Spatial Statistics - Salisbury University

Spatial StatisticsGEOG 419: Lembo Arthur J. Lembo, Jr.Salisbury UniversityPoint Pattern Analysis Global methods to analyze point patterns acrossentire study region (or a map)– Quantitative tools for examining a spatialarrangement of point locations on the landscape Two common types of analysis– spacing of individual points – nearest neighboranalysis Ex. fire stations locations – random or dispersed– Goal: equitable service throughout region– Design new configuration (e.g., relocating, new stations)– More or less dispersed than original configuration– nature of overall point pattern – are locationsdispersed or clustered Ex. diseased trees in a national forest– Widespread aerial spraying versus concentrated groundtreatment Arthur J. Lembo, Jr.Salisbury UniversityCenter PointEuclidean (straight-line) distance ArthurJ. Lembo,Jr. Total distance from all otherpointsis lowestSalisbury University1

Center Point Arthur J. Lembo, Jr.Salisbury UniversityMean Center mean center – average location of a setof points– Center of gravity of point pattern (spatialdistribution)– average X, Y values– equal weights Arthur J. Lembo, Jr.Salisbury UniversityMean Center Outliers .– add point (15, 13)– Average locationbut Arthur J. Lembo, Jr.Salisbury University2

Mean Center geographic “center of population” – pointwhere a rigid map of the country wouldbalance if equal weights (i.e., location ofeach person) were situated over it Arthur J. Lembo, Jr.Salisbury UniversityWeighted mean center Unequal weights applied to points– Ex. retail store volume, city populations,etc.– Weights analogous to frequencies Arthur J. Lembo, Jr.Salisbury UniversityWeighted mean center Arthur J. Lembo, Jr.Salisbury University3

Spatial measures ofdispersion standard distance – measures theamount of absolute dispersion in apoint distribution– spatial equivalent to standard deviation– calculate Euclidean distance from eachpoint to mean center Arthur J. Lembo, Jr.Salisbury UniversityStandard distance Arthur J. Lembo, Jr.Salisbury UniversityRelative Measure .Weighted standard distance Used with weighted mean center– Difference 1.54 vs. 1.70 Arthur J. Lembo, Jr.Salisbury University4

Standard Deviational Ellipse Extends standard distance to includeorientation of the point pattern– Calculated separately for X and Y Average distance points vary from meancenter on X and average distance points varyfrom mean center on Y axis Arthur J. Lembo, Jr.Salisbury UniversityStandard DeviationalEllipse cross of dispersion trigonometric function – angle of rotation– Rotated about mean center to minimize distancebetween both arms and points Arthur J. Lembo, Jr.Salisbury UniversityNearest Neighbor Analysis –(NNA) Distance of each point to its nearestneighbor is measured and mean distancefor all points is determined– Objective: describe the pattern of points in astudy region and make inferences about theunderlying process Arthur J. Lembo, Jr.Salisbury University5

Nearest Neighbor analysis –(NNA) Compare calculated value from point data totheoretical point distributions– Outcomes: random, clustered, dispersed– average nearest neighbor distance is anabsolute index Dependent on distance measure (ex. miles, km,meters, etc.) Minimum 0 (clustered), maximum is function ofpoint density– standardized nearest neighbor index (R) isoften used Comparison of data to random Arthur J. Lembo, Jr.Salisbury UniversityNearest Neighbor analysis – (NNA) Arthur J. Lembo, Jr.Salisbury UniversityNNA – R values Continuum – Result?– Descriptive test Arthur J. Lembo, Jr.Salisbury University6

FunctionalSWBResults? Arthur J. Lembo, Jr.Salisbury UniversityNearest neighbor analysis(nna) A difference test can be used to determine if theobserved nearest neighbor index (NNA) differssignificantly from the theoretical norm (NNAR)– H0: There is no difference between our distributionand a random distribution (Poisson) Arthur J. Lembo, Jr.Salisbury UniversityNearest neighbor analysis (nna)Example: Community Services inToronto Emergency services: fire and police Nonemergency services: polling sites and elementary schools– Seek dispersion to provide services equally– Seek clustering why? Arthur J. Lembo, Jr.Salisbury University7

Nearest neighbor analysis (nna) Example:Community Services in Toronto Result? Arthur J. Lembo, Jr.Salisbury UniversityNearest neighbor analysis (nna) Example:Community Services in TorontoEmergencyservices – moredispersedVoting locations –more clusteredElementaryschools - random Arthur J. Lembo, Jr.Salisbury UniversityNearest neighbor analysis (nna)Example: Community Services inToronto Issues to consider – Study area boundaries – political boundary or researchdelimited Doesn’t impact NNA distances but does impact area (pointdensity function)– Nearest feature – may be outside study area! Problem with using political boundaries– More advanced techniques available – Ripley’s K Evaluates more than one nearest neighbor Can define distances – How many police stations within 1km?2km? Arthur J. Lembo, Jr.Salisbury University8

General Issues in InferentialSpatial Statistics Geographers are interested inspatial patterns produced byphysical or cultural processes– Explain patterns of points and areas “global” overall arrangement– Random vs. Nonrandom spatial processes “local” concentrations or absences–Clusters – points or areas within largerarea» Groups of high values – “hot spots”» Groups of low values – “low spots” Arthur J. Lembo, Jr.Salisbury UniversityTypes of Spatial Patterns Compare existing pattern totheoretical pattern Clustered– Density of points variessignificantly from one part ofstudy area to another Points: retail locations nearhighway interchange Areas: registered majoritypolitical party affiliation– Patterns result from nonrandomfactors Accessibility, income, race,etc. Arthur J. Lembo, Jr.Salisbury UniversityTypes of Spatial Patterns Dispersed– Uniformly distributedacross study area Suggests systematicspatial process Area example:Central Place Theory– settlements areuniformly distributedacross landscape tobest serve needs of adispersed ruralpopulation Arthur J. Lembo, Jr.Salisbury University9

Types of Spatial Patterns Random– No dominant trendtoward clustering ordispersion Suggests spatiallyrandom process(Poisson) Ex. lightning strikes Geographic problems– Patterns typically appearas some combination ofthese three patterns Along continuum Arthur J. Lembo, Jr.Salisbury UniversitySpatial Autocorrelation Tobler’s Law – “Everything is related to everything else butnear things are more related than distant things” spatial autocorrelation: measures the degree to which ageographic variable is correlated with itself through space– Positive, negative or non-existent Positive spatial autocorrelation: objects near one another tend to besimilar– Features with high values are near other features with high values, featureswith medium values are near other features with medium values, etc. Negative spatial autocorrelation: objects near one another tend tohave sharply contrasting values– Features with high values near features with low values Most geographic phenomena exhibit positive spatialautocorrelation– Examples: rainfall amounts, home values, etc. Arthur J. Lembo, Jr.Salisbury UniversityVariogram Visualization of spatialautocorrelation variogram: scatterplot thatdisplay the differences invalues between geographiclocations against thedifferences in distancesbetween the geographiclocations– Y-axis: average variance(really half the variance) invalues for a set of geographicobjects– X-axis: distance betweenobjects– Use plot to determine averagedifference in values at specificdistancesGeographic locationsnear one anothertend to have smallerdifferences thangeographic locationsat greater distances(positiveautocorrelation)! Ex. 100 miles, 500 miles Arthur J. Lembo, Jr.Salisbury University10

Variogram Displayed as best-fittingcurve (function)– Differences in values withdistance noted and thendiminishes range - distance at whichthe difference in valuesare no longer correlated sill – average difference invalue where there is norelationship betweenlocation and value nugget – degree ofuncertainty whenmeasuring values forgeographic locations thatare very close to oneanotherNorelationshipValues becomesless similar withdistance– Effect of sampling,measurement error, etc.– Unlikely that two samplesnear each other will havethe exact same value Arthur J. Lembo, Jr.Salisbury UniversityVariogram Example: Last SpringFrost IN SE United states Two nearbystations, LSF datesshould be similar– 0 to 400 miles:distances betweenstations are large,dates are different– Beyond 400 miles,no longer spatiallyautocorrelated Arthur J. Lembo, Jr.Salisbury UniversitySpatial Autocorrelation: Importancein Geographic Research GIS – push of a button– Calculates relationship for any distances Is the test appropriate for any distance? Presence of spatial autocorrelation– Inferential statistics assume independent observations Example: last spring frost dates are spatially correlated! Impact: sample locations close together, just like taking the samesample– Sample size impacts size of standard error» Smaller standard error than warranted– Standard deviation calculation impacted» Even smaller standard error Global or local measurement– global – examine a distribution of subset (ex. ethnic group)across entire area (ex. city) One group more clustered, dispersed or random than another– local – compares each geographic object (ex. all groupmembers) with its surrounding neighbors Is area (ex. neighborhood) more clustered, dispersed or randomthan another? Arthur J. Lembo, Jr.Salisbury University11

Spatial Autocorrelation: Neighbor Definitions Measure of interaction between geographicfeatures– Defining neighbor adjacency – share common border– Binary: yes or no» Ex. New York and Pennsylvania, New York and California distance threshold – cut-off distance– Salisbury, MD – neighbor definition 60 miles Easton, Wilmington,DE? inverse-distance – strength of “neighborliness” between twoobjects as a function of distance separating them(1/distance)» New York City and Boston: 1/189 miles or .005,» NYC and LA: 1/2588 miles or .0004» Interaction measure (“neighborliness”) is 12 timesstronger between NYC and Boston versus NYC and LA– In equations/modeling, takes the form of weights wij : weight between geographic object i and j– Binary: 0 or 1– Inverse-distance: continuous value Arthur J. Lembo, Jr.Salisbury UniversitySpatial Autocorrelation First law of geography: “everything is relatedto everything else, but near things are morerelated than distant things” – Waldo Tobler Many geographers would say “I don’tunderstand spatial autocorrelation” Actually,they don’t understand the mechanics, theydo understand the concept. Arthur J. Lembo, Jr.Salisbury UniversitySpatial Autocorrelation Spatial Autocorrelation – correlation of avariable with itself through space.– If there is any systematic pattern in the spatialdistribution of a variable, it is said to be spatiallyautocorrelated– If nearby or neighboring areas are more alike,this is positive spatial autocorrelation– Negative autocorrelation describes patterns inwhich neighboring areas are unlike– Random patterns exhibit no spatialautocorrelation Arthur J. Lembo, Jr.Salisbury University12