空間統(tǒng)計ch9variogram-課件

Chapter9–(Semi)VariogrammodelsGivenageostatisticalmodel,Z(s),itsvariogramg(h)isformallydefinedaswheref(s,u)isthejointprobabilitydensityfunctionofZ(s)andZ(u).Foranintrinsicrandomfield,thevariogramcanbeestimatedusingthemethodofmomentsestimator,asfollows:wherehisthedistanceseparatingsamplelocationssiandsi+h,N(h)isthenumberofdistinctdatapairs.Insomecircumstances,itmaybedesirabletoconsiderdirectioninadditiontodistance.Inisotropiccase,hshouldbewrittenasascalarh,representingmagnitude.Note:Inliteraturethetermsvariogramandsemivariogramareoftenusedinterchangeably.Bydefinitiong(h)issemivariogramandthevariogramis2g(h).1Chapter9–(Semi)VariogrammoRobustvariogramestimatorVariogramprovidesanimportanttoolfordescribinghowthespatialdataarerelatedwithdistance.Aswehaveseenitisdefinedintermsofdissimilarityindatavaluesbetweentwolocationsseparatedbyadistanceh.Itisnotedthatthemomentestimatorgiveninthepreviouspageissensitivetooutliersinthedata.Thus,sometimesrobustestimatorsareused.ThewidelyusedrobustestimatorisgivenbyCressieandHawkins(1980):ThemotivationbehindthisestimatoristhatforaGaussianprocess,wehaveBasedontheBox-Coxtransformation,itisfoundthatthefourth-rootof12ismorenormallydistributed.*Cressie,N.andHawkins,D.M.1980.Robustestimationofthevariogram,I.JournaloftheInternationalAssociationforMathematicalGeology12:115-125.2Robustvariogramestimator2VariogramparametersThemaingoalofavariogramanalysisistoconstructavariogramthatbestestimatestheautocorrelationstructureoftheunderlyingstochasticprocess.Atypicalvariogramcanbedescribedusingthreeparameters:Nuggeteffect–representsmicro-scalevariationormeasurementerror.Itisestimatedfromtheempiricalvariogram

ath

=0.Range–isthedistanceatwhichthevariogram reachestheplateau,i.e.,thedistance(ifany) atwhichdataarenolongercorrelated.Sill–isthevarianceoftherandomfieldV(Z), disregardingthespatialstructure.Itisthe plateauwherethevariogramreachesatthe range,g(range).hg(h)02468100.0rangeh=5nugget=0.2sill=1.03Variogramparametershg(h)024685m20°20°0.5m0.5mSettingvariogramparametersConstructionofavariogramrequiresconsiderationofafewthings:Anappropriatelagincrementforh–Itdefinesthedistanceatwhichthevariogramiscalculated.Atoleranceforthelagincrement–Itestablishesdistancebinsforthelagincrementstoaccommodateunevenlyspacedobservations.Thenumberoflagsoverwhichthevariogramwillbecalculated–Thenumberoflagsinconjunctionwiththesizeofthelagincrementwilldefinethetotaldistanceoverwhichavariogramiscalculated.Atoleranceforangle–Itdetermineshowwidethebinswillspan.Twopracticalrules:Itisrecommendedthathischosenassuch thatthenumberofpairsisgreaterthan30.2. Thedistanceofreliabilityforan experimentalvariogramish<D/2,where

Disthemaximumdistanceoverthefieldofdata.45m20°20°0.5m0.5mSettingvarComputingvariogramsAnexperimentalvariogramiscalculatedusingtheRfunction(inpackagegstat):variogrm(pH~1,loc~gx+gy,soil87.dat) #gx:listorvectorofx-coordinates #gy:listorvectorofy-coordinates #pH:listorvectorofaresponsevariable5Computingvariograms5CovariogramandCorrelogramCovariogram(analogoustocovariance)andcorrelogram(analogoustocorrelationcoefficient)areanothertwousefulmethodsformeasuringspatialcorrelation.Theydescribesimilarityinvaluesbetweentwolocations.Covariogram:Itsestimator:whereisthesamplemean.Ath=0,?(0)issimplythefinitevarianceoftherandomfield.Itisstraightforwardtoestablishtherelationship:Thecorrelogramisdefinedas6CovariogramandCorrelogram6PropertiesofthemomentestimatorforvariogramItisunbiased:IfZ(s)isergodic, asn

.Thismeansthatthemomentestimatorapproachesthetruevalueforthevariogramasthesizeoftheregionincreases.Theestimatorisconsistent.Themomentestimatorconvergesindistributiontoanormaldistributionasn

,i.e.,itisapproximatelynormallydistributedforlargesamples.ForGaussianprocesses,theapproximatevariance-covariancematrixofisavailable(Cressie1985).*Cressie,N.1985.Fittingvariogrammodelsbyweightedleastsquares.MathematicalGeology17:563-586.7PropertiesofthemomentestimPropertiesofthemomentestimatorforcovarianceThecovariance:C(h)=cov(Z(si),Z(si+h))Themomentestimator:Properties:Themomentestimatorforthecovarianceisbiased.Thebiasarisesbecausethecovariancefunctionfortheresiduals, isnotthesameasthecovariancefunctionfortheerrors,Forasecond-orderstationaryrandomfield,themomentestimatorforthecovarianceisconsistent:?(h)C(h)almostsurelyasn.However,theconvergenceisslowerthanthevarigogram.Forasecond-orderstationaryrandomfield,themomentestimatorisapproximatelynormallydistributed.Properties1and2arethereasonswhythevariogramispreferredoverthecovariancefunction(andcorrelogram)inmodelinggeostatisticaldata.8PropertiesofthemomentestimpHFsrfxpyp01002003004000.00.4ypxp01002003004000.00.000.25VariogrambeforedetrendingVariogramafterdetrending01002003004005000200400600800010020030040050002004006008009pHFsrfxpyp01002003004000.00.10VariogrammodelsTherearetworeasonsweneedtofitamodeltotheempiricalvariogram:Spatialprediction(kriging)requiresestimatesofthevariogramg(h)forthoseh’swhicharenotavailableinthedata.Theempiricalvariogramcannotguaranteethevarianceofpredictedvaluestobepositive.Avariogrammodelcanensureapositivevariance.Variousparametricvariogrammodelshavebeenusedintheliterature.Thefollowsaresomeofthemostpopularones.Linearmodel–wherec0isthenuggeteffect.Thelinearvariogramhasnosill,andsothevarianceoftheprocessisinfinite.Theexistenceofalinearvariogramsuggestsatrendinthedata,soyoushouldconsiderfittingatrendtothedata,modelingthedataasafunctionofthecoordinates(trendsurfaceanalysis).hg(h)10Variogrammodelshg(h)10Powermodel-wherec0isthenuggeteffect.Thepowervariogramhasnosill,sothevarianceoftheprocessisinfinite.Thelinearvariogramisaspecialcaseofthepowermodel.Similarly,theexistenceofalinearvariogramsuggestsatrendinthedata,soyoushouldconsiderfittingatrendtothedata,modelingthedataasafunctionofthecoordinates(trendsurfaceanalysis).hg(h)a<1a>111Powermodel-hg(h)a<1a>11Exponentialmodel-wherec0isthenuggeteffect.Thesillisc0+c1.Therangefortheexponentialmodelisdefinedtobe3aatwhichthevariogramisof95%ofthesill.Gaussianmodel-wherec0isthenuggeteffect.c0+c1isthesill.Therangeis3a.ThismodeldescribesarandomfieldthatisconsideredtobetoosmoothandpossessesthepeculiarpropertythatZ(s)canbepredictedwithouterrorforanysontheplane.hg(h)hg(h)12Exponentialmodel-hg(h)hg(h)Logisticmodel(rationalquadraticmodel)-wherec0isthenuggeteffect.Thesillisc0+a/b.TherangefortheexponentialmodelisSphericalmodel-wherec0isthenuggeteffect.Thesillisc0+c1.Therangeforthesphericalmodelcanbecomputedbysettingg(h)=0.95(c0+c1).for0

h

aforh

ahg(h)hg(h)13Logisticmodel(rationalquadrParameterestimationTherearecommonlytwowaystofitavariogrammodeltoanempiricalvariogram.Assumethevariogrammodelg(h;q),whereqisanunknownparametervector.Forexample,fortheexponentialvariogrammodelq=(c0,c1,a).Ordinaryleastsquaresmethod–TheOLSestimatorforqisobtainedbyfindingthatminimizesTheOLSestimationcanbeeasilyimplementedinRusingfunctionoptimornls.Initialvaluesforqarerequired,thesevaluescanbeobtainedfromtheempiricalvariogram.Notes:

OLSestimationassumesthat -doesnotdependonthelagdistancehi -forallpairsoflagdistanceshihi.Bothassumptionsareviolated.Thevarianceandthecovariancedependonthenumberofpairsofsitesusedtocomputetheempiricalvariogram(seeCressie1985).Theseviolationsdonotcontributesignificantlytothebiasoftheparameterestimation.14Parameterestimation14WeightedleastsquaresestimatorTheWLSestimatorforqisobtainedbyfindingthatminimizeswhereSothat,TonotethattheWLSestimatorismoreprecise(hasasmallervariance)thantheOLSestimator.Modelselectioncriteria:SelectamodelwiththesmallestresidualsumofsquaresorAICorlog-likelihoodratio,butpayaparticularlyattentiontothegoodness-of-fitatshortdistancelags(importantforefficientspatialprediction).15WeightedleastsquaresestimatSplusimplementationFitvar.s(dt,c,sill,range,model,wt=F) #dt:listofdistancesandsamplevariogramsobtainedfromthefunction'variogrm' #c:initialestimateofthenuggeteffect #sill:initialestimateofthesill #range:initialestimateoftherange #model: ="exp",fitsanexponentialmodeltothesamplevariogram# ="gau",fitsaGaussianmodeltothesamplevariogram # ="sph",fitsasphericalmodeltothesamplevariogram # ="lin",fitsalinearmodeltothesamplevariogrampH.variog_variogrm(soil87.dat$gx,soil87.dat$gy,soil87.dat[,5],nint=30,dmax=400)pH.exp_fitvar.s(pH.variog,0.12,0.22,300,model="exp")x_seq(0,380,1)lines(expvar(x,pH.exp),col=5)Note:

Inthefunctionfitvar,wt=F(i.e.,OLS)eachsampleequallycontributestotheobjectivefunctionQ(q),whilewt=T(i.e.,WLS)Q(q)isweightedinproportiontothenumberofobs.usedincomputingthesamplevariance.Thus,locationsbasedonafewobs.willnotcarryasmuchweightcomparedtotheonebasedonalargenumberofobs.16Splusimplementation16Fractals–TheconceptofdimensionGeometricobjectsaretraditionallyviewedandmeasuredintheEuclideanspace,e.g.,line,rectangleandcube,withdimensionD=1,2,and3,respectively.However,manyphenomenainnature(e.g.,clouds,snowflakes,treearchitecture)cannotbesatisfactorilydescribedusingEuclideandimensions.Todescribetheirregularityofsuchgeometricobjects(irregulargeometricobjectsarecalledfractals),weneedtogeneralizetheconceptofEuclideandimension.TheHausdorffDimension–IfwetakeanobjectresidinginEuclideandimensionDandreduceitslinearsizeby1/rineachspatialdirection,thenumberofreplicasoftheoriginalobjectwouldincreasetoN=rDtimes.D=log(N)/log(r),istheHausdorffdimension,namedaftertheGermanmathematician,FelixHausdorff.TheimportantpointisthatinfractaldimensionDneednotbeaninteger,itcouldbeafraction.Ithasprovedusefulfordescribingnaturalobjects.D=1D=2D=3r=1r=2r=3N=1N=1N=1N=4N=8N=2N=3N=9N=2717Fractals–TheconceptofdimeExamplesofgeometricobjectswithnon-integerdimensions1.Cantorset(dust)–Beginwithalineoflength1,calledinitiator.Thenremovethemiddlethirdoftheline,thisstepiscalledthegenerator,becauseitspecifiesarulethatisusedtogenerateanewform.Thegeneratorcoulditerativelyinfinitelybeappliedtotheremainingsegmentssothattogenerateasetof“dust”.Thedustsareobviouslyneitherpointsnorlines,butlaysomewherebetweenthem,thushasadimensionbetween0and1:D=log(N)/log(r)=log(2)/log(3)=0.6309.2.Kochcurve–D=log(4)/log(3)=1.2618.InitiatorGenerator3.Sierpinskitriangle–D=log(3)/log(2)=1.585018ExamplesofgeometricobjectsSelf-similarityandsmoothnessAnimportantpropertyofafractalisself-similarity,whichreferstoaninfinitenestingofstructureonallscales.Itmeansthatasubstructureresemblestheformofitssuperstructure,e.g.,leafshaperesemblesbranchshape,whereasbranchresemblestreeshape.AnotherimportantwaytounderstandfractaldimensionisthatDisasmoothnessmeasureofaspatialprocess/object(e.g.,surfacesmoothness/roughness).WhenD=1(aline),or=2(aplane),theobjectsaresmooth.ForthoseobjectswhoseD’sarebetween1or2(e.g.,KochcurveorSierpinskitriangle),theirsmoothnessvariesbetweenalineandaplane.Studysurfacegrowthandsmoothnessisincreasinglybecominganimportantphysicandbiologicalsubjects.Ithasmuchtodowithfractalgeometryandspatialstatistics.Anexampleisatechnology,calledmolecularbeamepitaxy,usedtomanufacturethinfilmsforcomputerchipsandothersemiconductordevices.Itisaprocesstodepositsiliconmoleculestocreateaverysmoothsisurface.*Manderlbrot,B.B.1982.ThefractalgeometryofNature.Freeman,SanFrancisco.*Meakin,P.2019.Fractals,scalingandgrowthfarfromequilibrium.CambridgeU.Press.*Barabási,A.-L.andStanley,H.E.2019.Fractalconceptsinsurfacegrowth.CambridgeU.Press.19Self-similarityandsmoothnessCalculatingfractaldimensionfromavariogramBecausethesmoothnessofaspatialprocessisdirectlyrelatedtothesmoothnessofthecovariancefunctionath

0,afractaldimensioncanbecalculatedfromavariogram.IfthenwesaytheprocessZ(s)iscontinuous.Foracontinuouscovariance,wehave orWhereo(ha)isatermofsmallerorderthanhaforhatneighborhood0.ThefractaldimensionofthesurfaceisD=2–a/2.acanbeestimatedfromanempiricalvariogramasfollows: log(g(h))=log(b)+alog(h).*Davies,S.&Hall,P.2019.Fractalanalysisofsurfaceroughnessbyusingspatialdata(withDiscussion).JRSS,B.61:3-37.*Palmer,M.W.1988.Fractalgeometry:atoolfordescribingspatialpatternsofplantcommunities.Vegetation75:91-102.20CalculatingfractaldimensionChapter9–(Semi)VariogrammodelsGivenageostatisticalmodel,Z(s),itsvariogramg(h)isformallydefinedaswheref(s,u)isthejointprobabilitydensityfunctionofZ(s)andZ(u).Foranintrinsicrandomfield,thevariogramcanbeestimatedusingthemethodofmomentsestimator,asfollows:wherehisthedistanceseparatingsamplelocationssiandsi+h,N(h)isthenumberofdistinctdatapairs.Insomecircumstances,itmaybedesirabletoconsiderdirectioninadditiontodistance.Inisotropiccase,hshouldbewrittenasascalarh,representingmagnitude.Note:Inliteraturethetermsvariogramandsemivariogramareoftenusedinterchangeably.Bydefinitiong(h)issemivariogramandthevariogramis2g(h).21Chapter9–(Semi)VariogrammoRobustvariogramestimatorVariogramprovidesanimportanttoolfordescribinghowthespatialdataarerelatedwithdistance.Aswehaveseenitisdefinedintermsofdissimilarityindatavaluesbetweentwolocationsseparatedbyadistanceh.Itisnotedthatthemomentestimatorgiveninthepreviouspageissensitivetooutliersinthedata.Thus,sometimesrobustestimatorsareused.ThewidelyusedrobustestimatorisgivenbyCressieandHawkins(1980):ThemotivationbehindthisestimatoristhatforaGaussianprocess,wehaveBasedontheBox-Coxtransformation,itisfoundthatthefourth-rootof12ismorenormallydistributed.*Cressie,N.andHawkins,D.M.1980.Robustestimationofthevariogram,I.JournaloftheInternationalAssociationforMathematicalGeology12:115-125.22Robustvariogramestimator2VariogramparametersThemaingoalofavariogramanalysisistoconstructavariogramthatbestestimatestheautocorrelationstructureoftheunderlyingstochasticprocess.Atypicalvariogramcanbedescribedusingthreeparameters:Nuggeteffect–representsmicro-scalevariationormeasurementerror.Itisestimatedfromtheempiricalvariogram

ath

=0.Range–isthedistanceatwhichthevariogram reachestheplateau,i.e.,thedistance(ifany) atwhichdataarenolongercorrelated.Sill–isthevarianceoftherandomfieldV(Z), disregardingthespatialstructure.Itisthe plateauwherethevariogramreachesatthe range,g(range).hg(h)02468100.0rangeh=5nugget=0.2sill=1.023Variogramparametershg(h)024685m20°20°0.5m0.5mSettingvariogramparametersConstructionofavariogramrequiresconsiderationofafewthings:Anappropriatelagincrementforh–Itdefinesthedistanceatwhichthevariogramiscalculated.Atoleranceforthelagincrement–Itestablishesdistancebinsforthelagincrementstoaccommodateunevenlyspacedobservations.Thenumberoflagsoverwhichthevariogramwillbecalculated–Thenumberoflagsinconjunctionwiththesizeofthelagincrementwilldefinethetotaldistanceoverwhichavariogramiscalculated.Atoleranceforangle–Itdetermineshowwidethebinswillspan.Twopracticalrules:Itisrecommendedthathischosenassuch thatthenumberofpairsisgreaterthan30.2. Thedistanceofreliabilityforan experimentalvariogramish<D/2,where

Disthemaximumdistanceoverthefieldofdata.245m20°20°0.5m0.5mSettingvarComputingvariogramsAnexperimentalvariogramiscalculatedusingtheRfunction(inpackagegstat):variogrm(pH~1,loc~gx+gy,soil87.dat) #gx:listorvectorofx-coordinates #gy:listorvectorofy-coordinates #pH:listorvectorofaresponsevariable25Computingvariograms5CovariogramandCorrelogramCovariogram(analogoustocovariance)andcorrelogram(analogoustocorrelationcoefficient)areanothertwousefulmethodsformeasuringspatialcorrelation.Theydescribesimilarityinvaluesbetweentwolocations.Covariogram:Itsestimator:whereisthesamplemean.Ath=0,?(0)issimplythefinitevarianceoftherandomfield.Itisstraightforwardtoestablishtherelationship:Thecorrelogramisdefinedas26CovariogramandCorrelogram6PropertiesofthemomentestimatorforvariogramItisunbiased:IfZ(s)isergodic, asn

.Thismeansthatthemomentestimatorapproachesthetruevalueforthevariogramasthesizeoftheregionincreases.Theestimatorisconsistent.Themomentestimatorconvergesindistributiontoanormaldistributionasn

,i.e.,itisapproximatelynormallydistributedforlargesamples.ForGaussianprocesses,theapproximatevariance-covariancematrixofisavailable(Cressie1985).*Cressie,N.1985.Fittingvariogrammodelsbyweightedleastsquares.MathematicalGeology17:563-586.27PropertiesofthemomentestimPropertiesofthemomentestimatorforcovarianceThecovariance:C(h)=cov(Z(si),Z(si+h))Themomentestimator:Properties:Themomentestimatorforthecovarianceisbiased.Thebiasarisesbecausethecovariancefunctionfortheresiduals, isnotthesameasthecovariancefunctionfortheerrors,Forasecond-orderstationaryrandomfield,themomentestimatorforthecovarianceisconsistent:?(h)C(h)almostsurelyasn.However,theconvergenceisslowerthanthevarigogram.Forasecond-orderstationaryrandomfield,themomentestimatorisapproximatelynormallydistributed.Properties1and2arethereasonswhythevariogramispreferredoverthecovariancefunction(andcorrelogram)inmodelinggeostatisticaldata.28PropertiesofthemomentestimpHFsrfxpyp01002003004000.00.4ypxp01002003004000.00.000.25VariogrambeforedetrendingVariogramafterdetrending010020030040050002004006008000100200300400500020040060080029pHFsrfxpyp01002003004000.00.10VariogrammodelsTherearetworeasonsweneedtofitamodeltotheempiricalvariogram:Spatialprediction(kriging)requiresestimatesofthevariogramg(h)forthoseh’swhicharenotavailableinthedata.Theempiricalvariogramcannotguaranteethevarianceofpredictedvaluestobepositive.Avariogrammodelcanensureapositivevariance.Variousparametricvariogrammodelshavebeenusedintheliterature.Thefollowsaresomeofthemostpopularones.Linearmodel–wherec0isthenuggeteffect.Thelinearvariogramhasnosill,andsothevarianceoftheprocessisinfinite.Theexistenceofalinearvariogramsuggestsatrendinthedata,soyoushouldconsiderfittingatrendtothedata,modelingthedataasafunctionofthecoordinates(trendsurfaceanalysis).hg(h)30Variogrammodelshg(h)10Powermodel-wherec0isthenuggeteffect.Thepowervariogramhasnosill,sothevarianceoftheprocessisinfinite.Thelinearvariogramisaspecialcaseofthepowermodel.Similarly,theexistenceofalinearvariogramsuggestsatrendinthedata,soyoushouldconsiderfittingatrendtothedata,modelingthedataasafunctionofthecoordinates(trendsurfaceanalysis).hg(h)a<1a>131Powermodel-hg(h)a<1a>11Exponentialmodel-wherec0isthenuggeteffect.Thesillisc0+c1.Therangefortheexponentialmodelisdefinedtobe3aatwhichthevariogramisof95%ofthesill.Gaussianmodel-wherec0isthenuggeteffect.c0+c1isthesill.Therangeis3a.ThismodeldescribesarandomfieldthatisconsideredtobetoosmoothandpossessesthepeculiarpropertythatZ(s)canbepredictedwithouterrorforanysontheplane.hg(h)hg(h)32Exponentialmodel-hg(h)hg(h)Logisticmodel(rationalquadraticmodel)-wherec0isthenuggeteffect.Thesillisc0+a/b.TherangefortheexponentialmodelisSphericalmodel-wherec0isthenuggeteffect.Thesillisc0+c1.Therangeforthesphericalmodelcanbecomputedbysettingg(h)=0.95(c0+c1).for0

h

aforh

ahg(h)hg(h)33Logisticmodel(rationalquadrParameterestimationTherearecommonlytwowaystofitavariogrammodeltoanempiricalvariogram.Assumethevariogrammodelg(h;q),whereqisanunknownparametervector.Forexample,fortheexponentialvariogrammodelq=(c0,c1,a).Ordinaryleastsquaresmethod–TheOLSestimatorforqisobtainedbyfindingthatminimizesTheOLSestimationcanbeeasilyimplementedinRusingfunctionoptimornls.Initialvaluesforqarerequired,thesevaluescanbeobtainedfromtheempiricalvariogram.Notes:

OLSestimationassumesthat -doesnotdependonthelagdistancehi -forallpairsoflagdistanceshihi.Bothassumptionsareviolated.Thevarianceandthecovariancedependonthenumberofpairsofsitesusedtocomputetheempiricalvariogram(seeCressie1985).Theseviolationsdonotcontributesignificantlytothebiasoftheparameterestimation.34Parameterestimation14WeightedleastsquaresestimatorTheWLSestimatorforqisobtainedbyfindingthatminimizeswhereSothat,TonotethattheWLSestimatorismoreprecise(hasasmallervariance)thantheOLSestimator.Modelselectioncriteria:SelectamodelwiththesmallestresidualsumofsquaresorAICorlog-likelihoodratio,butpayaparticularlyattentiontothegoodness-of-fitatshortdistancelags(importantforefficientspatialprediction).35WeightedleastsquaresestimatSplusimplementationFitvar.s(dt,c,sill,range,model,wt=F) #dt:listofdistancesandsamplevariogramsobtainedfromthefunction'variogrm' #c:initialestimateofthenuggeteffect #sill:initialestimateofthesill #range:initialestimateoftherange #model: ="exp",fitsanexponentialmodeltothesamplevariogram# ="gau",fitsaGaussianmodeltothesamplevariogram # ="sph",fitsasphericalmodeltothesamplevariogram # ="lin",fitsalinearmodeltothesamplevariogrampH.variog_variogrm(soil87.dat$gx,soil87.dat$gy,soil87.dat[,5],nint=30,dmax=400)pH.exp_fitvar.s(pH.variog,0.12,0.22,300,model="exp")x_seq(0,380,1)lines(expvar(x,pH.exp),col=5)Note:

Inthefunctionfitvar,wt=F(i.e.,OLS)eachsampleequallycontributestotheobjectivefunctionQ(q),whilewt=T(i.e.,WLS)Q(q)isweightedinproportiontothenumberofobs.usedincomputingthesamplevariance.Thus,locationsbasedonafewobs.willnotcarryasmuchweightcomparedtotheonebasedonalargenumberofobs.36Splusimplementation16Fractals–TheconceptofdimensionGeometricobjectsaretraditionallyviewedandmeasuredintheEuclideanspace,e.g.,line,rectangleandcube,withdimensionD=1,2,and3,respectively.However,manyphenomenainnature(e.g.,clouds,snowflakes,treearchitecture)cannotbesatisfactorilydescribedusingEuclideandimensions.Todescribetheirregularityofsuchgeometricobjects(irregulargeometricobjectsarecalledfractals),weneedtogeneralizetheconceptofEuclideandimension.TheHausdorffDimension–IfwetakeanobjectresidinginEuclideandimensionDandreduceitslinearsizeby1/rineachspatialdirection,thenumberofreplicasoftheoriginalobjectwouldincreasetoN=rDtimes.D=log(N)/log(r),istheHausdorffdimension,namedaftertheGermanmathematician,FelixHausdorff.TheimportantpointisthatinfractaldimensionDneednotbeaninteger,itcouldbeafraction.Ithasprovedusefulfordescribingnaturalobjects.D=1D=2D=3r=1r=2r=3N=1N=1N=1N=4N=8N=2N=3N=9N=2737Fractals–Theconceptof