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1、TheAnnalsofStatistics2004,Vol.32,No.2,407499©InstituteofMathematicalStatistics,2004LEASTANGLEREGRESSIONBYBRADLEYEFRON,1TREVORHASTIE,2IAINJOHNSTONE34ANDROBERTTIBSHIRANIStanfordUniversityThepurposeofmodelselectionalgorithmssuchasAllSubsets,ForwardSelectionan
2、dBackwardEliminationistochoosealinearmodelonthebasisofthesamesetofdatatowhichthemodelwillbeapplied.Typicallywehaveavailablealargecollectionofpossiblecovariatesfromwhichwehopetoselectaparsimonioussetfortheefficientpredictionofaresponsevariable.LeastAngleRegr
3、ession(LARS),anewmodelselectionalgorithm,isausefulandlessgreedyversionoftraditionalforwardselectionmethods.Threemainpropertiesarederived:(1)AsimplemodificationoftheLARSalgorithmimplementstheLasso,anattractiveversionofordinaryleastsquaresthatconstrainsthesum
4、oftheabsoluteregressioncoefficients;theLARSmodificationcalculatesallpossibleLassoestimatesforagivenproblem,usinganorderofmagnitudelesscomputertimethanpreviousmethods.(2)AdifferentLARSmodificationefficientlyimplementsForwardStagewiselinearregression,anotherprom
5、isingnewmodelselectionmethod;thisconnectionexplainsthesimilarnumericalresultspreviouslyobservedfortheLassoandStagewise,andhelpsusunderstandthepropertiesofbothmethods,whichareseenasconstrainedversionsofthesimplerLARSalgorithm.(3)Asimpleapproximationforthede
6、greesoffreedomofaLARSestimateisavailable,fromwhichwederiveaCpestimateofpredictionerror;thisallowsaprincipledchoiceamongtherangeofpossibleLARSestimates.LARSanditsvariantsarecomputationallyefficient:thepaperdescribesapubliclyavailablealgorithmthatrequiresonly
7、thesameorderofmagnitudeofcomputationaleffortasordinaryleastsquaresappliedtothefullsetofcovariates.1.Introduction.Automaticmodel-buildingalgorithmsarefamiliar,andsometimesnotorious,inthelinearmodelliterature:ForwardSelection,BackwardElimination,AllSubsetsre
8、gressionandvariouscombinationsareusedtoauto-maticallyproducegoodlinearmodelsforpredictingaresponseyonthebasisofsomemeasuredcovariatesx1,x2,...,xm.Goodnessisoftendefinedintermsofpredictionaccuracy,butparsimonyi
