrelation. However, the joint treatment of distance and direction when assessing vector patterns and the correspondence of clusters of linked point locations is a field that has received very limited attention to date. Berglund and Karlstrom (1999) conducted one of the few studies of vector patterns (flow data) that consider both distance and direction components. Their analysis of spatial association of flow data is based on the aggregate measurement of “outflows”and “inflows”between zones of a predefined partition of the study area. It applies the Gilocal statistic introduced by Getis and Ord (1992) and Ord and Getis (1995) and generalizes it to flow data. In the generalized Gistatistic, each spatial weight is an index of similarity between vectors that captures either the topological relationships of flows or their distance and direction properties. Proposed similarity

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measures are designed to depict a few vector patterns only, such as identical flow origins or destinations,but they fail to register the differences between other dissimilar patterns. Moreover, they are sensitive to the size and spatial configuration of zones in the study area. In sum, the approach advanced by Berglund and Karlstrom (1999) is not a complete evaluation of the spatial autocorrelation of vectors based on the measurement of distance and direction attributes. For example, the flows in Figure 2 may show high spatial association according to Berglund and Karlstrom (1999), since the flows are all directed from the same zone (zone 1)into adjacent zones (zone 2 and zone 3 ) .It is clear however that the directions and distances of the flows are very different, and that these flows are definitely not associated with spatially autocorrelated vectors. When a distribution of vectors exhibits positive spatial autocorrelation, vectors whose origins (from-points) are in close proximity have destinations (to-points) close to one another as well. In this situation, to-points associatedwith any cluster of frompoints also form a cluster, and vice versa. Spatial dependence among points is preserved along vectors. We define the pattern of spatial dependence within clusters of from-points and to-points belonging to predetermined multi-location events as cluster correspondence. Therefore, the properties of “vector autocorrelation” and “cluster correspondence” exhibited by any spatial pattern of paired point locations are perfectly equivalent, even though they describe this pattern by means of different constructs-vectors and paired points, respectively (see Table 2). The former emphasizes the vectors linking paired points, while the latter evaluates the spatial match between sets of vector endpoints. Both properties examine spatial patterns from the perspective of inter-pair spatial correspondence of paired point locations (see Table 1).The cluster correspondence approach holds over its dual-vector autocorrelation- the advantage of “simplifjmg”the spatial autocorrelation analysis of onedimensional objects to the combined treatment of multiple sets of zero-dimensional objects by means of well-developed point pattern spatial statistics. Cluster correspondence brings to the study of inter-pair spatial relations what Tobler’s (1994) contribution set to accomplish with bidirectional regression on intrapair relations. By linking vector spatial autocorrelation with cluster correspondence between endpoints of vectors, the proposed approach transposes the task of investigating spatial autocorrelation of one-dimensional vectors into that of examining spatial patterns of paired zero-dimensional points. Cluster correspondence analysis investigates the spatial patterns of two sets of endpoints on the condition that each point is paired to a point from the other set. The distance and direction properties of vectors are observed implicitly through the conservation of paired point locations.

FIG.2. Illustration of Vectors That May Show High Degrees of Spatial Association but No Spatid Autocorrelation

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TABLE 2 Typology of Cluster Correspondence and Vector Autocorrelation Significance Levels of Clustering From-points

High

High

To-points

I 1

Medium LOW

Medium

High

I

Correspondmce of Clusters/ Vector Autocorrelation

High

High

High

Medium

Example Illustration

I41

Fair None

3

Fair

Medium

Low

Medium

Low

None

Low

High

None

Low

Medium

Low

Low

___, \

None None

Compared to the vector autocorrelation approach, the implementation of the cluster correspondence approach is more straightforward because traditional univariate point pattern analysis techniques can be extended with some adaptation to evaluate the closeness among points of each paired set. 3. APPROACHES TO CLUSTER CORRESPONDENCE ANALYSIS

It is well recognized that geographic processes and relationships between distinct processes are often not spatially stationary (Fotheringham 1992;Anselin 1995;Fotheringham, Brunsdon, and Charlton 2000). Local variations in relationships may point to “hotspots”of certain types of events and associations, and to the existence of dfferent causal processes at work in different locales. Along this line of thought, we propose a computational approach to second-order, or local, analysis that measures spatial association between multi-location events by means of local statistics.The algorithm serves to detect the local correspondencebetween clusters across sets of linked point events. The location of any cluster of multi-location events embedded in the dataset is pinpointed. Although the discussion is carried out on the case of paired-location events, it can readily be generalized to events encompassing more than two locations.

3.1. Typology of Cluster Correspondence and Its Relationship to Vector Autocorrelation As discussed above, the spatial correspondence of paired points can be tested by either examining one-dimensional objects (vectors)linking point pairs or transposing the assessment of vectors into the analysis of two sets of zero-dimensional objects

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(vector endpoints). In particular, this holds for local indicators of spatial correspondence, which can detect pockets of local non-stationarity of the process. Spatial process results in N ordered pairs of from-points and to-points. Let F denote the point set consisting of all from-points and T denote the point set consisting of all to-points. Let us consider a singular point location i in the study region. The goal of an analysis of cluster correspondence at location i is to test for the presence of a cluster of ordered location pairs whose from-points belong to the neighborhood around i and whose to-points cluster close to each other as well. Let R, denote the relevant neighborhood around i and ni be the number of from-points within Ri.The set of all the from-points within R, is denoted as Fjwhile the set of their paired to-points is Ti. Cluster correspondence analysis evaluates and compares the clustering of points in Fi and Tirespectively, and concludes on the spatial dependence of the process that associates them. Depending on the significance level of clustering of the from-points in Fiand of the to-points in Ti, the two point sets may show difference in closeness among their elements. For example, vectors having their from-point significantly clustered together may have to-points clustered more, less, or not at all. Cluster correspondence analysis needs to examine how different the corresponding subsets of points are when the closeness among respective point elements is measured. Possible cases are summarized in the typology of cluster correspondence and vector autocorrelation defined in Table 2. Nine cases are identified and illustrated by their archetype. When the measures of clustering of two corresponding point sets exhibit a high level of statistical significance, there is close correspondence between the clusters of from-points and that of to-points (namedfrom-cluster and to-cluster, respectively)and vectors are said to be highly correlated. As the statistical confidence for from-cluster and/or tocluster declines, so does the confidence in the presence of cluster correspondence and of a cluster of vectors around location i. In the proposed framework, it is the combination of the two univariate point patterns that determines the overall typology of cluster correspondence and vector autocorrelation.Typologies with a larger number of classes can be developed on the same principles, all of which are surrogates of the continuous reality of local vector clustering. According to Table 2, high cluster correspondencebetween location pairs indicates that both groups of linked endpoints are significantlyclustered; the vectors show significant spatial autocorrelation as well. Fair cluster correspondence means that one group of endpoints is highly clustered while the other is fairly clustered; correspondingly, the vectors show pattern of fair spatial autocorrelation. Vectors show no spatial autocorrelation,and there is no cluster correspondence if one group of endpoints exhibits a low level of clustering. When both groups of endpoints are moderately clustered, the vectors are said to be clustered at a low level, and so is the level of cluster correspondence of the location pairs. Therefore, the assessment of cluster correspondence, and also vector autocorrelation, is truly dependent upon the assessment of the clustering situation among locations forming the location pairs of interest. To quantify cluster correspondence and vector autocorrelation, one must first quantify the spatial autocorrelation of from-points and that of to-points, respectively. And this is where univariate spatial analysis can support the analysis of paired points spatial patterns. By combining the spatial autocorrelation assessment for both groups of endpoints, and by referring to the typologies defined in Table 2, the evaluation of cluster correspondence and vector autocorrelation is enabled. 3.2. Algorithmfor Local Analysis of Cluster Correspondence

Let us consider location i and a neighborhood R, around it. The null hypothesis for paired points cluster correspondence analysis can be stated in two different but equivalent ways:

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Ho: There is no local non-stationarity among vectors linkmg the from-point set Fi H;:

and to-point set Ti. There is no correspondence of clustering between subsets of points Fiand

Ti.

Although they are stated from a vector perspective and from a paired points perspective respectively, the two null hypotheses refer to same spatial pattern and are equivalent. Rejection of either of them would suggest the presence of local non-stationarity in the pattern of vectors linking the paired locations. Put another way, rejection of the null hypothesis would imply the preservation of clusteiing between two sets of vectors endpoints through the pairing process-points in Ti cluster when their paired points in F icluster, and vice versa. Two different approaches can be adopted for defining neighborhood Riaround location i. A simple distance criterion can be applied to define a buffer of predefined radius that is centered on location i. Alternatively, with a count criterion, the neighborhood is progressively enlarged from location i until m from-points are enclosed. The choice of one or the other approach depends on the specific case under study. The local analysis of cluster correspondence for from-point set Fiand to-point set Tientails the following four general steps:

(1) Evaluate the closeness of the points in Fc, and estimate the significance level of this measure as SFi; (2) Define the corresponding to-point set, T,, by identifjmg the to-points paired through process q to the from-points in Fi; ( 3 ) Evaluate the closeness of the points in Ti, and estimate the significance level of this measure as STi; and (4)Compare SFi and STi, and refer to the typology definition in Table 2 to conclude on the presence of a local cluster of vectors with from-points in Ri. The first step is to evaluate spatial autocorrelation of from-pointswithin a neighborhood around location i and estimate its statistical significance. Traditional univariate point clustering techniques (e.g., Openshaw et al. 1987; Fotheringham and Zhan 1996; Murray and Estivill-Castro 1998)can be applied to assess the level of clustering among from-points. Two general types of approaches are commonly used to measure the degree of clustering of a point set-either by the count of points in a predefined set or by the average distance between points in the set. Statistical inference can be challenging in point pattern analysis because the distribution properties of local association statistics may be hard to obtain. A common altemative (e.g., Openshaw et al. 1987;Anselin 1995; Fotheringham and Zhan 1996)consists of using Monte Carlo simulation. Monte Carlo simulation approaches can produce pseudo significancelevels by repeated randomization. An empirical distribution function is derived from a large number of simulated patterns. This function then serves as a benchmark against which the observed statistic of closeness is compared to establish the significancelevel. The second step of the analysis can easily be conducted using the basic functions of a database management system. The third step differs from ordinary point cluster analysis in that the outer envelope of the set of to-points may be a polygon covering a large part of the entire study region. Furthermore, this polygon may enclose to-points whose paired from-points are outside R , although these to-points are not in Ti,the subset of to-points to be evaluated for local non-stationarity of paired points. These properties invalidate some common measures of clustering such as the count criterion mentioned above. And the final step considers the statistical significance of clusters in the from- and to-point sets, and infers from the typology of cluster correspondence (Table 2) on the presence of hotspots of vectors anchored in region Ri.

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From the description given above, it is apparent that the analysis of cluster correspondence between paired points does not boil down to performing two single-pointset cluster analyses. The functional relationship between from- and to-points matters and has to be captured by an appropriate methodology to avoid a bias in testing the null hypothesis of no local spatial association between multi-location events. In this regard, two issues need to be stressed. First, the evaluation of clustering among topoints is dependent upon prior pattern analysis of from-points. The definition of region Ridetermines the selection of from-points, Fi,which further determines the corresponding to-points, Ti, to be evaluated for clustering. Secondly, it can generally be anticipated that from- and to-points of the vectors are generated by different geographical processes. It may therefore be necessary for the measurements assessing the spatial autocorrelation in each point set to reflect the respective underlying geographical processes. Among other considerations, complete spatial randomness may be tested for a point pattern that might have its incidence at any location in the study area while a conditional spatial randomness should be considered when the point pattern is restricted to the distribution of an at-risk population. For example, when looking at journey-to-work, the home base location pattern is restricted by the population distribution in general while the pattern of their workplaces is related to the general distribution of employment opportunities. Therefore, the algorithm proposed above provides a description of general steps for cluster correspondence analysis; the technical details need to be specified during the implementation of the algorithm for specific distributions. The following section demonstrates an implementation of the proposed generic algorithm on cluster correspondence analysis of vehicle theft and corresponding recovery locations in Buffalo, New York. 4. CLUSTER CORRESPONDENCE IN VEHICLE THEFT AND RECOVERY LOCATION PAIRS

For cleared vehicle theft crime cases (i.e., vehicle theft cases with the stolen vehicles recovered), a recovery location is associated to each vehicle theft location. Location pairs consisting of a theft location and the corresponding recovery location can easily be established. This section of the paper examines cluster correspondence of paired vehicle theft and recovery locations in the city of Buffalo. The purpose of this application is two-fold. Firstly and most importantly,it demonstrates an implementation of the proposed approach for cluster correspondence analysis on a relatively large dataset. Secondly and on the practical side, findings of the analysis should provide new insights into patterns of urban vehicle theft that will serve law enforcement agencies when fighting vehicle theft crime. The null hypothesis under consideration is that no local spatial autocorrelation exists among criminals’journey from the vehicle theft location to the location of recovery. Rejection of this hypothesis purports that recovery sites of vehicles stolen in the same geographic area are more clustered than one would expect to happen by chance. 4.1. The Spatial Dimension of Vehicle Theft in Buffalo

Spatial statistics and geographic information systems (GIS) techniques have revealed the existence of significant clustering in the distribution of vehicle theft locations and vehicle recovery locations in the city of Buffalo (e.g., Lu 2000; 2001) and other areas as well (e.g., Clarke 1983; Eck and Spelman 1988; Roncek and Maier 1991; Copes 1999). Moreover, it is widely believed that most stolen vehicles that are not recovered end up in chop shops (LaVigne,Fleury, and Szakas 2000; NICB 2000). From a crime prevention and suppression perspective, cluster correspondence analysis of vehicle theft and recovery sites can be used to uncover the possible locations of one anchor point of journey-with-stolen-vehiclebased on information about another anchor point, i.e., possible recovery locations can be recommended based on infor-

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mation of vehicle theft location and vice versa. Furthermore, a consistent pattern of cluster correspondence between location pairs may suggest the existence of gang activities or chop shops. According to the crime reporting system maintained by the Buffalo Police Department, there were 3,271 vehicle theft offenses within the city limits in 1998.The same data source recorded 2,284 vehicle recoveries in the same year. Through a series of GIS and database operations, 2,065 vehicle theft and recovery location pairs can be identified.l Location pairs with theft and recovery locations sharing the same address are subsequently eliminated from further analysis based on the following considerations. According to the police, vehicle theft cases with identical theft and recovery locations are more likely to be falsely reported cases. In addition, the cluster correspondence analysis is really about spatial patterns of location pairs whose anchor points are spatially separated. Finally, a total of 1,600 vehicle theft and recovery location pairs are selected and geo-coded onto an enhanced TIGElULine street map of Buffalo* for the analysis of cluster correspondence. The distribution of these 1,600 vehicle theft and recovery locations is displayed in Figure 3. Vectors can be built to link each theft location with its corresponding recovery location. Figure 4 illustrates the northeast part of the study are when such vectors are actually drawn on map. The average distance between each vehicle theft location and the nearest theft location is 294 feet (with a standard deviation of 289 feet). For vehicle recovery locations, the corresponding statistics are 450 feet and 1,803 feet. 4.2. Algorithm Implementation

This section demonstrates the implementation of the general algorithm proposed in section 3.2 for cluster correspondence analysis of vehicle theft and recovery location pairs in Buffalo, New York. The algorithm tests for local non-stationarity of paired locations when the theft location is within a defined neighborhood around some reference location i. In the present study, a reference location is taken to be a reported vehicle theft location. The algorithm is applied to each of the 1,600 vehicle theft locations in the dataset. A similar analysis can be done for reference locations that are either randomly or evenly distributed on a grid across the entire study region. The neighborhood Riaround each location i is defined as a 1000 feet circular buffer centered on i.3This buffer size corresponds to that of the average block group within the City of Buffalo. Block groups are displayed in Figure 3. Closeness among theft locations in the neighborhood of i is measured by the count of offense cases in Ri.Let n, denote this statistic. For N = 1600, the average of n, is 9 and the median is 7 (with a standard deviation of 6). Vehicle theft should not be expected to happen with equal likelihood throughout the study area due to the n0.n-random distribution of offense opportunities (e.g., Clarke 1983; Eck and Spelman 1988; Roncek and Maier 1991; Copes 1999). However, given the limited knowledge of offense opportunities in the study area and considering that the spatial analysis of offense opportunities is not a major concern of this research, the theoretical spatial distribution is taken to be completely spatial random. The expected number of of1. Records from two types of police crime re orts are linked together usin a ke field, “Complaint related stolen vehicle Desk Number,” which is uniquely assigned to eaci vehicle theft offense and to recoveT record. 2. T e TIGElULine eocoding is a street quality match, which means that street addresses were used during the process of ad%ressmatching. Manual interactive address matching was erformed and a field check was conducted when necessary and applicable. Details of the process is avaiible from the authors upon request. 3. It can be conjectured that results of‘the anal sis would be sensitive to buffer size since this size is related to the eographic scale at which criminal beiavior operates. Given that this application is mainly intended to ilkstrate an implementation of cluster correspondence analysis, we do not address the scale issue in the paper. Future work will address this matter.

8e

x

!

FIG.3. Distribution of Vehicle Theft Locations and Recovery Locations in Buffalo, New York

N

t

A/

~emdtLocattono Sample Recovay LOC8tlms V e d m Unklng LocalonPalm I aty B0U-w

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0

0.5

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1.5 Mil-

FIG.4.Illustration of Vectors Linking Selected Vehicle Theft and Recovery Location Pairs

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fense cases per unit area is given by Eni = N / A , where A is the area of the city of Buffalo. The probability for observing exactly y cases in Rj of area a is approximately Poisson distributed (Getis and Boots 1978; Besag and Newell 1991):

Following Fotheringham and Zhan (1996),values rendered by this distribution function are used to measure the statistical significance of a cluster among the offenses in 4. Closeness among recovery locations of the n, stolen vehicles is measured by the average distance among the n, recovery locations. Let didenote this statistic. The theoretical distribution function is not readily available for this statistic. Indeed the itinerary followed by criminals with stolen cars is known to be non-random. Decisions on where to go with stolen vehicles are restricted by the spatial distribution of opportunities where payoff for the offense can materialize. However, due to our sketchy knowledge of the processes generating the opportunities for stealing vehicles, for concealing and disposing of stolen vehicles, and for the chances of recovering a stolen vehicle, the task of establishing the theoretical distribution of recovery locations is insurmountable. To describe the real distribution of all possible recovery locations, we use the observed distribution of the 1,600known recovery locations as a fair approximation. Monte Carlo simulation is then adopted to estimate the theoretical distribution function of average distance among n, recovery locations under conditional spatial randomness. A total of 999 sample datasets of recovery locations are simulated by independently and randomly sampling n, locations from all 1,600 known recovery locations. The average distance among the n, observed recovery locations is compared to the distribution of averages for simulated locations to obtain a significance level. Statistical significance of from- and to-point clusters is said to be high if a 5 0.01, medium if 0.01 < a 5 0.05, and low if 0.05 < a.4 By assessing the significance of clusters for both vehicle theft locations and their corresponding recovery locations, the algorithm facilitates the comparison and association of cluster patterns from the paired location groups. Only those vectors having a high degree of clustering for both endpoints show signs of spatial autocorrelation and cluster correspondence. In other words, the vectors linkmg vehicle theft locations to recovery locations are spatially autocorrelated, and their location pairs show correspondence of clusters only when the related journeys start from locations clustered together and end at locations cluster together as well. 4.3. Results

Results of statistical tests of local clusters among vehicle theft locations using the procedure described above are reported in Figure 5. Results for local clustering among stolen vehicle recovery locations are presented in Figure 6. From Figure 5, it is clear that, at significance levels of 0.01 and 0.05, many offense locations are recognized as significant clusters. In Figure 6, the significance level of clustering of a subset of recovery locations is assigned to the reference offense location i, which is the center of neighborhood Ri. Areas showing a high level of statisticallysignificant clustering in Figure 6 are such that vehicles stolen in Ri around reference offense location i tend to be recovered at locations clumped together. However, this does not neces4. The details of implementing the algorithm roposed in Section 3.2 on vehicle theft and recove location air analysis is presented in the Appendix. {he related data and intermediate products are avaixble frome!It authors upon request.

FIG.5. Clustering among Vehicle Theft Locations

..

Block Oroup aing umng Rrovsly Locations signmcancs ~ e * a lr=90 signmcmceLeva 95 98.9 signincame L& c 95

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0

0.5

1

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FIG.6. Clustering among Paired Vehicle Recovery Locations

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sarily mean that a large number of vehicles are involved in this clustering. Furthermore, highly significant clustering among some recovery locations does not automatically imply a clustering of paired offense locations. A comparison of Figures 5 and 6 easily reveals areas having few vehicle thefts in Ri(i.e., low or no clustering of vehicle thefts in the neighborhood of reference offense location i ) while showing highly significant clustering of recovery locations (e.g., some areas in the southern and eastern portions of the city). This suggests that while stolen vehicles may get recovered at locations clustered together, they may be stolen from locations far apart from one another. In other words, without linking the clustering situation of certain recovery locations with that of the corresponding theft locations, one cannot get a complete image of cluster correspondence between these location pairs. Applying the typology of cluster correspondence defined in Table 2 with 1 percent and 5 percent significance levels enables us to identify non-stationarities in the spatial pairing of offense and recovery locations. The spatial distribution of hot spots of offense-recovery location pairs is reported on Figure 7. Statistics on the frequencies of each type of cluster correspondence situation are given in Table 3. According to the analysis, 90 subsets of theft-recovery location pairs are significant clusters at 1 percent. The criminals’ trips starting at vehicle theft locations in the neighborhoods centered on the 90 offense locations are considered as having “high cluster correspondence”on Figure 7 and show strong positive vector spatial autocorrelation. Furthermore, 106 neighborhoods are classified as exhibiting “fair cluster correspondence,’’which means that location pairs anchored in these areas have one endpoint highly clustered (1percent significance)and the other fairly clustered (5 percent significance).All together, offense-recovery cases that exhibit either high or fair spatial autocorrelation account for 12.3 percent of all cases in the Buffalo study area.

FIG.7. Clustering Correspondence of Vehicle Theft and Recovery Location Pairs Using 1%and 5% Significance Levels

306 / Geographical Analysis While all location pairs showing high or fair cluster correspondence have offense locations in neighborhoods with vehicle theft offenses clustered together, a significant number of offenses within the high-crime areas show low cluster correspondence (i.e., type 3 in Table 3). As many as 522 neighborhoods have highly clustered offense locations,while there is no significant evidence of clustering among the associated recovery locations. From a crime analysis perspective, location pairs of this type describe crime situations where a significant large number of vehicles are stolen in a certain neighborhood and are driven to scattered areas before being recovered. Statistical results reported in Figure 7 and Table 3 assume that the analyst is rather averse to falsely detecting clusters of paired locations. Because many potentially promising leads may be discarded by resorting to stringent statistical significance levels, field safety officers may favor using less stringent significance levels. This would be accomplished at the price of a greater number of false leads. The proposed approach to cluster correspondence allows for more or less stringent significance levels (e.g., 5 percent and 10 percent significance levels). Under these conditions, the number of clusters deemed to be highly significant swells from 90 to 219. This increase in hot spots is concentrated in and around the CBD, where many of the clusters were previously found, as well as in the northeastern section of city. The clear pattern in this respect should prompt the analyst to conduct some confirmatory analysis of criminal behavior in these areas.

TABLE 3 Cluster Correspondence between Vehicle Theft and Recovery Location Pairs in the City of Buffalo Using 1% and 5% Significance Levels TYPe

Significance Level (from-paintsto-points)

Number of Observations

Cluster Correspondence I Vector Autocorrelation

High-high High-medium High-low Medium-high Medium-medium Medium-low Low-high Low-medium Low-low

90 76 522 30 23 139 77 81 562

High Fair None Fair LOW None None None None

TABLE 4 Cluster Correspondence between Vehicle Theft and Recovery Location Pairs in the Clty of Buffalo Using 5% and 10% Significance Levels TYP

1 2 3

4 5 6 7 8 9

Significance Level (from-po,nts-to-points)

High-high High-medium High-low Medium-high Medium-medium Medium-low Low-high Low-medium Low-low

Number of Observatlons

Cluster correspondence 1 Vector Autccorrelahon

219 81 580 78 14 204 80 32 312

High Fair None Fair LOW

None None None None

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5 . CONCLUSION

It is not uncommon that complex geographic events associate multiple point locations together. A majority of existing point pattern analysis methods investigate spatial patterns of single point sets only. This paper discusses some fundamental properties of spatial autocorrelationof multi-location events-also called vectors-and proposes a procedure for the detection of hot spots of multi-location events. When a distribution of vectors exhibits positive spatial autocorrelation,vectors whose from-points are in close proximity also have to-points close to one another. Reciprocally, distant topoints are associated with distant from-points. This spiitial pattern is said to exhibit cluster correspondence. The properties of “vector autocorrelation”and “cluster correspondence” of any spatial pattern of paired point locations are perfectly equivalent, although they describe this pattern by means of different constructs-vectors and clusters, respectively. Two aspects set cluster correspondence analysis of point pairs apart from a simple combination of two independent cluster analyses of single point set. Firstly, the two subsets of points must be made up of points corresponding to each other. Secondly, the detection of correspondence between local clusters of points rests on the joint considerationof levels of clustering in both subsets. Because of the complexity of spatial features under study, several archetypical clustering situations can emerge, each one pointing to dfferent domain-dependent processes that mark the transition from exploratory to confirmatory analysis. The algorithm proposed for the local test of cluster correspondence shows great potential for application to a large number of complex geographic events, including crime analysis, spatial epidemiology, personal and freight movements, and many others. It accommodates domain-specificcircumstances that may require empirical estimation of probability density functions. The application of this algorithm to vehicle theft and recovery in Buffalo illustrates the use of this tool for spatial exploratory analysis. The empirical analysis also evidences the intrinsic limitations of a point pattern analysis that would not explicitly recognize the pairing of elemental point events forming complex spatial objects. Future research on the spatial analysis of multi-location events should be pursued in several directions. First, a significant aspect of cluster correspondenceis the definition of the relevant neighborhood around each singular location. As with other spatial statistics, conclusions on the rejection of the hypothesis of no spatial dependence may critically rest of the operational size of this neighborhood. This is not a limitation of the proposed approach, however. On the contrary, since the size of this neighborhood can be varied, the opportunity exists for gaining direct insight on the geographic scale at which geographic processes linking locations together operate. Research on the stability of cluster correspondence across geographic scales is a high-priority matter. Furthermore, it is not well established at this time what local univariate statistics are best suited to handle multi-location data from diverse application domains. The nature of processes generating multi-location events being quite diverse, one can expect that domain-specific circumstances will dictate that different measurement and computation methods be used to capture the association between point events. In addition, while the proposed general approach to vector spatial autocorrelation is broad enough to analyze events composed of more than two points, more complex algorithms may be necessary to handle the added dimensionalityof such problems. Finally, our work has so far primarily focused on local properties of spatial patterns of vectors. Global properties of such patterns remain to be investigated. Like Anselin (1995)established a formal link between so-called local indicators of spatial association and global statistics of spatial autocorrelation, such a bridge between local and global analysis would foster spatial analysis of complex spatial events.

308 / Geographical Analysis APPENDIX THE ALGORITHM FOR ASSESSING CLUSTER CORRESPONDENCE BETWEEN VEHICLE THEFT AND RECOVERY LOCATION PAIRS

The detailed implementation of the algorithm proposed in section 3.2 for vehicle theft and recovery location pair analysis can be described as follows: (1) Evaluate the closeness'among the vehicle theft locations in neighborhood R, centered on location i : (a) Identify all vehicle theft locations within a 1,000-footradius of location i; (b) Compute the number ni of theft locations within the circular buffer; ( c ) Compute the Poisson probability distribution functionf4 given by equation

(4); (d) Assign a significance level (high, medium, or low) to offense location i. (2) Identify the corresponding recovery locations: (a) Identify actual vehicle recovery locations of all ni offense locations. ( 3 ) Evaluate the closeness of the corresDondine recovery locations: Calculate the average distance amocg the n, dbserved recovery locations; Randomly and independently select n,locations from the whole set of N observed recovery locations; Calculate the average distance among n, recovery locations simulated under (b); Repeat steps (b) and (c) 999 times and generate the empirical probability distribution function; Compare the average distance diamong the ni observed recovery locations to the empirical probability distribution function generated under (d); Assign a significance level (high, medium, or low) to offense location i for clustering among recovery locations. (4) Defining the type of cluster correspondence in the neighborhood of i: (a) Combine the findings from step (Id) and step (30 to describe the situation of cluster correspondence in the neighborhood of i by referring to the typology defined in Table 2.

ai

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