multivariate regression depthnew

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1、MultivariateRegressionDepthMarshallBern∗DavidEppstein†AbstractTheregressiondepthofahyperplanewithrespecttoasetofnpointsinRdistheminimumnumberofpointsthehyperplanemustpassthroughinarotationtovertical.Wegeneralizehyperplaneregressiondepthtok-flatsforanykbetween0andd−1.Thek=0casegivestheclassicalnotio

2、nofcenterpoints.Weprovethatforanykandd,deepk-flatsexist,thatis,foranysetofnpointstherealwaysexistsak-flatwithdepthatleastaconstantfractionofn.Asaconsequence,wederivealinear-time(1+ǫ)-approximationalgorithmforthedeepestflat.1IntroductionLinearregressionasksforanaffinesubspace(aflat)thatfitsasetofdatapoin

3、ts.Themostfa-miliarcaseassumesd−1independentorexplanatoryvariablesandonedependentorresponsevariable,andfitsahyperplanetoexplainthedependentvariableasalinearfunctionoftheinde-pendentvariables.Quiteoften,however,theremaybemorethanonedependentvariable,andthemultivariateregressionproblemrequiresfittinga

4、lower-dimensionalflattothedatapoints,perhapsevenasuccessionofflatsofincreasingdimensions.Multivariateleast-squaresregressioniseasilysolvedbytreatingeachdependentvariableseparately,butthisisnotcorrectforothercommonformsofregressionsuchasleastabsolutedeviation[8]orleastmedianofsquares[12].Rousseeuwand

5、Hubert[14]introducedthenotionofregressiondepthasarobustcriterionforlinearregression.TheregressiondepthofahyperplaneHfittingasetofnpointsistheminimumnumberofpointswhoseremovalmakesHintoanonfit.Anonfitisahyperplanethatcanberotatedtovertical(thatis,paralleltothedependentvariable'saxis)withoutpassingthro

6、ughanypoints.TheintuitionbehindthisdefinitionisthataverticalhyperplanepositsnorelationshipbetweenthearXiv:cs/9912013v1[cs.CG]20Dec1999dependentandindependentvariables,andhencemanypointsshouldhavetobeinvalidatedinordertomakeagoodregressionhyperplanecombinatoriallyequivalenttoaverticalhyperplane.Sinc

7、ethisdefinitiondoesnotmakeuseofthesizeoftheresiduals,butonlyusestheirsigns,itisrobustinthefaceofskewedorheteroskedastic(data-dependent)errormodels.Regressiondepthalsohasanumberofothernicepropertiesincludinginvaria