x0000ytrainasmatrixConcretetrain9ကxtestasmatrixConcretetrain18ကytestasmatrixConcretetrain9ကdrmavemaveytrainxtrainmethodMEANOPGmaxdim8ကdrmaveCallma ID: 818459
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ShortIntroductiontotheUsageofPakageMAVEW
ShortIntroductiontotheUsageofPakageMAVEWeiqiangHangHongfanZhangYingcunXiaNationalUniversityofSingapore,SingaporeOctober18,20191IntroductionPackageMAVEprovidesseveralmethods,includingMAVEandOPGmethodsproposedby[4,5,6],tondthecentralspace(CS)andthecentralmeanspace(CMS).Italsoimplementsslicedinverseregressionofakernelversion;see[1,4].Formaldenitionofthecentralspaceandthecentralmeanspacecanbefoundin[2,3].Forcomparison,apackagedrinCRANalsocontainsothersucientdimensionreductionmethods[7].ThemainpartofpackageMAVEiswritteninC++basedonRcppArmadillopackage.Ifthereisanyproblemduringinstallation,pleaseupdateyourRcpppackageandinstallRcppArmadillopackageandtryagain.2UsageTheprimaryfunctioninthispackageisMAVE.TheinputargumentsincludeannpcovariatematrixX,ann1respondmatrixY,andthemethodargumentfordimensionreduction.Theoptionsforthemethodargumentare'csopg','cs-mave','meanopg','meanmave'and'ksir',andthedefaultis'csopg'.'csopg'and'cs-mave'aremethodsofndingCSbyOPGandMAVErespectively,'meanopg'and'meanmave'aremethodsofndingCMSbyOPGandMAVE,'ksir'istheslicedinverseregressionofkernelversion.SinceOPGmethodistime-savingcomparedwithMAVEandtheresultofOPGisasgoodasthatofMAVE,werecommend
usingOPG.Argumentmax.dimsetsthemaximumdi
usingOPG.Argumentmax.dimsetsthemaximumdimensionformavetocompute.Thedefaultvalueis10meaningthatitwillonlycalculatedimensionreductionspacesofdimensionupto10.Itwillhelpusersavecomputationtimewhenthedataishighdimensional.Wewilluseexamplestoillustratetheusageofthepackage.�library(MAVE)�data(Concrete)�set.seed(1234)�train-sample(1:1030)[1:500]ကx.train-as.matrix(Concrete[train,1:8])1�y.train-as.matrix(Concrete[train,9])ကx.test-as.matrix(Concrete[-train,1:8])ကy.test-as.matrix(Concrete[-train,9])ကdr.mave-mave(y.train~x.train,method='MEANOPG',max.dim=8)ကdr.maveCall:mave(formula=y.train~x.train,method="MEANOPG",max.dim=8)centralmeanspaceofdimensions12345678arecomputedTheobjectreturnedbymaveormave.computecontainsinformationofcall,dataandbasismatricesofdimensionreductionspaceswithdifferentdimensions.Thebasismatrixofagivendimension,2,forexample,canbeobtainedbyကdir2-coef(dr.mave,dim=2)Thenthereduceddataisobtainedfromoriginaldatamultipliedbythebasismatrixofdimensionreductionspace.Thereduceddatacanbecalculatedbymave.data.Thefollowingisanexampletoapplymarsinpacakgemdatothereduceddataforprediction.ကlibrary(mda)ကx.train.mave-mave.data(dr.mave,x=x.train,dim=
2)ကx.test.mave-mave.data(dr.mave,x=x.t
2)ကx.test.mave-mave.data(dr.mave,x=x.test,dim=2)ကmodel.mars-mars(x.train.mave,y.train,degree=2)ကy.pred.mars-predict(model.mars,x.test.mave)ကmean((y.pred.mars-y.test)^2)[1]113.0373Forconvenience,thepackageprovidespredicttoimplementtheaboveprocedurewithsomemodifications.Theargumentdegreewillbepassedtomarswhichspecifiesthemaximuminteractiondegree.Moreargumentslikethreshorpenaltycanbepassedtomarsbyplacingthemafterdiminthepredictmethod.ကy.pred-predict(dr.mave,newx=x.test,dim=2,degree=2)ကmean((y.pred-y.test)^2)[1]88.48931InMAVEpackageofversion1.3.8,mavefunctionallowsmutiplereponse,whichmeansthatargumentycananqmatrix.mave.dimimplementstheselectionofdimensionoftheCSorCMSdiscussedinsection??.Itreturnsanobjectwithadditionalinformationofcross-validationvaluesofdifferentdimensions.Belowisasimpleexampletoillustrateitsusage.2�set.seed(12345)�n=800�x-matrix(rnorm(n*5),n,5)ကbeta1-matrix(c(0.717,0.717,0,0,0))ကbeta2-matrix(c(0,0,0.717,0.717,0))ကbeta3-matrix(c(0,0,0,0,1))ကerr1-matrix(rnorm(n))ကerr2-matrix(rnorm(n))ကy1-as.matrix((x%*%beta1)/(1+2*(x%*%beta2)^2)+(x%*%beta3)*err1)ကy2-as.matrix((x%*%beta3)^2)+err2ကy=cbind(y1,y2)ကdr.mave-mave(y~x,method='CSO
PG')ကdr.mave.dim-mave.dim(dr.mave)ကd
PG')ကdr.mave.dim-mave.dim(dr.mave)ကdr.mave.dimCall:mave.dim(dr=dr.mave)Thecross-validationisrunondimensionsof012345Dimension012345CV-value0.290.260.240.250.260.27Theselecteddimensionofcentralspaceis2Thecodebelowcanbeusedtofindtheselecteddimensionwithminimumcross-validationvalue.ကwhich.min(dr.mave.dim$cv)[1]2Fromtheresult,theestimateddimensionreductionspacewithdimension3hastheminimumcross-validationvalue.TheestimatedbasisvectorsofCSofdimension3canbeaccessedbycoefmethod.Fromtheresult,theestimatedbasisvectorfallsinthelinearspacegeneratedby(1;2;3)withsmalldeviation.Althoughinthisexample,theestimatedbasisvectorsarecloseto(1;2;3),weshouldnotethattheoriginalbasisvectorslike(1;2;3)areunidentifiable,onlythespacegeneratedbythetheoriginalbasisvectorsisidentifiable.ကcoef(dr.mave,dim=3)dir1dir2dir3x1-0.0022907430.71487066-0.12210258x20.0056505790.69502519-0.01138889x3-0.006105577-0.07085736-0.63225614x40.034851445-0.02713024-0.76417882x50.9993552530.011962970.035272503InMAVEpackageofversion1.3.8,weusescreeningmethodtoselectimportvariablesinhighdimensionaldata.Thedefaultnumberofvariablesretainedafterscreeningisn=log(n).Thefollowingisanexampleaboutit
.�set.seed(12345)�n-200က
.�set.seed(12345)�n-200ကp-500ကx-matrix(rnorm(n*p),n,p)ကy-x[,1]+x[,2]+rnorm(n)ကdr.mave-mave(y~x,method='MEANOPG')screeningmethodisusingtoselectimportvariables.ကdr.mave.dim-mave.dim(dr.mave)ကdr.mave.dimCall:mave.dim(dr=dr.mave)Thecross-validationisrunondimensionsof012345678910Dimension012345678910CV-value1.470.730.620.630.740.870.921.021.071.111.14Theselecteddimensionofcentralmeanspaceis2References[1]Li,K.C.(1991).Slicedinverseregressionfordimensionreduction.JournaloftheAmericanStatisticalAssociation,86(414),316-327.[2]Cook,R.D.(1998),RegressionGraphics.NewYork:Wiley[3]Cook,R.D.,andLi,B.(2002).Dimensionreductionforconditionalmeaninregression.AnnalsofStatistics,455-474.[4]Xia,Y.,Tong,H.,Li,W.K.,andZhu,L.X.(2002).Anadaptiveestimationofdimensionreductionspace.JournaloftheRoyalStatisticalSociety:SeriesB(StatisticalMethodology),64(3),363-410.[5]Xia,Y.(2007)Aconstructiveapproachtotheestimationofdimensionreductiondirections.AnnalsofStatistics,35(3),2654-26904[6]Wang,H.,andXia,Y.(2008).Slicedregressionfordimensionreduction.JournaloftheAmericanStatisticalAssociation,103(482),811-821.[7]Weisberg,S.(2002).DimensionreductionregressioninR.JournalofStatisticalSoftware,7(1),1-22