1. calibrating the size of the kernel or search window to the amount of spatial autocorrelation found in the attributes of the data being examined;

Transcription

1 WithApplicationtoDeprivationIndices Background GeographicallyWeightedRegression(GWR)(Fotheringham,Brunsdon,&Charlton2002),likemany othermethodsofspatialanalysis,ischaracterisedbymultiplerepeattestingasthedataaredivided intogeographicalregionsandalsorandomlyredistributedmanytimestosimulatethelikelihoodthat theresultsobtainedfromtheanalysisareactuallyduetochance.eachofthesetestsrequires computertimeso,givenalargedatasetsuchastheukcensusstatistics,runningtheanalysisona standardmachinecantakealongtime intheorderofdaysorweeks.thisisfarfromidealwhen thepurposeofmanyspatialstatisticsistobeexploratory:allowingtheusertointeractwithdata andfindspatialpatternsofassociationwithinthem. Consequently,theapplicationofhighperformancecomputingtospatialanalysishaslongbeenof interesttosocialandgeographicalscientists.ofparticularnoteisthepioneeringworkundertaken bystanopenshawattheuniversityofnewcastleandatthecentreforcomputationalgeographyat LeedsUniversity,ofwhichanexemplaristheGeographicalAnalysisMachine(GAM)(Openshawet al.1987).morerecently,martin(2005)hasidentifiedthepotentialforgeocomputationtodevelop undertherubricofhighperformancecomputer(grid)networksande(electronic)socialscience.he identifiesfouressentialresearchissuesforesocialscience:automateddatamining;visualizationof spatialdatauncertainty;incorporationofanexplicitlyspatialdimensionintosimulationmodeling; andneighborhoodclassificationfrommultisourcedistributeddatasets. MissingfromMartin'slististheexplicituseofparallelizationtospeedupthecalculationsassociated withspatialstatistics.whatgam,gwrandothermethodsofspatiallylocalizedanalysishavein commonisageneralsequenceof: 1.calibratingthesizeofthekernelorsearchwindowtotheamountofspatialautocorrelationfound intheattributesofthedatabeingexamined; 2.creatingspatiallyoverlappingsubsetsofthedatatoreflectthis; 3.allowingthekerneltopassfromonesubsettothenext,applyingastatisticaltestineach; 4.simulatingconfidenceintervalsforthestatisticalresultbydetachingthedataattributesfromthe geographicalcoordinatesatwhichtheywerecaptured,thenrepeatedlyreattachingtheattributesto randomlyselectedlocationsandapplyingthetestagain. Formanyspatialstatisticalprocedures,eachofthestagesofcalibration,fittingandassessing significancecanbeparallelizedwithprocessesthatwilloperatewithoutcommunicationwiththe others(since,forexample,theoutcomeofamodelfittedtoonespatialsubsetofthedatadoesnot affectormodifytheoutcomeofamodelfittedtoanother).thus,eachoftheprocessescanbesent toseparatecomputationalnodes,theiroutputspooledandthenincorporatedintotheoverall calculation.

2 By gridenabling GWRwehopetobalanceitscomputationallyintensiverequirementswiththe needofusersforfasterruntimes,andtoshowcaseitasanexampleofmethodsofspatialanalysis canbeoperatedontheuk snationalgridinfrastructure. Objectives Theoverarchingobjectiveoftheresearchwas: Todevelopaprototype,gridenabledimplementationofGWRthatcanbeusedbyother researchersandwhichbuildsuponexistingescienceinfrastructure(thenationalgridservice). Thisobjectivewascompletedinfull. Otheraimswere: (1)todemonstratetheuseofGWRandgridtechnologieswithregardstotheimportantsocialand policyissueofunderstandingandmeasuringthespatiallydistributedcorrelatesofdeprivation; (2)toworkcollaborativelyacrossinstitutions(theUniversityofBristol,theUniversityofLeicester andthenationaluniversityofireland,maynooth)andacrossdisciplines(geographicaland computationalscience),tofosterresearchnetworks; (3)to connect andliaisewithcurrentlyfundedresearchprojectsthatcomplementthisproposal; and (4)tofosterthecontinuedprofessionaldevelopmentoftheresearchteamandparticularlythe researcheremployedtocarryforwardanddevelopesciencewithinthesocialsciences. Ofthese,(3)and(4)werefullyaddressed,(2)metinkindand(1)inpartandisongoing(seeResults sectionbelow). Veryearlyintheproject slife,collaborationhadbeenexploredwiththelancasteruniversitycentre forescience.anopportunitytoformalisethisarrangementcamewiththeunexpecteddepartureof theresearchassistantatbristoltootheremployment,andthesubsequentfailuretorecruita suitablyreplacement.collaborationthereforefocusedonthreeinstitutions:theuniversityofbristol, theuniversityofleicesterandtheuniversityoflancaster,thoughtheworkwasalsopresentedat thenationaluniversityofireland,maynooth. Methods AttheoutsetoftheprojectfourversionsofGWRwereavailabletoustodevelop.Thefirstwasa Windowsbasedversionwithagraphicaluserinterface.ThisisthesoftwareproducedbytheGWR developmentteamatthenationaluniversityofireland,maynooth.thesecondwasthe raw Fortran77underpinningtheWindowsdelivery.ThethirdwasanexistingimplementationofGWR writteninrbyoneoftheresearchteamandoriginatorsofgwr(professorchrisbrunsdon).the fourthwasanopensourcelibraryforrunningr:thespgwrpackagedevelopedbybivandandyuand hostedonthecomprehensiverarchivenetwork.

3 ThespgwrpackageforRprovidesfunctionsforcalibrationofthebandwidthandcalculationofthe regressionparametersusingthemethodsofgeographicallyweightedregression.clearly,itwould beadvantageoustoreusethiswellusedandsupportedpackageasmuchaspossiblewhen developingaparallelversionofthegwrmethods.doingsowouldminimizetheamountof additionalskillsrequiredbyexistingusersofspgwrwhenadaptingtousingaparallel implementation.inaddition,itwouldreducetheoveralldevelopmenteffortrequiredtoimplement parallelgwr. Infact,thisapproachtoparallelisingGWRhasalreadybeentakenbytheauthorsofthespgwr packagetomakegwravailableonmultiprocessorsystems.thiswasachievedusingthesnow packagewhichprovidesasetofmethodsforevaluatingrfunctionsinparallelusingpvm,socketsor threads.however,snowdoesnotprovidethemeansofemployingalargenumberofdistributed systemssuchasaretypicallyencounteredinagridenvironment. Risanopensourcepackageforstatisticalcomputingandgraphicsandhasalargeandgrowinguser base,manyofwhomprovidelibraries(or addins )extendingitsfunctionality.apriornational Centreforesocialscience(NCeSS)projectcalledSABREinRhadinvolvedtheLancasterUniversity CentreforeSciencedevelopingaparallelimplementationofSABRE(aprogramforthestatistical analysisofbinary,ordinalandcountrecurrentevents)asrobjects.thatprojecthadusedgrowl toprovideuserfriendlyaccesstogridresourcesforapplicationsaccessiblefromdesktop computer ( thatcouldruntheexistingspgwrlibraryonadesktopcomputerusingrbutdotheprocessing remotelyonthenationalgridinfrastructurebecamethemethodofchoice. Apackage,entitledmultiR,wasdevelopedforthispurpose,usingGROWLtechnology.Unlikesnow, multirdoesprovideaclientrinterfaceforparallelcomputinginahighthroughputdistributed computingenvironment.thepackage,multir,isaclient/serversystemwhichprovidesameansof submittingagroupoftasksforprocessingonmultiplesystemsthatareremotefromtheclient system.theremotesystemscouldbeprocessorsonalocalhighperformancecluster,acondorpool orcombinationsoftheseandpossiblymanyothertypesofsystem.themultirclientinterfaceis distributedasapackageforranditsusageissimilarinmanyrespectstothatoftherfunction lapply.themultirconceptistoprovideameansofspecifyingarfunctionformultipleinvocation withvaryingargumentswherethefunctionisevaluatedonmultipleprocessors.bydoingsoitallows Rtobecomeaprogrammingenvironmentforcoursegrainedparallelprocessing. ThemultiRclient/serversystemisbasedonathreetierarchitecture.Itisimplementedinthisway becausesuchanarchitecturaldesignpatternovercomesmanyofthedifficultiesassociatedwith providingandadministratingasecureservicewheretheresourcesemployedtoimplementthe servicearemanifold,variedandconstantlychanging.figure1outlinestheprincipleofthe architecture.clientsusertodefinethefunctionsthatrequireevaluationandusemultirtosubmita job(thefunctioninvocations)tothemultirserver.themultirserverthendelegatesthesetasksto whateverresourcesitemploys.theprogressofjobsthathavebeensubmittedbyaclientmaybe monitoredwithinrandtheresults harvested bycommandsprovidedwithinthemultirpackage. TheevaluationofthefunctioninvocationswhichcomprisethejobareevaluatedwithinRsessions invokedonthehostsystemswhichactasproxiesfortheclientrsession.

4 Figure1.Thethreetierclient/serverarchitectureemployedbymultiR. Results Themain result oftheresearchwasthespgwr.distandmultirpackagesforrwhichwerethenused tofitagwrmodelofcarnonownershipusing165,665datapoints.thesearenowdescribed. Thespgwr.distpackageforRcontainsthefunctionsrequiredforgridenabledGWR(distisan abbreviationofdistributed,i.e.itisdesignedfordistributedcomputing).itusesafurtherrpackage calledmultirwhichisinstalledlocallybutsetsuprtorunonadistributedcomputingplatformby identifyingaremotemultirserverby name andbytheportnumberonwhichtheserviceishosted. ThemultiRpackageandserverarethemiddlewarebetweentheuser sdesktopandthegridsystem onwhichthegwranalysiswillbecompleted.themultirsessionrequiresthreesecuritycredentials tobesupplied:amultirproxycertificate,acertificatevalidatingthemultirserverandtheuser s proxycredentialsforthenationalgridservice(ngs).thelastoftheseisgeneratedusingmultir s create.proxyfunctionfromtheuser scertificatekeypairissuedbytheukesciencecertification Authority.(Theactualcertificateobtainedfromhttps://ca.gridsupport.ac.ukisexportedfromaweb browserin.p12format;thatfilethenneedsconvertingintotwoseparatebutpairedfilesbyusing theopenssltoolkit:seewww.gridsupport.ac.uk/content/view/67/184/fordetail).specifyingthe multircertificatewillshortlybecomeunnecessaryandtheassociatedargumentwillbedeprecated infutureversionsofmultir. Currently,atypicalsessioninRbeginsas:

5 > library(spgwr.dist) # loads the spgwr.dist and multir packages > session <- multir.session("stats-grid.hpc.lancs.ac.uk", "50000", + "~/multir.ca.pem", "~/grid.proxy.pem") Theanalysisthencontinuesinmuchthesamewayasfortheexistingspgwrpackgage.Where,in spgwr,thebandwidthforgwriscalculatedontheuser sdesktopusingafunctionoftheform > bw = gwr.sel(y~x, data, coords) forthegridenabledversionweuse > bw = gwr.sel.dist(session, y~x, data, coords, max.processors) Similarly,wherethemodelisfittedinspgwrusing > gwr.model = gwr(y~x, data, coords, bw) itisfittedinspgwr.distusing > gwr.model = gwr.dist(session, y~x, data, coords, bw, + max.processors) Theonlydifference,fromtheuser sperspective,isthattheadditionalparameter session contains theinformationrequiredtoconnecttothemultirserver,andtheparameter max.processors (whichisoptional)specifiesamaximumnumberofprocessorsthegwrfitshouldrunon. Imagineacommadelimitedfilecalled census.csv containingsixcolumnsofdata.thefirstare attributedata,headedy,x1,x2,x3,andtheremainingtwodefineapointcoordinateassociated withwherethedatawerecollected.thoseareheadedeastingandnorthing.tofitagwrmodelon thegridsystematlancaster,exploringthegeographicallyvaryingrelationshipof y x x x theprocesswouldbe: (i,j) 1(i, j) 2(i, j) 3(i, j) > mydata = read.csv( census.csv, header=true) > locations = cbind(mydata$easting, mydata$northing) > bw = gwr.sel.dist(session, Y~X1+X2+X3, data=mydata, + coords=locations, max.processors=20) > gwr.model = gwr.dist(session, Y~X1+X2+X3, data=mydata, + coords=locations, bandwidth=bw, max.processors=20) Thereislittlesenseinusingthespgwr.distpackagefor small datasetsofabout1000observations orless.forthose,thewindowsbasedsoftwareortheexistingspgwrpackageinrwillbeabetter choice:faster,becauseofthegreaterneedforcommunicationanddataexchangethattheuseofa distributedsystemintroduces.

6 However,GWRdoesnotscalewell.ThereasonisthatGWRfitsadistanceweightedregression model,usuallyoftheform y i x i i i i 0(u,v) k k(u,v) ik toeachofmpointswithinacontinuous, geographicspace:(u i,v i)denotesthegeographiccoordinatesoftheithofthempoints.foramodel thatexaminesaregressionrelationshipateachof100,000censuszones,n=100,000anditwould appearthatthereare100,000regressionsurfaceswhichneedtobecalculated.whilsttrue,there arealsopriorcalculationstobecompleted. First,becausetheregressionisdistanceweighted,thedistancesbetweenthepointsneedtobe calculated.intheexample,anbynmatrixisrequired.moregenerally,becausethefitpointsneed notbethesamelocationsasthoseforwhichthedataarecollected,thengivenagwrmodelwithn datapointsandmfitpoints,thedistancematrix,disofsizembyn.nevertheless,thenumberof 2 2 calculationsrequiredtoobtainthedistancematrixapproximatestotheorderofn,d:o(n ). Havingcalculatedthedistancematrix,them(orn)regressionmodelsarefitted.However,thisisnot sufficient.firstthebandwidthcontrollingthedistanceweightingmustbefoundandoptimised (usingacrossvalidationtechniqueorbasedontheakaikeinformationcriteria,aic).ifittakesg iterationsfortheoptimisationproceduretoconvergeonapreferredbandwidth,thentheare actuallyg mregressionmodelstofit. Returningtotheexampleofcensuszones,wherem=n=100,000(whichisabouttwothirdsofthe totalnumberof2001censusoutputareasinenglandandwales),weestimatethatusingadesktop implementationofgwritwouldtakeabouthalfadaytoderivedandabouttwoweekstoobtain thebandwidth.thisis doable butconflictswiththenotionofusinggwrasatoolforexploratory dataanalysis(toinsomesense interact withthedata).asthetimesforthevariousstagessuggest, themainbottleneckisnotinfindingthedistancematrixbutincalibratingthebandwidth:each 3 iterationisoforder,o(n ). Itisunsurprisingtodiscoverthatpriortothisresearch(andtothebestofourknowledge)thelargest datasetforwhichgwrhasbeenattemptedwasofsizen=12,493(fotheringham,brunsdon,& Charlton2002).Here,wehavedemonstratedthepotentialforgridenabledGWRbyusingadataset withgreaterthantentimesthatnumberofobservations. Specifically,asimpleanalysishasbeenundertakentopredicttheproportionofhouseholdswithout acar(orvan)inn=165,665outputareasusingdatadrawnfromthe2001census.thepredictor variablesincorporatesocial,economic,demographicandethnicityinformationandare: X :Proportionofpersonsofworkingageunemployed 1 X :Proportionofhouseholdsinpublichousing 2 X :Proportionofhouseholdsthatareloneparenthouseholds 3 X :Proportionofpersons16orabovethataresingle 4 X :Proportionofpersonsthatare whitebritish 5 Thereasonformodellingcarnonownershipisthatitgenerallyisregardedasanindicatorof materialandsocialdisadvantage(reflectinganinabilitytoaffordandinsureavehiclewhichboth

7 causesandsustainsdisadvantageinthejobmarketwhereaccesstoemploymentbecomesanissue) (Clark&Wang2005).However,thatisnottrue,everywhere:carownershipislowerinLondon,for example,presumablybecausepublictransportoffersacrediblealternative(harris,sleight,& Webber2005,p ). Theregressioncoefficientsforastandard,ordinaryleastsquaresregressionmodelfittedtoallofthe 165,665observationsare1.61,0.46,0.32,0.38and0.07,respectively.Eachissignificantata greaterthan99%confidencebutthisishardlysurprisingandnotespeciallyinstructive:ita consequenceofthesizeofn(itbeinglarge). Moreinterestingishowthecoefficientsvaryspatially,asestimatedbyGWRandindicatedinTable1 bytheinterquartilerangeforeach k(u,v).forexample,whereasthegeneralmodelpredictsa10% increaseintheproportionofloneparenthouseholdswouldbeassociatedwithanaveragedecrease incarnonownershipof3.2%,thegwrmodelsuggestsadecreaseinthe(interquartile)rangefrom 9.6%to1.5%.Becauseofthedoublenegative,itiseasiertointerprettheresultsasshowingthatas ratesofloneparenthoodincreasesotoodoratesofcarownership,butthattheeffectisgreaterin someplacesmorethanothers. Global GWR (u, v): Q1 Median Mean Q3 IQR intercept unemployment public housing lone parents single white British Table1.ComparingthecoefficientsofastandardlinearmodelandaGWR modelpredictingcarnonownershipforn=165,665censusoutputareasin EnglandandWales. Figures2and3showsomeofthespatialvariationinthecoefficientfortheloneparentvariable. Figure2isforLondon,andFigure3isforBirminghamandCoventry.Botharecartograms. CartogramsareproducedbywarpingaEuclideanviewofgeographicspacetopermitthesizeofeach circletobeproportionaltothepopulationdensityatthelocationthatcirclerepresents(dorling 1996).Consequently,thepositionsofthemotorwaysareindicative,includedonlytoaid interpretationofthemaps.

8 Figure 2. A cartogram showing the spatial variation in the lone parent coefficient across London. Figure 3. A cartogram showing the spatial variation in the lone parent coefficient across Birmingham and Coventry.

9 Theinterestingareasarethoseshadedyelloworred,asthesearetheplaceswhereanincreasein loneparenthoodisleastassociatedwithincreasedcarownership.inbirminghamandcoventry theseplacesareneartothecitycentres;inlondontheyaremoredispersedbutprevalenttothe Eastofthecity.Ifthereisanadvantageinthejobmarkettobehadbyowningacar,thentheresults mightsuggestratherdifferentexperiences(ormeanings)ofloneparenthoodacrossgeographical space. TheGWRmodelforthen=m=165,665fitpointstookaboutthreehourstocalculateusingthe NorthWestGridService(atLancaster).Clearlythisisnot immediate butalsonotunreasonable fromtheuser sperspective(especiallygiventhatitisnotrunningorconsumingresourcesontheir ownpc). Inasense,however,we cheated.weestimatethatittakesabout1.5secondstofitasingle regressionsurfaceusinggeneralisedgeographicallyweightedregression.ifittakes50iterationsto findthegwrbandwidthfor100,000fitpointsandthecalculationisdistributedover100processors, thenthetotaltimetoobtainthemodelwouldbeabout (100,000/100)seconds about 20hours.Whetheritisreallynecessarytocalibratethebandwidthusingallthefitpointsisamoot pointandanareaforfurtherstudytheeffectsofsamplingongwrneedtobemorefully understood.inanycase,arandomsampleofabout50,000wasused(thegwr.sel.distfunctioncan generatearandomsampleofthepointsifdesired). Anumberofpointsfollow: Ageneralisedgeographicallyweightedregressionwasusedtofitthemodelofcarnon ownership(primarilytocheckitworked).however,itisthemorebasic(weightedleast squaresandgaussian)modelwhichisdescribedwiththespgwr.distpackage,above.itwill runfaster.(infact,ittakesabout14hourstorunontheentiredataset aboutonethird faster). Ifitsatisfactorytouseasamplingstrategywhencalibratingthebandwidththenitmayalso besufficientwheninvestigatingspatialvariationintheregressioncoefficients.thiswould seemappropriatefortheexploratorystagesofananalysis. Recallthattheprocessing bottleneck istheregressionfit.manyotherspatialstatistics(for examplevarioustypesofhotspotanalysis)aresimplerthangwrwhere,basically,they comparetherate,incidenceordensityofaneventorfeatureatoneplaceagainstthe correspondingvaluesforotherplacesacrossthestudyregion.thederivationofsuch statisticscanstillbetreatedasembarrassinglyparallel(withdifferentprocessorsoperating ondifferentsubsetsofthedata)andbecausetheyaremoredescriptivethanexplanatory, theywillrunconsiderablyfaster thereisnoregressionrequired. Activities Theresearchandprojecthavebeenpresentedatthesecondandthirdinternationalconferenceson esocialscience(manchester2006andannarbor,michigan2007,respectively),andatthe9th InternationalConferenceonGeoComputation(Maynooth,Ireland,2007).Ithasalsobeenpresented attherecentncessshowcase(manchester2008)andwillbeattheforthcomingdigitalgeography

10 inaweb2.0worldconference(london2008)aswellattheruser sconference(dortmund, Germany2008).AfreetrainingworkshopinusinggridenabledGWRwasundertaken(Lancaster 2007). Theresearchwasgenuinelycollaborative,involvingmembersoftheUniversityofLeicester ssplint (SpatialLiteracyinTeaching)groupand,especially,theLancasterUniversityCentreforeScience. Thelattercollaborationwasnotenvisionedintheoriginalproposalandwaslargelyserendipitous;it wasalsoextremelysuccessfulandmayrepresentsomethingofamodelbywhichcomputerand socialscientistsmaycollaborative. WealsogratefulfortheinputofProfessorRogerBivand,amemberoftheRcoredevelopment team,withwhomtimewasspentinbergen,norway. Outputs PapersarebeingpreparedfortheInternationalJournalofGeographicalInformationScience, focusingonthemoretechnicalaspectsofhowspatialstatisticsmaybegridenabled,andalsoforthe TransactionsinGISjournal,providingamoreappliedcasestudy.Afurtherpaperisbeingprepared forthejournalofstatisticalsoftwareandwehopetoproduceashortfeatureforthescientific ComputingWorldmagazine. Nevertheless,themainoutputsarethemultiRandspgwr.distpackagesforRwhicharebeing cleaned tomakethemfreelyaccessibleoncran(thethecomprehensiverarchivenetwork).beta versionsmayberequestedfrommembersoftheprojectteam. Thetrainingmanualwillbeuploadedtoasuitablewebsite initiallybeupdatingthecontentat Impacts ThedevelopmentofthemultiRpackageandserverisnotspecifictoGWRbutprovidesamore generallinkbetween(desktop)randgridresources.itisadevelopmentoftheexistinggrowl softwareandfurtherenhancestheuseofthenorthwestgridasahubforstatisticaloperationsof relevancetosocialscientists. Futureresearchpriorities Therearefourlinesofprioritywhicharisefromtheproject. Methodological:theimpactofsamplingonGWRneedstobebetterunderstood,asmaythe impactofmulticollinearityandcorrelationamonglocalregressioncoefficientsin geographicallyweightedregression(wheeler&tiefelsdorf2005).morepositively,thereisa possibilitytoresolveoneofthesimplifyingassumptionsofbasicgwr:thatasinglemeasure ofspatialautocorrelation(onebandwidth)issufficientfortheentirestudyregion.the simplepossibilityistoregionalisethedata,processitseparately,andcomparethe bandwidths.

11 Developmental:theapplicationofmultiRisnotlimitedtogwr.Atoolboxofstatistical operationscouldbeofferedrunninginargridenvironment,includingtypesofhotspot analysisandgeostatisticaloperationsincludingkriging:infact,almostanyprocessthatcan beseparatedintosubsets(notnecessarilyspatial)ofthedata. Datalinkage:tocensusandotherdataviatheNationalGridService.SeetheGEMSprojectat Collaborative:toextendthecollaborativemodelofworkingbetweencomputerandsocial scientists,forexampleby disciplinehopping funding. References Clark,W.A.V.&Wang,W.W.,2005.JobAccessandCommutePenalties:BalancingWorkand ResidenceinLosAngeles.UrbanGeography,26(7),p Dorling,D.,1996.AreaCartograms:TheirUseandCreation,Norwich:EnvironmentalPublications. Fotheringham,A.S.,Brunsdon,C.,&Charlton,M.,2002.GeographicallyWeightedRegression:The AnalysisofSpatiallyVaryingRelationships,Chichester:JohnWiley&Sons. Harris,R.,Sleight,P.,&Webber,R.,2005.Geodemographics:GISandNeighbourhoodTargeting, Chichester:JohnWiley&Sons. Martin,D.,2005.SocioeconomicGeoComputationandESocialScience.TransactionsinGIS,9(1), p.13. Openshaw,S.etal.,1987.AMarkIGeographicalAnalysisMachinefortheAutomatedAnalysisof PointDatasets.InternationalJournalofGeographicalInformationSystems,1(4),p Wheeler,D.&Tiefelsdorf,M.,2005.Multicollinearityandcorrelationamonglocalregression coefficientsingeographicallyweightedregression.journalofgeographicalsystems,7(2), p