第八讲-空间自相关分析

第八讲空间自相关分析

SpatialAutocorrelation1SpatialAutocorrelation:

Moran’sI2SpatialAutocorrelationandMoran’sISeveraltestsexistformeasuringthespatialautocorrelationrelatingtoareasorpoints.OnesuchmeasurehasbeendevisedbyMoran(1950)andcanbeappliedtoareapatternsandtopointpatterns.ForarealdatatheequationforMoran’scoefficientis:WhereI=Moran’sspatialautocorrelationcoefficient n=thenumberofareasinthestudyregion J=thenumberofjoins X=avalueforanarea(ordinalorinterval) Xi,Xj=aretwocontiguousareas(oneithersideofajoin) c=apairofcontiguousareas3HypotheticalStudyRegion4CalculationsforMoran’s

SpatialAutocorrelationCoefficientI5CalculationsforMoran’s

SpatialAutocorrelationCoefficientI6CalculatedMoran’sIMoran’sICalculated:Moran’scoefficient(I)is-0.183,althoughthisvalueonitsownisnotverymuchuseindescribingthedegreeofspatialautocorrelationinavariable.TherangeofpossiblevaluesofIdependsonthespatialstructureoftheparticularregion.TodeterminewhatthevalueofIimpliesitisnecessarytocarryoutasignificancetest.7SignificanceThesignificancetestinvolvescalculatingthestandardnormaldeviatefromthecalculatedvalueofI,theexpectedvalueI,anditsstandarddeviation.Therearetwopossibleformsofthenullhypothesis: normalityandrandomizationNormality:Thenullhypothesisisthattheobservedvaluesofthevariablearetheresultofarandomsamplefromanormallydistributedpopulationofvalues.Randomization:Thequestionaskedis“givenaparticularsetofvaluesX,whatisthepossibilitythattheycouldhavebeenarrangedintheobservedwaybychance?Thenullhypothesisisthatthespatialdistributionisrandom.8NormalityTheequationfortheexpectedvalueofIunderthenullhypothesisofnormalityis:Theequationforthestandarddeviationofthisvalueis:Where n=thenumberofareasinthestudyregion J=thenumberofjoins L=thenumberofareastowhichanareaisjoined

9NormalitySubstitutingvaluesintotheformulaweget:Theequationforthestandarddeviationofthisvalueis:ThepreviouslycalculatedvalueofIcannowbeconvertedintoastandardnormaldeviateusingthefollowingequation:10NormalityNotethattheexpectedvalueofIforarandomarrangementissmallandnegative(-0.2)Asmallervalue,onefurtherfromzerointhenegativedirectionimpliesdispersion.Positivevaluesimplyclustering.AfterconvertingtheobservedItoastandardnormaldeviate,itssignificancecanbeassessedbyreferencetoatableofcriticalvalues.11NormalityAdoptingthe0.05significancelevel,thetwo-tailedcriticalvalueforapositivestandardnormaldeviateis1.96.Atwo-tailedtestisappropriate,sincenospecificdirectionofdepartureishypothesized.Theobservedvalue(0.061)islessthanthecriticalvalue,sothenullhypothesismaynotberejected.Theobservedarrangementofvaluesisnotsignificantlydifferentfromrandom(randomlysamplingfromanormaldistribution).Itcouldhaveeasilyoccurredunderthenullhypothesisofrandomsamplingfromanormallydistributedpopulation.12RandomizationTheequationfortheexpectedvalueofIunderthenullhypothesisofrandomizationis:Theequationforthestandarddeviationofthisvalueisrathermorecomplex:Where k=kurtosisKurtosisisameasureofpeaknessofthedistributionofX13CalculationofKurtosisforRandomizationSignificanceTestofI14RandomizationSubstitutingvaluesintotheformulaweget:Theequationforthestandarddeviationofthisvalueis:ThepreviouslycalculatedvalueofIcannowbeconvertedintoastandardnormaldeviateusingthefollowingequation:At0.01significancelevelthetwo-tailedcriticalvalueis2.576. Sincethecalculatedvalueislessthanthecriticalvaluethenullhypothesisisnotrejected.Theobservedarrangementisnotsignificantlydifferentfromrandom.Itcouldhaveoccurredbychance.15CometruewithArcGISArcToolbox>SpatialStatisticsTools>AnalyzingPatterns>SpatialAutocorrelation(Moran’sI)OrHigh-LowClustering(Getis-OrdGeneralG)(吉瑞C)吉瑞C在[0，2]之间吉瑞C：0-1表示空间正相关吉瑞C：1-2表示空间负相关吉瑞C=1表示相互独立Moran’sI在[0，1]之间Moran’sI接近于1，表示空间正相关，即高高相邻，低低相邻Moran’sI接近于-1，表示空间负相关，即高低相邻，低高相邻Moran’sI接近于0，表示空间无相关性，即随机分布16ExercisewithBeijingtown17FurtherTopicsinSpatialAutocorrelation18SpatialAutocorrelationforPointDataMeasuresofspatialautocorrelationcanbeextendedtosituationsofpointvalues.Withpointdata,insteadofconsideringtherelationshipbetweenpairsofcontiguousareavalues,itisnecessarytomeasuretherelationshipbetweenallpairsofpointvalues,takingintoaccountthedistancesseparatingthem.Iftherearenpoints,therewillben(n-1)/2possiblepairsofpoints.Withasfewas20pointsthismeans190pairsofvaluestobemultipliedandsummed.Thetechniquecanbequiteeasilycomputerized(verytediousbyhand).Wewillreviewasimpleexample.19RevisedMoran’sIforPointPatternsThetestforspatialautocorrelationinpointpatternsisarevisedversionofMoran’scoefficient:WhereI=Moran’sspatialautocorrelationcoefficient n=thenumberofpoints Wij=theweightgiventotherelationshipbetween twopointsiandj p=apairofpointsTheweightisusuallythereciprocalofthedistancebetweenthetwopoints.Thedistancebetweenpointsiandjaredefinedasdij,thusWij=1/dij.Eachweightismeanttobeameasureoftheinfluenceexertedbyonepointonanother.Theuseofthereciprocalofdistanceasaweightimpliesthattheinfluencedecreaseswithdistance.20SignificanceTest:NormalityTheequationfortheexpectedvalueofIunderthenullhypothesisofnormalityis:Theequationforthestandarddeviationofthisvalueis:where

Thenullhypothesisofnormalityinvolvestheassumptionthatthepointvalueswithinthestudyregioncanberegardedasarandomsampleofvaluesdrawnfromanormallydistributedpopulation.21StepsforcalculatingsignificancefornormalityThecalculationofthisexpressioncanbethoughtofintermsofanumberofsteps:Foreachpointadduptheweightsbetweenitandallotherpointstoget:Squarethetotal,toget foreachpoint.Addupallthesesquaredtotals,togetWewillexamineanapplicationofthesestepsforthistest.22SignificanceTest:RandomizationTheequationfortheexpectedvalueofIunderthenullhypothesisofnormalityis:Theequationforthestandarddeviationofthisvalueis:whereTherandomizationnullhypothesisonlytakesintoaccounttheparticularsetofpointswithinthestudyregionIngeneral,randomizationisthesaferchoicesinceitinvolvesfewerassumptions23HypotheticalPointPattern24Calculationsforallpoints25Calculationsforallpairsofpoints26Wij=1/dijCalculationsrelatingtothematrixofweights27CalculatedMoran’sIMoran’sICalculated:Moran’scoefficient(I)is-0.0825,althoughthisvalueonitsownisnotverymuchuseindescribingthedegreeofspatialautocorrelationinavariable.TherangeofpossiblevaluesofIdependsonthespatialstructureoftheparticularpointpattern.TodeterminewhatthevalueofIimpliesitisthereforenecessarytocarryoutasignificancetest.28SignificanceTest:NormalityTheequationfortheexpectedvalueofIunderthenullhypothesisofnormalityis:Theequationforthestandarddeviationofthisvalueis:Therefore

Assumingatwo-tailedtestatthe0.05significancelevel,theobserveddegreeofspatialautocorrelationisnotsignificant(criticalvaluez=1.96)29SignificanceTest:RandomizationTheequationfortheexpectedvalueofIunderthenullhypothesisofnormalityis:Theequationforthestandarddeviationofthisvalueis:Therefore

Assumingatwo-tailedtestatthe0.05significancelevel,theobserveddegreeofspatialautocorrelationisnotsignificant(criticalvaluez=1.96)30FromGlobaltoLocal

MeasuresofSpatialPattern31

HepatitisRatesofCaliforniaCountiesin1998(per100,000pop.)

33LocalSpatialStatisticsGeneraltestsaredesignedtoprovideasinglemeasureofoverallpatternforamapconsistingofpointlocationsThesegeneraltestsprovideatestofthenullhypothesisthatthereisnounderlyingpattern,ordeviationfromrandomness,amongthesetofpoints.Examples:nearestneighbortest,thequadratmethod,andMoran’sIThesearecalledGLOBALstatistics–asinglesummaryvalue.LocalSpatialStatisticsSometimestheresearcherwantstoknowifthereisaclusterofeventsaroundasingleorsmallnumberoffoci.Forexample,doesdiseaseclusteraroundatoxicwastesite,crimeclusteraroundexoticdancingestablishments.Sometimeswewanttohaveamethodtodetectclustering.Noaprioriideajustaneedtodetermineifclustersexist.ThesearecalledLOCALspatialstatistics.LocalMoran’sILocalMoran’sIisalocalspatialautocorrelationstatisticbasedontheMoran’sIstatistic.ItwasdevelopedbyAnselin(1995)asalocalindicatorofspatialassociation(LISAstatistic)AnselindefinesLISAashavingthefollowingproperties:TheLISAforeachobservationgivesanindicationoftheextentofsignificantspatialclusteringofsimilarvaluesaroundthatobservation;ThesumofLISAsforallobservationsisproportionaltoaglobalindicatorofspatialassociation.36AnalysisAnalysisisverysimilartothatofglobalMoran’sI.ValuesofIithatexceedE(Ii)indicatepositivespatialautocorrelation,inwhichsimilarvalues,eitherhighvaluesorlowvaluesarespatiallyclusteredaroundpointi.ValuesofIibelowE(Ii)indicatenegativespatialautocorrelation,inwhichneighboringvaluesaredissimilartothevalueatpointi.Again,anormallydistributedZstatistic(2-tailed)iscalculatedtodeterminesignificance.37Spatialweightingmethodstheinputcanalsobeaweight,m,thatthedistanceisraisedinordertoshowtheinfluenceofdistance.Anexampleofthismightbewhichdisraisedtothepowerofm=2.Forthistypeofweightingscheme,thestatisticiscalculatedforbandsonly.BearinmindthateachIivalueforagivensiteIrepresentsassociationbetweentheithsiteandonlythejvaluesinagivenband.38FormulaforLocalMoran’sITheformulais:Where andPerhapsRemember,whenthisweightingschemeisused,thestatisticiscalculatedforbandsonly.Aspatialweightsmatrixmayalsobeused.39RandomizationHypothesisTheExpectedvalueis:Thevarianceis: Where: 40LocalSpatialStatistics:Getis’sGiStatisticTherearetwovariationsofthisstatistic,dependingonwhethertheunit(observation)iaroundwhichtheclusteringismeasuredisincludedinthecalculations.Gidoesnotincludetheobservationaroundwhichthemeasureisbeingcalculated.

Gi*doesincludetheobservationaroundwhichthemeasureisbeingcalculated.Getis’GiPurpose:totestwhetherclusteringexistsaroundacertainlocation(i)Where GiisthemeasureoflocalclusteringofattributeXaroundi, XjisthevalueofXatj, Wijrepresentsthestrengthofthespatialrelationshipbetween unitsiandjwhichcanbemeasuredaseitherabinary contiguityvariableoracontinuousdistance-decaymeasureIfhighvaluesofXareclusteredaroundi,Giwillbehigh.IflowvaluesofXareclusteredaroundi,Giwillbelow.Noclusteringofvaluesaroundi,Giwillbeintermediate.Getis’GiTheexpectedvalueofGiis:where Andthevarianceis:Wherethesubscriptiindicatesthecalculationofthemeanandvarianceofxexcludingthevalueati.43Getis’sG*iStatisticPurpose:totestwhetheraparticularlocationianditssurroundingregionhavehigherthanaveragevaluesonavariableofinterest.OrdandGetis(19