Package`huge'April1,2020TypePackageTitleHigh-DimensionalUndirectedGrap - PDF document

Package`huge'April1,2020TypePackageTitle
Package`huge'April1,2020TypePackageTitleHigh-DimensionalUndirectedGraphEstimationVersion1.3.4.1AuthorHaomingJiang,XinyuFei,HanLiu,KathrynRoeder,JohnLafferty,LarryWasserman,XingguoLi,andTuoZhaoMaintainerHaomingJiang&#xjian;&#xghm.;&#xustc;&#x@gma;&#xil.c;&#xom00;DependsR&#xjian;&#xghm.;&#xustc;&#x@gma;&#xil.c;&#xom00;(=3.0.0)ImportsMatrix,igraph,MASS,grDevices,graphics,methods,stats,utils,RcppLinkingToRcpp,RcppEigenDescriptionProvidesageneralframeworkforhigh-dimensionalundirectedgraphestimation.Itintegratesdatapreprocessing,neighborhoodscreening,graphestimation,andmodelselectiontechniquesintoapipeline.Inpreprocessingstage,thenonparanormal(npn)transformationisappliedtohelprelaxthenormalityassumption.Inthegraphestimationstage,thegraphstructureisestimatedbyMeinshausen-Buhlmanngraphestimationorthegraphicallasso,andbothmethodscanbefurtheracceleratedbythelossyscreeningrulepreselectingtheneighborhoodofeachvariablebycorrelationthresholding.Wetargetonhigh-dimensionaldataanalysisusuallyd&#xjian;&#xghm.;&#xustc;&#x@gma;&#xil.c;&#xom00;&#xjian;&#xghm.;&#xustc;&#x@gma;&#xil.c;&#xom00;n,andthecomputationismemory-optimizedusingthesparsematrixoutput.Wealsoprovideacomputationallyefcientapproach,correlationthresholdinggraphestimation.Threeregularization/thresholdingparameterselectionmethodsareincludedinthispackage:(1)stabilityapproachforr

egularizationselection(2)rotationinforma
egularizationselection(2)rotationinformationcriterion(3)extendedBayesianinformationcriterionwhichisonlyavailableforthegraphicallasso.LicenseGPL-2RepositoryCRANNeedsCompilationyes12huge-packageRoxygenNote6.1.1EncodingUTF-8Date/Publication2020-04-0105:40:25UTCRtopicsdocumented:huge-package........................................2huge.............................................4huge.ct............................................7huge.generator.......................................8huge.glasso.........................................10huge.inference........................................11huge.mb...........................................13huge.npn...........................................14huge.plot..........................................15huge.roc...........................................16huge.select.........................................17huge.tiger..........................................19plot.huge..........................................20plot.roc...........................................21plot.select..........................................22plot.sim...........................................22print.huge..........................................23print.roc...........................................23print.select..........................................24print.sim...........................................

24stockdata.............................
24stockdata..........................................25Index26huge-packageHigh-DimensionalUndirectedGraphEstimationDescriptionApackageforhigh-dimensionalundirectedgraphestimationDetailsPackage:hugeType:PackageVersion:1.2.7Date:2015-09-14License:GPL-2LazyLoad:yeshuge-package3Thepackage"huge"provides8mainfunctions:(1)thedatageneratorcreatesrandomsamplesfrommultivariatenormaldistributionswithdifferentgraphstructures.Pleaserefertohuge.generator.(2)thenonparanormal(npn)transformationhelpsrelaxthenormalityassumption.Pleaserefertohuge.npn.(3)Thecorrelationthresholdinggraphestimation.Pleaserefertohuge.(4)TheMeinshausen-Buhlmanngraphestimation.Pleaserefertohuge.(5)ThegraphicalLassoalgorithmusinglosslessscreeningrule.Pleasereferandhuge.**Both(4)and(5)canbefurtheracceleratedbythelossyscreeningrulepreselectingtheneigh-borhoodofeachnodeviathresholdingsamplecorrelation.(6)Themodelselectionusingthestabilityapproachtoregularizationselection.Pleaserefertohuge.select.(7)Themodelselectionusingtherotationinformationcriterion.Pleaserefertohuge.select.(8)ThemodelselectionusingtheextendedBayesianinformationcriterion.Pleaserefertohuge.select.Author(s)TuoZhao,HanLiu,HaomingJiang,KathrynRoeder,JohnLafferty,andLarryWassermanMaintainers:HaomingJiang&#xhjia;&#xng98;&#x@g50;atech.edu;References1.T.ZhaoandH.Liu.ThehugePackageforHigh-dimensionalUndirec

tedGraphEstimationinR.JournalofMachineLe
tedGraphEstimationinR.JournalofMachineLearningResearch,20122.H.Liu,F.Han,M.Yuan,J.LaffertyandL.Wasserman.HighDimensionalSemiparametricGaussianCopulaGraphicalModels.AnnalsofStatistics,20123.D.WittenandJ.Friedman.Newinsightsandfastercomputationsforthegraphicallasso.Jour-nalofComputationalandGraphicalStatistics,toappear,2011.4.HanLiu,KathrynRoederandLarryWasserman.StabilityApproachtoRegularizationSelection(StARS)forHighDimensionalGraphicalModels.AdvancesinNeuralInformationProcessingSystems,2010.5.R.FoygelandM.Drton.Extendedbayesianinformationcriteriaforgaussiangraphicalmodels.AdvancesinNeuralInformationProcessingSystems,2010.6.H.Liu,J.LaffertyandL.Wasserman.TheNonparanormal:SemiparametricEstimationofHighDimensionalUndirectedGraphs.JournalofMachineLearningResearch,20097.J.FanandJ.Lv.Sureindependencescreeningforultra-highdimensionalfeaturespace(withdiscussion).JournalofRoyalStatisticalSocietyB,2008.8.O.Banerjee,L.E.Ghaoui,A.d'Aspremont:ModelSelectionThroughSparseMaximumLikeli-hoodEstimationforMultivariateGaussianorBinaryData.JournalofMachineLearningResearch,2008.9.J.Friedman,T.HastieandR.Tibshirani.RegularizationPathsforGeneralizedLinearModelsviaCoordinateDescent.JournalofStatisticalSoftware,2008.10.J.Friedman,T.HastieandR.Tibshirani.Sparseinversecovarianceestimationwiththelasso,Biostatistics,2007.11.N.MeinshausenandP.Buhlmann.H

igh-dimensionalGraphsandVariableSelectio
igh-dimensionalGraphsandVariableSelectionwiththeLasso.TheAnnalsofStatistics,2006.4hugeSeeAlsohuge.generator,huge.npn,huge,huge.plotandhuge.rochugeHigh-dimensionalundirectedgraphestimationDescriptionThemainfunctionforhigh-dimensionalundirectedgraphestimation.Threegraphestimationmeth-ods,including(1)Meinshausen-Buhlmanngraphestimation(mb)(2)graphicallasso(glasso)(3)correlationthresholdinggraphestimation(ct)and(4)tuning-insensitivegraphestimation(tiger),areavailablefordataanalysis.Usagehuge(x,lambda=NULL,nlambda=NULL,lambda.min.ratio=NULL,method="mb",scr=NULL,scr.num=NULL,cov.output=FALSE,sym="or",verbose=TRUE)ArgumentsxThereare2options:(1)xisannbyddatamatrix(2)adbydsamplecovariancematrix.Theprogramautomaticallyidentiestheinputmatrixbycheckingthesymmetry.(nisthesamplesizeanddisthedimension).lambdaAsequenceofdecreasingpositivenumberstocontroltheregularizationwhenmethod="mb","glasso"or"tiger",orthethresholdinginmethod="ct".Typicalusageistoleavetheinputlambda=NULLandhavetheprogramcomputeitsownlambdasequencebasedonnlambdaandlambda.min.ratio.Userscanalsospecifyasequencetooverridethis.Whenmethod="mb","glasso"or"tiger",usewithcare-itisbettertosupplyadecreasingsequencevaluesthanasingle(small)value.nlambdaThenumberofregularization/thresholdingparameters.Thedefaultvalueis30formethod="ct"and10formethod="mb","glasso"or"tiger".lambd

a.min.ratioIfmethod="mb","glasso"or"tige
a.min.ratioIfmethod="mb","glasso"or"tiger",itisthesmallestvalueforlambda,asafractionoftheupperbound(MAX)oftheregularization/thresholdingpa-rameterwhichmakesallestimatesequalto0.Theprogramcanautomati-callygeneratelambdaasasequenceoflength=nlambdastartingfromMAXtolambda.min.ratio*MAXinlogscale.Ifmethod="ct",itisthelargestsparsitylevelforestimatedgraphs.Theprogramcanautomaticallygeneratelambdaasasequenceoflength=nlambda,whichmakesthesparsitylevelofthegraphpathincreasesfrom0tolambda.min.ratioevenly.Thedefaultvalueis0.1whenmethod="mb","glasso"or"tiger",and0.05method="ct".methodGraphestimationmethodswith4options:"mb","ct","glasso"and"tiger".Thedefaultvalueis"mb".huge5scrIfscr=TRUE,thelossyscreeningruleisappliedtopreselecttheneighborhoodbeforethegraphestimation.ThedefaultvalueisFALSE.NOTapplicablewhenmethod="ct","mb",or"tiger".scr.numTheneighborhoodsizeafterthelossyscreeningrule(thenumberofremainingneighborspernode).ONLYapplicablewhenscr=TRUE.Thedefaultvalueisn-1.Analternativevalueisn/log(n).ONLYapplicablewhenscr=TRUEandmethod="mb".cov.outputIfcov.output=TRUE,theoutputwillincludeapathofestimatedcovariancematrices.ONLYapplicablewhenmethod="glasso".Sincetheestimatedco-variancematricesaregenerallynotsparse,pleaseuseitwithcare,oritmaytakemuchmemoryunderhigh-dimensionalsetting.ThedefaultvalueisFALSE.symSymmetrizetheoutputgraphs.Ifsym=

"and",theedgebetweennodeiandnodejisselec
"and",theedgebetweennodeiandnodejisselectedONLYwhenbothnodeiandnodejareselectedasneighborsforeachother.Ifsym="or",theedgeisselectedwheneithernodeiornodejisselectedastheneighborforeachother.Thedefaultvalueis"or".ONLYapplicablewhenmethod="mb"or"tiger".verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.DetailsThegraphstructureisestimatedbyMeinshausen-Buhlmanngraphestimationorthegraphicallasso,andbothmethodscanbefurtheracceleratedviathelossyscreeningrulebypreselectingtheneigh-borhoodofeachvariablebycorrelationthresholding.Wetargetonhigh-dimensionaldataanalysisusuallyd»n,andthecomputationismemory-optimizedusingthesparsematrixoutput.Wealsoprovideahighlycomputationallyefcientapproachescorrelationthresholdinggraphestimation.ValueAnobjectwithS3class"huge"isreturned:dataThenbyddatamatrixordbydsamplecovariancematrixfromtheinputcov.inputAnindicatorofthesamplecovariance.ind.matThescr.numbykmatrixwitheachcolumncorrespondingtoavariableinind.groupandcontainstheindicesoftheremainingneighborsaftertheGSS.ONLYapplicablewhenscr=TRUEandapprox=FALSElambdaThesequenceofregularizationparametersusedinmborthresholdingparame-tersinct.symThesymfromtheinput.ONLYapplicablewhenmethod="mb"or"tiger".scrThescrfromtheinput.ONLYapplicablewhenmethod="mb"or"glasso".pathAlistofkbykadjacencymatricesofestimatedgraphsasagraphpathco

rre-spondingtolambda.sparsityThesparsity
rre-spondingtolambda.sparsityThesparsitylevelsofthegraphpath.icovAlistofdbydprecisionmatricesasanalternativegraphpath(numericalpath)correspondingtolambda.ONLYapplicablewhenmethod="glasso"or"tiger".6hugecovAlistofdbydestimatedcovariancematricescorrespondingtolambda.ONLYapplicablewhencov.output=TRUEandmethod="glasso"methodThemethodusedinthegraphestimationstage.dfIfmethod="mb"or"tiger",itisakbynlambdamatrix.Eachrowcontainsthenumberofnonzerocoefcientsalongthelassosolutionpath.Ifmethod="glasso",itisanlambdadimensionalvectorcontainingthenumberofnonzerocoefcientsalongthegraphpathicov.loglikAnlambdadimensionalvectorcontainingthelikelihoodscoresalongthegraphpath(icov).ONLYapplicablewhenmethod="glasso".Foranestimatedin-versecovarianceZ,theprogramonlycalculateslog(det(Z))-trace(SZ)whereSistheempiricalcovariancematrix.Forthelikelihoodfornobservations,pleasemultiplybyn/2.NoteThisfunctionONLYestimatesthegraphpath.Formoreinformationabouttheoptimalgraphselec-tion,pleaserefertohuge.select.SeeAlsohuge.generator,huge.select,huge.plot,huge.roc,andhuge-package.Examples#generatedataL=huge.generator(n=50,d=12,graph="hub",g=4)#graphpathestimationusingmbout1=huge(L$data)out1plot(out1)#Notalignedplot(out1,align=TRUE)#Alignedhuge.plot(out1$path[[3]])#graphpathestimationusingthesamplecovariancematrixastheinput.#out1=huge(cor(L$data),method="g

lasso")#out1#plot(out1)#Notaligned#plot(
lasso")#out1#plot(out1)#Notaligned#plot(out1,align=TRUE)#Aligned#huge.plot(out1$path[[3]])#graphpathestimationusingct#out2=huge(L$data,method="ct")#out2#plot(out2)#graphpathestimationusingglasso#out3=huge(L$data,method="glasso")#out3huge.ct7#plot(out3)#graphpathestimationusingtiger#out4=huge(L$data,method="tiger")#out4#plot(out4)huge.ctGraphestimationviacorrelationthresholding(ct)DescriptionSeemoredetailsinhugeUsagehuge.ct(x,nlambda=NULL,lambda.min.ratio=NULL,lambda=NULL,verbose=TRUE)ArgumentsxThereare2options:(1)xisannbyddatamatrix(2)adbydsamplecovariancematrix.Theprogramautomaticallyidentiestheinputmatrixbycheckingthesymmetry.(nisthesamplesizeanddisthedimension).nlambdaThenumberofregularization/thresholdingparameters.Thedefaultvalueis30formethod="ct"and10formethod="mb","glasso"or"tiger".lambda.min.ratioIfmethod="mb","glasso"or"tiger",itisthesmallestvalueforlambda,asafractionoftheupperbound(MAX)oftheregularization/thresholdingpa-rameterwhichmakesallestimatesequalto0.Theprogramcanautomati-callygeneratelambdaasasequenceoflength=nlambdastartingfromMAXtolambda.min.ratio*MAXinlogscale.Ifmethod="ct",itisthelargestsparsitylevelforestimatedgraphs.Theprogramcanautomaticallygeneratelambdaasasequenceoflength=nlambda,whichmakesthesparsitylevelofthegraphpathincreasesfrom0tolambda.min.ratioevenly.Thedefaultvalueis0.1whenmethod="m

b","glasso"or"tiger",and0.05method="ct".
b","glasso"or"tiger",and0.05method="ct".lambdaAsequenceofdecreasingpositivenumberstocontroltheregularizationwhenmethod="mb","glasso"or"tiger",orthethresholdinginmethod="ct".Typicalusageistoleavetheinputlambda=NULLandhavetheprogramcomputeitsownlambdasequencebasedonnlambdaandlambda.min.ratio.Userscanalsospecifyasequencetooverridethis.Whenmethod="mb","glasso"or"tiger",usewithcare-itisbettertosupplyadecreasingsequencevaluesthanasingle(small)value.verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.8huge.generatorSeeAlsohuge,andhuge-package.huge.generatorDatageneratorDescriptionImplementsthedatagenerationfrommultivariatenormaldistributionswithdifferentgraphstruc-tures,including"random","hub","cluster","band"and"scale-free".Usagehuge.generator(n=200,d=50,graph="random",v=NULL,u=NULL,g=NULL,prob=NULL,vis=FALSE,verbose=TRUE)ArgumentsnThenumberofobservations(samplesize).Thedefaultvalueis200.dThenumberofvariables(dimension).Thedefaultvalueis50.graphThegraphstructurewith4options:"random","hub","cluster","band"and"scale-free".vTheoff-diagonalelementsoftheprecisionmatrix,controllingthemagnitudeofpartialcorrelationswithu.Thedefaultvalueis0.3.uApositivenumberbeingaddedtothediagonalelementsoftheprecisionmatrix,tocontrolthemagnitudeofpartialcorrelations.Thedefaultvalueis0.1.gFor"cluster"or"hub"graph,gisthenu

mberofhubsorclustersinthegraph.Thedefaul
mberofhubsorclustersinthegraph.Thedefaultvalueisaboutd/20ifd�=40and2ifd40.For"band"graph,gisthebandwidthandthedefaultvalueis1.NOTapplicableto"random"graph.probFor"random"graph,itistheprobabilitythatapairofnodeshasanedge.Thedefaultvalueis3/d.For"cluster"graph,itistheprobabilitythatapairofnodeshasanedgeineachcluster.Thedefaultvalueis6*g/difd/g30and0.3ifd/g&#x=-25;�30.NOTapplicableto"hub"or"band"graphs.visVisualizetheadjacencymatrixofthetruegraphstructure,thegraphpattern,thecovariancematrixandtheempiricalcovariancematrix.ThedefaultvalueisFALSEverboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.huge.generator9DetailsGiventheadjacencymatrixtheta,thegraphpatternsaregeneratedasbelow:(I)"random":Eachpairofoff-diagonalelementsarerandomlysettheta[i,j]=theta[j,i]=1fori!=jwithprobabilityprob,and0otherwise.Itresultsinaboutd*(d-1)*prob/2edgesinthegraph.(II)"hub":Therow/columnsareevenlypartitionedintogdisjointgroups.Eachgroupisassociatedwitha"center"rowiinthatgroup.Eachpairofoff-diagonalelementsaresettheta[i,j]=theta[j,i]=1fori!=jifjalsobelongstothesamegroupasiand0otherwise.Itresultsind-gedgesinthegraph.(III)"cluster":Therow/columnsareevenlypartitionedintogdisjointgroups.Eachpairofoff-diagonalelementsaresettheta[i,j]=theta[j,i]=1fori!=jwiththeprobabilityprobifbothiandjbelongtothesamegroup,and0o

therwise.Itresultsinaboutg*(d/g)*(d/g-1)
therwise.Itresultsinaboutg*(d/g)*(d/g-1)*prob/2edgesinthegraph.(IV)"band":Theoff-diagonalelementsaresettobetheta[i,j]=1if1and0otherwise.Itresultsin(2d-1-g)*g/2edgesinthegraph.(V)"scale-free":ThegraphisgeneratedusingB-Aalgorithm.Theinitialgraphhastwocon-nectednodesandeachnewnodeisconnectedtoonlyonenodeintheexistinggraphwiththeprobabilityproportionaltothedegreeoftheeachnodeintheexistinggraph.Itresultsindedgesinthegraph.Theadjacencymatrixthetahasalldiagonalelementsequalto0.Toobtainapositivedeniteprecisionmatrix,thesmallesteigenvalueoftheta*v(denotedbye)iscomputed.Thenwesettheprecisionmatrixequaltotheta*v+(|e|+0.1+u)I.Thecovariancematrixisthencomputedtogeneratemultivariatenormaldata.ValueAnobjectwithS3class"sim"isreturned:dataThenbydmatrixforthegenerateddatasigmaThecovariancematrixforthegenerateddataomegaTheprecisionmatrixforthegenerateddatasigmahatTheempiricalcovariancematrixforthegenerateddatathetaTheadjacencymatrixoftruegraphstructure(insparsematrixrepresentation)forthegenerateddataSeeAlsohugeandhuge-packageExamples##bandgraphwithbandwidth3L=huge.generator(graph="band",g=3)10huge.glassoplot(L)##randomsparsegraphL=huge.generator(vis=TRUE)##randomdensegraphL=huge.generator(prob=0.5,vis=TRUE)##hubgraphwith6hubsL=huge.generator(graph="hub",g=6,vis=TRUE)##hubgraphwith8clustersL=huge.generator(graph="cluster",g=8,vis=TRUE)##s

cale-freegraphsL=huge.generator(graph="s
cale-freegraphsL=huge.generator(graph="scale-free",vis=TRUE)huge.glassoThegraphicallasso(glasso)usingsparsematrixoutputDescriptionSeemoredetailsinhugeUsagehuge.glasso(x,lambda=NULL,lambda.min.ratio=NULL,nlambda=NULL,scr=NULL,cov.output=FALSE,verbose=TRUE)ArgumentsxThereare2options:(1)xisannbyddatamatrix(2)adbydsamplecovariancematrix.Theprogramautomaticallyidentiestheinputmatrixbycheckingthesymmetry.(nisthesamplesizeanddisthedimension).lambdaAsequenceofdecreasingpositivenumberstocontroltheregularizationwhenmethod="mb","glasso"or"tiger",orthethresholdinginmethod="ct".Typicalusageistoleavetheinputlambda=NULLandhavetheprogramcomputeitsownlambdasequencebasedonnlambdaandlambda.min.ratio.Userscanalsospecifyasequencetooverridethis.Whenmethod="mb","glasso"or"tiger",usewithcare-itisbettertosupplyadecreasingsequencevaluesthanasingle(small)value.lambda.min.ratioIfmethod="mb","glasso"or"tiger",itisthesmallestvalueforlambda,asafractionoftheupperbound(MAX)oftheregularization/thresholdingpa-rameterwhichmakesallestimatesequalto0.Theprogramcanautomati-callygeneratelambdaasasequenceoflength=nlambdastartingfromMAXtolambda.min.ratio*MAXinlogscale.Ifmethod="ct",itisthelargestsparsitylevelforestimatedgraphs.Theprogramcanautomaticallygeneratelambdaasahuge.inference11sequenceoflength=nlambda,whichmakesthesparsitylevelofthegraphpathincreasesf

rom0tolambda.min.ratioevenly.Thedefaultv
rom0tolambda.min.ratioevenly.Thedefaultvalueis0.1whenmethod="mb","glasso"or"tiger",and0.05method="ct".nlambdaThenumberofregularization/thresholdingparameters.Thedefaultvalueis30formethod="ct"and10formethod="mb","glasso"or"tiger".scrIfscr=TRUE,thelossyscreeningruleisappliedtopreselecttheneighborhoodbeforethegraphestimation.ThedefaultvalueisFALSE.NOTapplicablewhenmethod="ct","mb",or"tiger".cov.outputIfcov.output=TRUE,theoutputwillincludeapathofestimatedcovariancematrices.ONLYapplicablewhenmethod="glasso".Sincetheestimatedco-variancematricesaregenerallynotsparse,pleaseuseitwithcare,oritmaytakemuchmemoryunderhigh-dimensionalsetting.ThedefaultvalueisFALSE.verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.SeeAlsohuge,andhuge-package.huge.inferenceGraphinferenceDescriptionImplementstheinferenceforhighdimensionalgraphicalmodels,includingGaussianandNonpara-normalgraphicalmodelsWeconsidertheproblemsoftestingthepresenceofasingleedgeandthehypothesisisthattheedgeisabsent.Usagehuge.inference(data,T,adj,alpha=0.05,type="Gaussian",method="score")ArgumentsdataTheinputnbyddatamatrix(nisthesamplesizeanddisthedimension).TTheestimatedinverseofcorrelationmatrixofthedata.adjTheadjacencymatrixcorrespondingtothegraph.alphaThesignicancelevelofhypothesis.Thedefaultvalueis0.05.typeThetypeofinputdata.Thereare2optio

ns:"Gaussian"and"Nonparanormal".Thedefau
ns:"Gaussian"and"Nonparanormal".Thedefaultvalueis"Gaussian".methodWhenusingnonparanormalgraphicalmodel.Testmethodwith2options:"score"and"wald".Thedefaultvalueis"score".12huge.inferenceDetailsForNonparanormalgraphicalmodelweprovideScoretestmethodandWaldTest.HoweveritisreallyslowforinferencingonNonparanormalmodel,especiallyforlargedata.ValueAnobjectisreturned:dataThenbyddatamatrixfromtheinput.pThedbydp-valuematrixofhypothesis.errorThetypeIerrorofhypothesisatalphasignicancelevel.References1.QGu,YCao,YNing,HLiu.Localandglobalinferenceforhighdimensionalnonparanormalgraphicalmodels.2.JJankova,SVanDeGeer.Condenceintervalsforhigh-dimensionalinversecovarianceestima-tion.ElectronicJournalofStatistics,2015.SeeAlsohuge,andhuge-package.Examples#generatedataL=huge.generator(n=50,d=12,graph="hub",g=4)#graphpathestimationusingglassoest=huge(L$data,method="glasso")#inferenceofGaussiangraphicalmodelat0.05significancelevelT=tail(est$icov,1)[[1]]out1=huge.inference(L$data,T,L$theta)#inferenceofNonparanormalgraphicalmodelusingscoretestat0.05significancelevelT=tail(est$icov,1)[[1]]out2=huge.inference(L$data,T,L$theta,type="Nonparanormal")#inferenceofNonparanormalgraphicalmodelusingwaldtestat0.05significancelevelT=tail(est$icov,1)[[1]]out3=huge.inference(L$data,T,L$theta,type="Nonparanormal",method="wald")#inferenceofNonparanormalgraphic

almodelusingwaldtestat0.1significancelev
almodelusingwaldtestat0.1significancelevelT=tail(est$icov,1)[[1]]out4=huge.inference(L$data,T,L$theta,0.1,type="Nonparanormal",method="wald")huge.mb13huge.mbMeinshausen&BuhlmanngraphestimationDescriptionSeemoredetailsinhugeUsagehuge.mb(x,lambda=NULL,nlambda=NULL,lambda.min.ratio=NULL,scr=NULL,scr.num=NULL,idx.mat=NULL,sym="or",verbose=TRUE)ArgumentsxThereare2options:(1)xisannbyddatamatrix(2)adbydsamplecovariancematrix.Theprogramautomaticallyidentiestheinputmatrixbycheckingthesymmetry.(nisthesamplesizeanddisthedimension).lambdaAsequenceofdecreasingpositivenumberstocontroltheregularizationwhenmethod="mb","glasso"or"tiger",orthethresholdinginmethod="ct".Typicalusageistoleavetheinputlambda=NULLandhavetheprogramcomputeitsownlambdasequencebasedonnlambdaandlambda.min.ratio.Userscanalsospecifyasequencetooverridethis.Whenmethod="mb","glasso"or"tiger",usewithcare-itisbettertosupplyadecreasingsequencevaluesthanasingle(small)value.nlambdaThenumberofregularization/thresholdingparameters.Thedefaultvalueis30formethod="ct"and10formethod="mb","glasso"or"tiger".lambda.min.ratioIfmethod="mb","glasso"or"tiger",itisthesmallestvalueforlambda,asafractionoftheupperbound(MAX)oftheregularization/thresholdingpa-rameterwhichmakesallestimatesequalto0.Theprogramcanautomati-callygeneratelambdaasasequenceoflength=nlambdastartingfromMAXtolambda.min.

ratio*MAXinlogscale.Ifmethod="ct",itisth
ratio*MAXinlogscale.Ifmethod="ct",itisthelargestsparsitylevelforestimatedgraphs.Theprogramcanautomaticallygeneratelambdaasasequenceoflength=nlambda,whichmakesthesparsitylevelofthegraphpathincreasesfrom0tolambda.min.ratioevenly.Thedefaultvalueis0.1whenmethod="mb","glasso"or"tiger",and0.05method="ct".scrIfscr=TRUE,thelossyscreeningruleisappliedtopreselecttheneighborhoodbeforethegraphestimation.ThedefaultvalueisFALSE.NOTapplicablewhenmethod="ct","mb",or"tiger".scr.numTheneighborhoodsizeafterthelossyscreeningrule(thenumberofremainingneighborspernode).ONLYapplicablewhenscr=TRUE.Thedefaultvalueisn-1.Analternativevalueisn/log(n).ONLYapplicablewhenscr=TRUEandmethod="mb".idx.matIndexmatrixforscreening.14huge.npnsymSymmetrizetheoutputgraphs.Ifsym="and",theedgebetweennodeiandnodejisselectedONLYwhenbothnodeiandnodejareselectedasneighborsforeachother.Ifsym="or",theedgeisselectedwheneithernodeiornodejisselectedastheneighborforeachother.Thedefaultvalueis"or".ONLYapplicablewhenmethod="mb"or"tiger".verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.SeeAlsohuge,andhuge-package.huge.npnNonparanormal(npn)transformationDescriptionImplementstheGausianizationtohelprelaxtheassumptionofnormality.Usagehuge.npn(x,npn.func="shrinkage",npn.thresh=NULL,verbose=TRUE)ArgumentsxThenbyddatamatrixrepresentingnobservationsind

dimensionsnpn.funcThetransformationfunct
dimensionsnpn.funcThetransformationfunctionusedinthenpntransformation.Ifnpn.func="truncation",thetruncatedECDFisapplied.Ifnpn.func="shrinkage",theshrunkenECDFisapplied.Thedefaultis"shrinkage".Ifnpn.func="skeptic",thenonparanormalskepticisapplied.npn.threshThetruncationthresholdusedinnonparanormaltransformation,ONLYappli-cablewhennpn.func="truncation".Thedefaultvalueis1/(4*(n^0.25)*sqrt(pi*log(n))).verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.DetailsThenonparanormalextendsGaussiangraphicalmodelstosemiparametricGaussiancopulamod-els.Motivatedbysparseadditivemodels,thenonparanormalmethodestimatestheGaussiancopulabymarginallytransformingthevariablesusingsmoothfunctions.Computationally,theestimationofanonparanormaltransformationisveryefcientandonlyrequiresonepassofthedatamatrix.huge.plot15ValuedataAdbydnonparanormalcorrelationmatrixifnpn.func="skeptic",andAnbyddatamatrixrepresentingnobservationsindtransformeddimensionsotherwise.SeeAlsohugeandhuge-package.Examples#generatenonparanormaldataL=huge.generator(graph="cluster",g=5)L$data=L$data^5#transformthedatausingtheshrunkenECDFQ=huge.npn(L$data)#transformthenon-GaussiandatausingthetruncatedECDFQ=huge.npn(L$data,npn.func="truncation")#transformthenon-GaussiandatausingthetruncatedECDFQ=huge.npn(L$data,npn.func="skeptic")huge.plotGraphvisual

izationDescriptionImplementsthegraphvis
izationDescriptionImplementsthegraphvisualizationusingadjacencymatrix.Itcanautomaticorganize2Dembed-dinglayout.Usagehuge.plot(G,epsflag=FALSE,graph.name="default",cur.num=1,location=NULL)ArgumentsGTheadjacencymatrixcorrespondingtothegraph.epsflagIfepsflag=TRUE,savetheplotasanepsleinthetargetdirectory.ThedefaultvalueisFALSE.graph.nameThenameoftheoutputepsles.Thedefaultvalueis"default".cur.numThenumberofplotssavedasepsles.Onlyapplicalewhenepsflag=TRUE.Thedefaultvalueis1.locationTargetdirectory.Thedefaultvalueisthecurrentworkingdirectory.16huge.rocDetailsTheusercanchangecur.numtoplotseveralguresandselectthebestone.Theimplementationisbasedonthepopularpackage"igraph".SeeAlsohugeandhuge-package.Examples##visualizethehubgraphL=huge.generator(graph="hub")huge.plot(L$theta)##visualizethebandgraphL=huge.generator(graph="band",g=5)huge.plot(L$theta)##visualizetheclustergraphL=huge.generator(graph="cluster")huge.plot(L$theta)#showworkingdirectorygetwd()#plot5graphsandsavetheplotsasepsfilesintheworkingdirectoryhuge.plot(L$theta,epsflag=TRUE,cur.num=5)huge.rocDrawROCCurveforagraphpathDescriptionDrawsROCcurveforagraphpathaccordingtothetruegraphstructure.Usagehuge.roc(path,theta,verbose=TRUE)ArgumentspathAgraphpath.thetaThetruegraphstructure.verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTR

UE.DetailsToavoidthehorizontaloscillatio
UE.DetailsToavoidthehorizontaloscillation,falsepositiveratesisautomaticallysortedintheascentorderandtruepositiveratesalsofollowthesameorder.huge.select17ValueAnobjectwithS3class"roc"isreturned:F1TheF1scoresalongthegraphpath.tpThetruepositiveratesalongthegraphpathfpThefalsepositiveratesalongthegraphpathsAUCAreaundertheROCcurveNoteForalassoregression,thenumberofnonzerocoefcientsisatmostn-1.If��dn,evenwhenregularizationparameterisverysmall,theestimatedgraphmaystillbesparse.Inthiscase,theAUCmaynotbeagoodchoicetoevaluatetheperformance.SeeAlsohugeandhuge-package.Examples#generatedataL=huge.generator(d=200,graph="cluster",prob=0.3)out1=huge(L$data)#drawROCcurveZ1=huge.roc(out1$path,L$theta)#MaximumF1scoremax(Z1$F1)huge.selectModelselectionforhigh-dimensionalundirectedgraphestimationDescriptionImplementstheregularizationparameterselectionforhighdimensionalundirectedgraphestima-tion.Theoptionalapproachesarerotationinformationcriterion(ric),stabilityapproachtoregular-izationselection(stars)andextendedBayesianinformationcriterion(ebic).Usagehuge.select(est,criterion=NULL,ebic.gamma=0.5,stars.thresh=0.1,stars.subsample.ratio=NULL,rep.num=20,verbose=TRUE)18huge.selectArgumentsestAnobjectwithS3class"huge".criterionModelselectioncriterion."ric"and"stars"areavailableforall3graphes-timationmethods.ebicisonlyapplicablew

henest$method="glasso"inhuge().Thedefaul
henest$method="glasso"inhuge().Thedefaultvalueis"ric".ebic.gammaThetuningparameterforebic.Thedefaultvalueis0.5.Onlyapplicablewhenest$method="glasso"andcriterion="ebic".stars.threshThevariabilitythresholdinstars.Thedefaultvalueis0.1.Analternativevalueis0.05.Onlyapplicablewhencriterion="stars".stars.subsample.ratioThesubsamplingratio.Thedefaultvalueis10*sqrt(n)/nwhen�n144and0.8whenn,wherenisthesamplesize.Onlyapplicablewhencriterion="stars".rep.numThenumberofsubsamplingswhencriterion="stars"orrotationswhencriterion="ric".Thedefaultvalueis20.NOTapplicablewhencriterion="ebic".verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.DetailsStabilityapproachtoregularizationselection(stars)isanaturalwaytoselectoptimalregularizationparameterforallthreeestimationmethods.Itselectstheoptimalgraphbyvariabilityofsubsam-plingsandtendstooverselectedgesinGaussiangraphicalmodels.Besidesselectingtheregulariza-tionparameters,starscanalsoprovideanadditionalestimatedgraphbymergingthecorrespondingsubsampledgraphsusingthefrequencycounts.ThesubsamplingprocedureinstarsmayNOTbeveryefcient,wealsoprovidetherecentdevelopedhighlyefcient,rotationinformationcriterionapproach(ric).Insteadoftuningoveragridbycross-validationorsubsampling,wedirectlyes-timatetheoptimalregularizationparameterbasedonrandomRotations.However,

ricusuallyhasverygoodempiricalperformanc
ricusuallyhasverygoodempiricalperformancesbutsuffersfromunderselectionssometimes.Therefore,wesug-gestifuseraresensitiveoffalsenegativerates,theyshouldeitherconsiderincreasingr.numorapplyingthestarstomodelselection.ExtendedBayesianinformationcriterion(ebic)isanothercompetitiveapproach,buttheebic.gammacanonlybetunedbyexperience.ValueAnobjectwithS3class"select"isreturned:refitTheoptimalgraphselectedfromthegraphpathopt.icovTheoptimalprecisionmatrixfromthepathonlyapplicablewhenmethod="glasso"opt.covTheoptimalcovariancematrixfromthepathonlyapplicablewhenmethod="glasso"andest$covisavailable.mergeThegraphpathestimatedbymergingthesubsamplingpaths.Onlyapplicablewhentheinputcriterion="stars".huge.tiger19variabilityThevariabilityalongthesubsamplingpaths.Onlyapplicablewhentheinputcriterion="stars".ebic.scoresExtendedBICscoresforregularizationparameterselection.Onlyapplicablewhencriterion="ebic".opt.indexTheindexoftheselectedregularizationparameter.NOTapplicablewhentheinputcriterion="ric"opt.lambdaTheselectedregularization/thresholdingparameter.opt.sparsityThesparsitylevelof"refit".andanythingelseincludedintheinputestNoteThemodelselectionisNOTavailablewhenthedatainputisthesamplecovariancematrix.SeeAlsohugeandhuge-package.Examples#generatedataL=huge.generator(d=20,graph="hub")out.mb=huge(L$data)out.ct=huge(L$data,method="ct")out.glasso=h

uge(L$data,method="glasso")#modelselecti
uge(L$data,method="glasso")#modelselectionusingricout.select=huge.select(out.mb)plot(out.select)#modelselectionusingstars#out.select=huge.select(out.ct,criterion="stars",stars.thresh=0.05,rep.num=10)#plot(out.select)#modelselectionusingebicout.select=huge.select(out.glasso,criterion="ebic")plot(out.select)huge.tigerTuning-insensitivegraphestimationDescriptionSeemoredetailsinhuge20plot.hugeUsagehuge.tiger(x,lambda=NULL,nlambda=NULL,lambda.min.ratio=NULL,sym="or",verbose=TRUE)ArgumentsxThereare2options:(1)xisannbyddatamatrix(2)adbydsamplecovariancematrix.Theprogramautomaticallyidentiestheinputmatrixbycheckingthesymmetry.(nisthesamplesizeanddisthedimension).lambdaAsequenceofdecreasingpositivenumberstocontroltheregularizationwhenmethod="mb","glasso"or"tiger",orthethresholdinginmethod="ct".Typicalusageistoleavetheinputlambda=NULLandhavetheprogramcomputeitsownlambdasequencebasedonnlambdaandlambda.min.ratio.Userscanalsospecifyasequencetooverridethis.Whenmethod="mb","glasso"or"tiger",usewithcare-itisbettertosupplyadecreasingsequencevaluesthanasingle(small)value.nlambdaThenumberofregularization/thresholdingparameters.Thedefaultvalueis30formethod="ct"and10formethod="mb","glasso"or"tiger".lambda.min.ratioIfmethod="mb","glasso"or"tiger",itisthesmallestvalueforlambda,asafractionoftheupperbound(MAX)oftheregularization/thresholding

pa-rameterwhichmakesallestimatesequalto0
pa-rameterwhichmakesallestimatesequalto0.Theprogramcanautomati-callygeneratelambdaasasequenceoflength=nlambdastartingfromMAXtolambda.min.ratio*MAXinlogscale.Ifmethod="ct",itisthelargestsparsitylevelforestimatedgraphs.Theprogramcanautomaticallygeneratelambdaasasequenceoflength=nlambda,whichmakesthesparsitylevelofthegraphpathincreasesfrom0tolambda.min.ratioevenly.Thedefaultvalueis0.1whenmethod="mb","glasso"or"tiger",and0.05method="ct".symSymmetrizetheoutputgraphs.Ifsym="and",theedgebetweennodeiandnodejisselectedONLYwhenbothnodeiandnodejareselectedasneighborsforeachother.Ifsym="or",theedgeisselectedwheneithernodeiornodejisselectedastheneighborforeachother.Thedefaultvalueis"or".ONLYapplicablewhenmethod="mb"or"tiger".verboseIfverbose=FALSE,tracinginformationprintingisdisabled.ThedefaultvalueisTRUE.SeeAlsohuge,andhuge-package.plot.hugePlotfunctionforS3class"huge"DescriptionPlotsparsitylevelinformationand3typicalsparsegraphsfromthegraphpath.plot.roc21Usage##S3methodforclass'huge'plot(x,align=FALSE,...)ArgumentsxAnobjectwithS3class"huge"alignIfalign=FALSE,3plottedgraphsarealigned...Systemreserved(Nospecicusage)SeeAlsohugeplot.rocPlotfunctionforS3class"roc"DescriptionPlottheROCcurveforanobjectwithS3class"roc".Usage##S3methodforclass'roc'plot(x,...)ArgumentsxAnobjectwithS3class"roc"...Systemreserved(Nospecicusage)SeeAlso

huge.roc22plot.simplot.selectPlotfunct
huge.roc22plot.simplot.selectPlotfunctionforS3class"select"DescriptionPlottheoptimalgraphbymodelselection.Usage##S3methodforclass'select'plot(x,...)ArgumentsxAnobjectwithS3class"select"...Systemreserved(Nospecicusage)SeeAlsohuge.selectplot.simPlotfunctionforS3class"sim"DescriptionVisualizethecovariancematrix,theempiricalcovariancematrix,theadjacencymatrixandthegraphpatternofthetruegraphstructure.Usage##S3methodforclass'sim'plot(x,...)ArgumentsxAnobjectwithS3class"sim"...Systemreserved(Nospecicusage)SeeAlsohuge.generatorandhugeprint.huge23print.hugePrintfunctionforS3class"huge"DescriptionPrinttheinformationaboutthemodelusage,thegraphpathlength,graphdimension,sparsitylevel.Usage##S3methodforclass'huge'print(x,...)ArgumentsxAnobjectwithS3class"huge"....Systemreserved(Nospecicusage)SeeAlsohugeprint.rocPrintfunctionforS3class"roc"DescriptionPrinttheinformationabouttruepositiverates,falsepositiverates,theareaundercurveandmaxi-mumF1score.Usage##S3methodforclass'roc'print(x,...)ArgumentsxAnobjectwithS3class"roc"....Systemreserved(Nospecicusage)SeeAlsohuge.roc24print.simprint.selectPrintfunctionforS3class"select"DescriptionPrinttheinformationaboutthemodelusage,graphdimension,modelselectioncriterion,sparsityleveloftheoptimalgraph.Usage##S3methodforclass'select'print(x,...)ArgumentsxAnobjectwithS3class"sel

ect"....Systemreserved(Nospecicusage
ect"....Systemreserved(Nospecicusage)SeeAlsohuge.selectprint.simPrintfunctionforS3class"sim"DescriptionPrinttheinformationaboutthesamplesize,thedimension,thepatternandsparsityofthetruegraphstructure.Usage##S3methodforclass'sim'print(x,...)ArgumentsxAnobjectwithS3class"sim"....Systemreserved(Nospecicusage)SeeAlsohuge.generatorstockdata25stockdataStockpriceofS&P500companiesfrom2003to2008DescriptionThisdatasetconsistsofstockpriceandcompanyinformation.Usagedata(stockdata)FormatTheformatisalistcontainingcontainstwomatrices.1.data-1258x452,representsthe452stocks'closepricesfor1258tradingdays.2.info-452x3:The1stcolumn:thequerysymbolforeachcompany.The2ndcolumn:thecategoryforeachcompany.The3rdcolumn:thefullnameofeachcompany.DetailsThisdatasetcanbeusedtoperformhigh-dimensionalgraphestimationtoanalyzetherelationshipsbetweenS&P500companies.SourceItwaspubliclyavailableatnance.yahoo,whichisnowoutofdateExamplesdata(stockdata)image(stockdata$data)stockdata$infoIndexTopicdatasetsstockdata,25_PACKAGE(huge-package),2huge,3,4,4,7–17,19–23huge-package,2huge.ct,7huge.generator,3,4,6,8,22,24huge.glasso,10huge.inference,11huge.mb,13huge.npn,3,4,14huge.plot,4,6,15huge.roc,4,6,16,21,23huge.select,3,6,17,22,24huge.tiger,19plot.huge,20plot.roc,21plot.select,22plot.sim,22print.huge,23print.roc,23print.select,24print.sim,