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1、MultivariateRegressionDepthMarshallBern?DavidEppstein?AbstractTheregressiondepthofahyperplanewithrespecttoasetofnpointsinRdistheminimumnumberofpointsthehyperplanemustpassthroughinarotationtovertical.Wegeneralizehyperplaneregressiondepthtok-?atsforanykbetween0andd?1.Thek=0casegivestheclassicalnotio
2、nofcenterpoints.Weprovethatforanykandd,deepk-?atsexist,thatis,foranysetofnpointstherealwaysexistsak-?atwithdepthatleastaconstantfractionofn.Asaconsequence,wederivealinear-time(1+?)-approximationalgorithmforthedeepest?at.1IntroductionLinearregressionasksforanaf?nesubspace(a?at)that?tsasetofdatapoin
3、ts.Themostfa-miliarcaseassumesd?1independentorexplanatoryvariablesandonedependentorresponsevariable,and?tsahyperplanetoexplainthedependentvariableasalinearfunctionoftheinde-pendentvariables.Quiteoften,however,theremaybemorethanonedependentvariable,andthemultivariateregressionproblemrequires?ttinga
4、lower-dimensional?attothedatapoints,perhapsevenasuccessionof?atsofincreasingdimensions.Multivariateleast-squaresregressioniseasilysolvedbytreatingeachdependentvariableseparately,butthisisnotcorrectforothercommonformsofregressionsuchasleastabsolutedeviation[8]orleastmedianofsquares[12].Rousseeuwand
5、Hubert[14]introducedthenotionofregressiondepthasarobustcriterionforlinearregression.TheregressiondepthofahyperplaneH?ttingasetofnpointsistheminimumnumberofpointswhoseremovalmakesHintoanon?t.Anon?tisahyperplanethatcanberotatedtovertical(thatis,paralleltothedependentvariable'saxis)withoutpassingthro
6、ughanypoints.Theintuitionbehindthisde?nitionisthataverticalhyperplanepositsnorelationshipbetweenthearXiv:cs/9912013v1[cs.CG]20Dec1999dependentandindependentvariables,andhencemanypointsshouldhavetobeinvalidatedinordertomakeagoodregressionhyperplanecombinatoriallyequivalenttoaverticalhyperplane.Sinc
7、ethisde?nitiondoesnotmakeuseofthesizeoftheresiduals,butonlyusestheirsigns,itisrobustinthefaceofskewedorheteroskedastic(data-dependent)errormodels.Regressiondepthalsohasanumberofothernicepropertiesincludinginvaria
