Spatial Pattern, Analysis ofLauren M Scott, Esri, Redlands, CA, USA

� 2015 Elsevier Ltd. All rights reserved.This article is a revision of the previous edition article by B. Boots, volume 22, pp. 14818–14822, � 2001, Elsevier Ltd.

Abstract

Spatial pattern analysis methods provide insights about where things occur, how the distribution of incidents or thearrangement of data aligns with other features in the landscape, and what the patterns may reveal about potentialconnections and correlations. Focusing primarily on vector data (points, lines, and areas), this article introduces a samplingof descriptive and inferential exploratory spatial pattern analysis methods relevant to a variety of real-world problems andquestions. Web links to additional resources for learning more about spatial pattern analysis are included in tables at the endof the article.

Spatial pattern analysis involves identifying, describing, andmeasuring the shape, arrangement, location, configuration,trend, or relationships in geographic data. Data are spatial orgeographic data when they are collected with, or assigned,locations such as an X and Y coordinates (longitude and lati-tude) or an address. When data are associated with locations,they can be mapped, and mapping geographic data is animportant first step in analyzing spatial patterns.

The next steps involve identifying, describing, and measuringthe spatial characteristics of data. Is there a discernible pattern,order, or structure associated with a disease outbreak, a series ofpublic protests, the results of state-wide test scores, or emergencycall events? Researchers and analysts look for spatial patterns indata because often understanding where things occur providesimportant clues about why things occur. Understanding why(i.e., cause–effect relationships) has been described as the ‘holygrail’ of science (Cressie and Wikle, 2011), because it offers thepotential for changing outcomes: remediation, intervention,alteration. When you understand the patterns associated witha series of crime events, for example, you have an importanttool for predicting future crimes. Examining the patternsassociated with consumer behavior can help focus marketingefforts to increase sales. Analyzing spatial variations inchildhood obesity can help inform policies and programsaimed at reducing this growing epidemic.

Sometimes looking at a map is all that is required to identifyand explain spatial patterns. It is important to recognize,however, that the cartographic decisions made when a map iscreated can have a big impact on what a map communicates(Monmonier, 1996). The maps below (Figure 1), for example,show standardized health care costs by hospital regions in thecontiguous United States. All three maps are based on the sameunderlying data; the only difference is the cartographic display(class breaks based on equal intervals, quantiles, or standarddeviations).

Consequently, when visual analysis is not sufficient toidentify or to explain spatial patterns or when you need toquantify or to make inferences about the patterns in spatialdata, you will want to adopt a more formal analyticalapproach. Focusing primarily on vector data (points, lines,and areas), this article presents a small sampling of the manyspatial pattern analysis methods that can be used to move

beyond visual analytics in order to describe, quantify, andeffectively map spatial patterns. The first section covers simpledescriptive pattern analysis methods. The second and thirdsections describe inferential methods for quantifying andmapping spatial patterns. The article concludes with somecomments on future directions and links to web pages withmore details, the math behind the methods, and the resourcesfor learning more.

Describing Spatial Patterns

Descriptive spatial pattern analysis methods summarize thekey characteristics and properties for a set of geographicfeatures and their associated attribute values. Table 1 providesexamples of some geographic features, how they might berepresented on a map, and potential attributes.

If asked to analyze a set of numbers such as householdincomes, one might begin by computing the mean and thestandard deviation. Similarly, analyzing spatial data often beginswith computing spatial versions of the same statistics: the meancenter and the standard deviational ellipse. Calculating themeancenter helps understanding the central tendency of the features,and calculating the standard deviational ellipse highlights thespatial distribution and orientation of features around the meancenter. Using these descriptive pattern analysis methods tosummarize important spatial properties in data is an effectivestrategy for comparing groups of features across space (e.g., race/ethnicity integration or segregation) or highlighting changes inspatial patterns over time (e.g., urbanization trends).

Central Tendency

Points and AreasMeasures of central tendency are simple descriptive statisticsthat can help answer questions such as:

l Which resource is most accessible?l Has the center of innovation changed over time?l What is the primary direction of migration?

The most common measure of central tendency is the meancenter statistic. It is computed as the average X coordinate and

178 International Encyclopedia of the Social & Behavioral Sciences, 2nd edition, Volume 23 http://dx.doi.org/10.1016/B978-0-08-097086-8.72064-2

the average Y coordinate for all features in a data set. Althoughit is a simple statistic, it has powerful applications. Epidemi-ologists, for example, might compute the mean center for a setof health events in order to gather clues about the potentialsources of a disease or to better understand mechanisms ofdisease transmission (Koch, 2005).

When the mean center is calculated using an attribute valueas a weight, the effect is to pull the mean center toward thefeatures with the largest attributes. A retail analyst looking forpotential warehouse locations to serve a chain of grocery stores,for example, might compute a weighted mean center using thesales volume attribute for each store. By including this attributein the mean center calculation, the X and Y coordinates forstores with the largest sales volumes will have the strongestimpact on the resulting location of the weighted mean center(Mitchell, 2005).

In Figure 2, the weighted mean center for the population ofCalifornia has been computed each decade between 1900 and2000. In the first part of the century, the mean center is near SanFrancisco, perhaps reflecting the maritime trade and earlierpopulation booms associated with the 1848 discovery of goldin the region. With each decade, however, the weighted meancenter moves south owing to stronger population growth insouthern California. This southern-moving trend is likelya reflection of the economic opportunities associated with theburgeoning movie and oil industries, as well as the increasingavailability of air-conditioning (Scott and Janikas, 2010).

LinesFor line features, central tendency is computed using a circularstatistic to summarize either the mean direction or the meanorientation. Given a set of lines representing animal move-ments, for example, calculating the mean direction willdistinguish between the spring and fall migration pathways.

Alternatively, calculating the mean orientation for data span-ning all four seasons would be appropriate if the main focus isidentifying the key corridors that migrating animals use. Whilea mean direction is often expressed as a line with one arrow-head, the mean orientation is general represented by a line withtwo arrowheads (Figure 3).

Other measures summarizing line features include themean line length, mean center point, and circular variance. Thecircular variance is analogous to a standard deviation. It rangesfrom 0 to 1 and indicates how similar or how different theangles for a set of lines are from each other. Consider ananalysis of lines at locations throughout southern Australiarepresenting primary wind directions in January. If everywherein the study area the wind blows consistently to the northeast,the circular variance will be near 0. If, on the other hand, thewind direction changes drastically from one location to thenext so that it blows in many different directions depending onlocation within the study area, the circular variance will benear 1.

Applications for mean orientation include analyzing roads,urban spatial structure, or fault lines (Mardia and Jupp, 2000;Gaile and Burt, 1980). Mean direction applications includeanalyzing the consistency of hurricane pathways, migrationroutes for animals or people, or regional wind patterns. Meandirection has been used to analyze geographic variationsin journey to work (Corcoran et al., 2009), to model journeyto crime (Van Patten and Delhauer, 2007), and, morebroadly, to describe patterns of transport, spatial interaction,flows, or diffusion (Gaile and Burt, 1980).

Spatial Distribution

While the mean center and linear mean pattern analysismethods provide information about central tendency,

Figure 1 Geographic variations in per capita health care costs for Medicare recipients by hospital referral region, 2010. Data source: http://www.iom.edu/Activities/HealthServices/GeographicVariation/Data-Resources.aspx.

Table 1 Examples of spatial features, geometric representation, and feature attributes

Features Typical geometry Example attributes

Cities Points representing the center of a city or areasrepresenting city boundaries

Unemployment rate, jobs, Hispanic population

Traffic accidents Points Severity, fatalities, date, timeRoads Lines Speed limit, traffic volume, road conditionParks Areas or points Acreage, entry fee, estimated annual visitorsCounties Areas Population, average home values, median incomes

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descriptive spatial statistics like the standard deviational ellipsesummarize the distribution and orientation of features aroundthe mean center. These pattern analysis tools help answerquestions like:

l How quickly and widely is a disease spreading through thevillage?

l Where are the core areas for particular language dialects?How integrated are they and which have the broadestspatial extent?

Figure 4 shows an analysis of gang territories based ongraffiti locations for cases where the gangs responsible could beidentified. Each color reflects a different gang affiliation.Creating a standard deviational ellipse for each set of thesegang-related incidents provides information about the coreextent of gang territories and how the territories relate to eachother and to various features in the landscape. Further, areaswhere the ellipses overlap are a good indication of locationswith higher risk for gang-related conflict. This type of spatialpattern analysis can be very useful for making decisions abouthow best to allocate scarce police resources.

The size of the ellipse created is a function of the standarddeviation of both the X and the Y feature coordinates, rotated tominimize the distance between the points and the axes of the

ellipse. Analytics using standard deviational ellipses includecreating a spatial index of segregation (Wong, 1999), targetingresource distribution for disease control (Eryando et al., 2012),and predicting crime (Kent and Leitner, 2007; Levine, 2002).

Quantifying Spatial Patterns

Beyond the simple descriptive pattern analysis methods,a number of inferential statistical tests can effectively quantify

Figure 2 The center of population moves steadily south between 1900 and 2000 as growth in southern California outpaces population growth innorthern California. Adapted with permission from Scott, Lauren M., Janikas, Mark V., 2010. Spatial statistics in ArcGIS. In: Fischer, M.M., Getis, A.(Eds.), Handbook of Applied Spatial Analysis: Software Tools, Methods and Applications. Springer-Verlag, Berlin. Data source: http://www.census.gov/population/cencounts/ca190090.txt.

Figure 3 The linear mean statistic can assess overall direction ororientation.

Figure 4 Standard deviational ellipses used to identify core gang terri-tories. Adapted with permission from Scott, Lauren M., Janikas, MarkV., 2010. Spatial statistics in ArcGIS. In: Fischer, M.M., Getis, A. (Eds.),Handbook of Applied Spatial Analysis: Software Tools, Methods andApplications. Springer-Verlag, Berlin. Data source: simulated based onexpert knowledge.

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spatial patterns. These methods compare the observed patternof features, or feature attributes, against a theoretical spatialpattern. Most often, the theoretical pattern used as a baseline iscomplete spatial randomness (CSR) and, consequently, thenull hypothesis for these statistical tests is that the observedpattern is the result of underlying spatial processes that arerandom in nature. These methods calculate the probability,expressed as a p-value, that the null hypothesis is correct. Asmall p-value (<0.05 or, more conservatively, <0.01) indicatesstatistically significant spatial patterns in the data, providingevidence that nonrandom spatial processes are at work. Onlynonrandom spatial processes suggest cause–effect relationshipsand, consequently, indicate possible opportunities for reme-diation or intervention.

In addition to calculating p-values, these tools often reportcorresponding z-scores to indicate how the observed spatialpatterns deviate from CSR. Spatial patterns that are statisticallysignificant because they are more clustered than CSR generallyreflect infectious, transmittable, or attracting types of under-lying local spatial processes. For example, it is not uncommonfor the retail, financial, and industrial activities in a city tocluster the result of zoning ordinances and/or to take advantageof economies of scale or economies of agglomeration. Spatialpatterns that are statistically significant because they are moredispersed than CSR generally result from some form ofcompetitive process where features repel each other in order tocreate distance. Only small trees can grow under big trees, forexample. The small trees will be clustered as a result of seedtransmission processes but will thin out over time, the result ofcompetition for local resources.

A number of inferential pattern analysis methods focusspecifically on the spatial pattern of point features. A common,decades-old method widely used in geography and ecology isthe average nearest neighbor statistic. It works by computingthe mean distance to each feature’s closest neighbor and thencomparing that distance to the distance that would be obtainedfor a random distribution of those same features (Mitchell,2005). Another method, quadrat analysis, overlays pointfeatures with a fishnet mesh and counts the number of pointsoccurring within each quadrat. A chi-square statistic is thenused to assess how likely it is that the observed distribution ofpoints among the quadrats might be the result of randomspatial processes (Burt et al., 2009). Kernel density techniquesextend the quadrat method by considering point intensitywithin a predefined radius, generally presenting results asa map-based surface (Murray, 2010). The K function isunique in that it looks at spatial clustering and dispersion fora range of distances, providing clues about the spatial scalesat which the processes promoting clustering or dispersion areoperating (Scott and Janikas, 2010; Csillag and Boots, 2005).These point pattern analysis methods can help answerquestions like:

l Which types of businesses are most concentrated?l Does the spatial pattern of a disease mirror the spatial

pattern of the population at risk?

More common in the social and behavioral sciences aremeasures that consider both location and feature attributes.These methods typically operate on area features such as censustracts, parcels, or cities, where data have been aggregated to

represent counts or rates. They provide a single summarystatistic that quantifies spatial autocorrelation or the degree towhich features nearby in space are similar with regard toa measured attribute (Getis, 2010). Consequently, globalmeasures of spatial autocorrelation such as the Moran’s Istatistic are especially effective for understanding broadspatial patterns and trends (Mitchell, 2005). They can helpanswer questions like:

l Is there an unexpected spike in the number of riots or publicprotests?

l Is the new virus remaining geographically fixed or is itspreading to surrounding communities?

The thematic maps below (Figure 5) illustrate relative percapita income, by county, for the state of New York over severalyears. The darker areas reflect higher incomes. Unfortunately,a visual inspection of these maps is not sufficient to determinewhether spatial segregation among the rich and poor countiesis increasing or decreasing. It is difficult to discern the answerbecause doing so requires visual detection of both global andlocal differences among multiple locations over several years.Using the global Moran’s I tool to quantify the degree ofclustering for each year, however, and then creating a line graphof the z-score results, makes the overall trend clear (Figure 5). Ahigher z-score reflects more intense clustering. Consequently,the resulting sinking line of the z-score graph indicates thatclustering is dissipating. This pattern analysis provides someevidence that spatial segregation among the New York countieshas decreased overall between 1969 and 2002 (Scott andJanikas, 2010).

Similar to the K function, the global Moran’s I statistic canbe used to assess clustering across multiple spatial scales. Byrunning the global Moran’s I statistic for increasing distancesand plotting the resultant z-scores, a picture of spatial auto-correlation emerges (Figure 6). Peak z-scores indicate distanceswhere the processes promoting spatial clustering are mostpronounced, and these distances often serve as appropriatescales of analysis for the local spatial autocorrelation methodsdiscussed next.

Mapping Spatial Patterns

Global spatial autocorrelation measures, like the Moran’s Istatistic discussed in the previous section, help to quantifybroad trends in data by computing a single indicator tosummarize the statistical likelihood that the observed datareflect a random spatial pattern. They answer the question: isthere a discernible spatial pattern or not? For geographicanalyses, however, the more interesting question is often:where are the statistically significant spatial patterns located?Where are the hot spots, cold spots, and spatial anomalies? Toaddress questions like these, local versions of the global spatialautocorrelation statistics, such as the Getis-Ord Gi* and theAnselin Local Moran’s I, have been developed (Anselin, 1995;Ord and Getis, 1995; Getis, 2010). These pattern analysismethods consider each feature within the context of nearbyneighboring features and compare local statistical propertiesof an attribute to global properties. If the local mean fora feature and its neighbors is unexpectedly large compared to

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the global mean (i.e., the mean value for all features in the dataset), it is indicative of a hot spot (a spatial cluster of high values).Similarly, an unexpectedly small local mean suggests a cold spot(a spatial cluster of small values). An individual feature witha large attribute value surrounded by features with smallvalues, or a feature with a small value surrounded by featureswith large values, reflects spatial anomalies. Accounting forthe number of features and global variation in the attributevalue being analyzed, these methods determine if thedifferences between the local and global properties arestatistically significant, and return a result (a p-value and/or z-score) for every feature in the data set. Mapping these resultseffectively illustrates the spatial patterns (Aldstadt, 2010).

Local spatial autocorrelation statistics are used in a numberof application areas including crime (Ratcliffe, 2010; Scott

and Warmerdam, 2005) and health (Haque et al., 2012;Getis, 2004). In Figure 7, the Anselin Local Moran’s Istatistic is applied to data from the 2008 United Statespresidential election. The map shows statistically significantclusters of the Republican and the Democratic political partyvictories as well as spatial anomalies. A political strategistcould use the results from this type of pattern analysis tohelp design more effective canvassing. Active campaigningin counties without strong allegiances to either party may bemore effective than investing campaign money in countieswhere the opposing party has had strong wins. The spatialoutlier (Figure 7) in the southern part of Texas reflectsa Republican Party win surrounded by counties stronglysupporting the Democratic Party. Understanding whatmakes the political leanings in this county so different from

Figure 6 Analyzing the spatial pattern of crime, the peak z-scores at 3500 and 5500 feet reflect distances where the processes promoting clusteringare most pronounced. Analysis is based on data obtained from https://data.cityofchicago.org/.

Figure 5 Relative per capita incomes by county for New York. Adapted with permission from Scott, Lauren M., Janikas, Mark V., 2010. Spatialstatistics in ArcGIS. In: Fischer, M.M., Getis, A. (Eds.), Handbook of Applied Spatial Analysis: Software Tools, Methods and Applications. Springer-Verlag, Berlin. Data source: Bureau of Economic Analysis, www.bea.gov/regional/.

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its neighbors may help strategists design targeted messaging towin votes there.

Another approach to cluster analysis identifies groups offeatures based on inherent similarities for a set of specifiedattributes. The most commonly used methods rely on K-Meansalgorithms, but a number of other methods have also beendeveloped (Murray, 2010; Duque et al., 2007), includingmethods that impose both spatial and temporal constraintson group membership (Esri, 2012). The goal of thesemethods is to classify features into natural groupings such thatboth the within-group similarity and the between-groupdifferences are maximized. These methods have been used tostudy regional economics (Duque et al., 2006) and have beenproposed for analyzing cancer rates (Guo and Wang, 2011).While these methods have been used most often in thephysical sciences (e.g., climate zoning, landscape analysis)and for image classification, they have much broaderapplication whenever there is a need to create homogeneousregions or to elucidate hidden patterns in spatial data (Miller,2009; Guo and Wang, 2011). With increased availability ofhigh-resolution spatial data, these types of analytics will beneeded for understanding human–environment interactionsand socioeconomic dynamics, and to address critical problemssuch as global climate change (Guo and Mennis, 2009).

Future Directions

Looking into the future of spatial pattern analysis methodsand applications is both exciting and overwhelming. Accordingto IBM (Selinger, 2013), 90% of the data today has been

generated in the past 2 years, and new data are being createdmore quickly than are algorithms capable of processing them.Sources of these data include purchase transactions, sensorsmonitoring climate, and other aspects of the physical world,vehicle electronic transmissions (cars, planes, and ships,for example), utility meter readings, a host of data generatedfrom social media sources (e.g., digital photographs, videos,blogs, tweets), and Global Positioning System-enabledmobile devices (Selinger, 2013). As a consequence, the abilityto effectively describe, quantify, and map spatial patternsis becoming more important than ever. The challenge isnot only dealing with this large volume of messy,dynamic, streaming data but also effectively analyzing itsmultidimensional characteristics.

Even prior to analytical considerations, however, methodsfor filtering data to eliminate noise, detect and remove errors,reduce redundancy, compensate for missing data, and dealwith uncertainty will need to be refined. These filteringmethods are a precursor to powerful analytics that must becapable of detecting and then predicting unexpected patterns,correlations, and trends. Effective pattern analysis techniquesare not yet well developed for dealing with space, time, andmobility simultaneously. Methods that can detect coevolvingspatial–temporal patterns – various event series that havesimilar tracks (similar spatial–temporal ‘fingerprints’) based onmutual locations, social networks, behaviors, opportunitiesand histories – also need to be developed (Shekhar et al.,2009).

Once these new pattern analysis methods are implemented,however, additional challenges surround how best tocommunicate the multidimensional results. An important goal

Figure 7 Hot spot analysis of the 2008 presidential election results by county for Texas. Data source: National Atlas of the United States.

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will be to extend descriptive cartographic visualizations (suchas animations) by developing mathematical and statisticalframeworks. These will be required to detect, for example,significant trends or anomalies from the increasing informationoverload in order to motivate effective action (Murray, 2010).In summary, development of new spatial pattern analysismethods, algorithms, and applications represents vital areasof research where scholars and analysts in the social andbehavioral sciences can make enormous contributions.

See also: Access: Geographical; Applied Geography: A USPerspective; Diffusion: Geographical Aspects; ElectoralGeography; Geographic Information Systems; Geography:Overview; Global Migration; Health, Geography of; MarketingStrategies; Mathematical Models in Geography; PopulationGeography; Race and Racism, Geography of; Regional Science;Residential Segregation: Geographic Aspects; Scale inGeography; Social Geography; Spatial Analysis in Geography;Spatial Association, Measures of; Spatial Autocorrelation;Spatial Data; Spatial–Temporal Modeling; Time-Space inGeography; Urban Geography; Urbanization.

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Relevant Websites

http://resources.arcgis.com/en/help/ – Math and additional details for the LocalMoran’s I statistic.

http://resources.arcgis.com/en/help/ – Math and additional details about spatiallyconstrained cluster analysis.

www.esriurl.com/spatialstats – Analyzing the Spatial Pattern of Piracy.http://www.census.gov/geo/www/cenpop/MeanCenter.html – US Census Bureau maps

and animations tracing the center of population over time.http://resources.arcgis.com/en/help/ – Math and additional details about the Global

Moran’s I statistic.http://resources.arcgis.com/en/help/ – Math and additional details about the local K

Function.http://resources.arcgis.com/en/help/ – Math and additional details about the Linear

Mean and Circular Variance statistics.http://resources.arcgis.com/en/help/ – Math and additional details about the Mean

Center and Weighted Mean Center statistic.http://resources.arcgis.com/en/help/ – Math and additional details for the Standard

Deviational Ellipse method.www.esriurl.com/spatialstats, https://geodacenter.asu.edu/eslides – Spatial pattern

analysis tutorials and free web seminars.http://resources.arcgis.com/en/help/ – Additional information about p-values and

z-scores.

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