应用多元统计分析1.ppt

Prefacetothe1stEdition,Mostoftheobservablephenomenafinminintheempirical(empirikl经验)sciencesareofamultivariatenature.Infinancialstudies,assetsinstockmarketsareobservedsimultaneouslyandtheirjointdevelopmentisanalyzedtobetterunderstandgeneraltendencies（趋势）andtotrackindices（路灯）.Theunderlyingtheoreticalstructureoftheseandmanyotherquantitativestudiesofappliedsciencesismultivariate.ThisbookonAppliedMultivariateStatisticalAnalysispresentsthetoolsandconceptsofmultivariatedataanalysiswithastrongfocusonapplications.Theaimofthebookistopresentmultivariatedataanalysisinawaythatisunderstandablefornon-mathematiciansandpractitionerswhoare（面对）bystatisticaldataanalysis.Thisisachievedbyfocusingonthepracticalrelevanceandthroughthee-bookcharacterofthistext.Allpracticalexamplesmayberecalculatedandmodifiedbythereaderusingastandardwebbrowserandwithoutreferenceorapplicationofanyspecificsoftware.,Mostoftheobservablephenomenafinminintheempirical(empirikl经验)sciencesareofamultivariatenature.Theunderlyingtheoreticalstructureoftheseandmanyotherquantitativestudiesofappliedsciencesismultivariate.ThisbookonAppliedMultivariateStatisticalAnalysispresentsthetoolsandconceptsofmultivariate,mltivereitdataanalysiswithastrongfocusonapplications.,Thebookisdividedintothreemainparts.Thefirstpartisdevotedtographicaltechniquesdescribingthedistributionsofthevariablesinvolved.Thesecondpartdealswithmultivariaterandomvariablesandpresentsfromatheoreticalpointofviewdistributions,estimatorsandtestsforvariouspracticalsituations.Thelastpartisonmultivariatetechniquesandintroducesthereadertothewideselectionoftoolsavailableformultivariatedataanalysis.Alldatasetsaregivenintheappendixandaredownloadablefromwww.md-.Thetextcontainsawidevarietyofexercisesthesolutionsofwhicharegiveninaseparatetextbook.Inadditionafullsetoftransparenciesonwww.md-isprovidedmakingiteasierforaninstructortopresentthematerialsinthisbook.Alltransparenciescontainhyperlinkstothestatisticalwebservicesothatstudentsandinstructorsalikemayrecomputeallexamplesviaastandardwebbrowser.,1-2week,UNIT-IDescriptiveTechniques(描述技术)1Comparison（对照）ofBatches1.1Boxplots41.2Histograms101.3Scatterplots171.4DataSet-BostonHousing35,1ComparisonofBatches,Multivariatestatisticalanalysisisconcernedwithanalyzingandunderstandingdatainhighdimensions.Wesupposethatwearegivenasetxini=1ofnobservationsofavariablevectorXinRp.Thatis,wesupposethateachobservationxihaspdimensions:xi=(xi1,xi2,.,xip),andthatitisanobservedvalueofavariablevectorXRp.Therefore,Xiscomposedofprandomvariables:X=(X1,X2,.,Xp)whereXj,forj=1,.,p,isaone-dimensionalrandomvariable.,1ComparisonofBatches,Multivariatestatisticalanalysisisconcernedwithanalyzingandunderstandingdatainhighdimensions.Howdowebegintoanalyzethiskindofdata?Beforeweinvestigatequestionsonwhatinferenceswecanreachfromthedata,weshouldthinkabouthowtolookatthedata.Thisinvolvesdescriptivetechniques.Questionsthatwecouldanswerbydescriptivetechniquesare:AretherecomponentsofXthataremorespreadoutthanothers?AretheresomeelementsofXthatindicatesubgroupsofthedata?ArethereoutliersinthecomponentsofX?How“normal”isthedistributionofthedata?,1.1Boxplots,1ComparisonofBatches,Genuinedenjuin真正的,X6,X1,Themedianandmeanbarsaremeasuresoflocations.Therelativelocationofthemedian(andthemean)intheboxisameasureofskewness.Thelengthoftheboxandwhiskersareameasureofspread.Thelengthofthewhiskersindicatethetaillengthofthedistribution.Theoutlyingpointsareindicatedwitha“”or“”dependingoniftheyareoutsideofFUL1.5dForFUL3dFrespectively.Theboxplotsdonotindicatemultimodalityorclusters.Ifwecomparetherelativesizeandlocationoftheboxes,wearecomparingdistributions.,Summary,Readingmaterial,1.2Histograms,h=0.4,Diagonal,Histogramsaredensity(denst)(密度)estimates(estimeits概算).Adensityestimategivesagoodimpressionofthedistributionofthedata.Incontrasttoboxplots,densityestimatesshowpossiblemultimodality(多模式；综合,mltimdliti)ofthedata.Theideaistolocallyrepresentthedatadensitybycountingthenumberofobservationsinasequenceofconsecutive（连续的）intervals(bins)（箱）withorigin（rn起源、原点）x0.LetBj(x0,h)denote(dinut,指示,表示)thebinoflengthhwhichistheelementofabingridstartingatx0:Bj(x0,h)=x0+(j1)h,x0+jh),jZ,where.,.)(squarebrackets)denotesaleftclosedandrightopeninterval(ntrvl间隔,右开区间).,Ifxini=1isani.i.d.samplewithdensityf,thehistogramisdefinedasfollows:Insum(1.7)thefirstindicatorfunctionIxiBj(x0,h)countsthenumberofobservationsfallingintobinBj(x0,h).ThesecondindicatorfunctionIisresponsiblefor“localizing”（luklizi局限）thecountsaroundx.Theparameterhisasmoothingorlocalizingparameterandcontrolsthewidth(wid)ofthehistogrambins.Anhthatistoolargeleadstoverybigblocksandthustoaveryunstructuredhistogram.Ontheotherhand,anhthatistoosmallgivesaveryvariableestimatewithmanyunimportantpeaks.,H=0.1,H=0.2,H=0.3,Diagonaldaignladj.对角线的,斜的n.对角线,斜线,H=0.4,TheeffectofhisgivenindetailinFigure1.6.Itcontainsthehistogram(upperleft)forthediagonalofthecounterfeitbanknotesforx0=137.8(theminimumoftheseobservations)andh=0.1.Increasinghtoh=0.2andusingthesameorigin,x0=137.8,resultsinthehistogramshowninthelowerleftofthefigure.Thisdensityhistogramissomewhatsmootherduetothelargerh.Thebinwidthisnextsettoh=0.3(upperright).Fromthishistogram,onehastheimpressionthatthedistributionofthediagonalisbimodalwithpeaksatabout138.5and139.9.Thedetectionofmodesrequiresafinetuningofthebinwidth.Usingmethodsfromsmoothingmethodology(medldi，n.方法学)onecanfindan“optimal”binwidthhfornobservations:,counterfeitkauntfitadj.假冒的,假装的,InFigure1.7,weshowhistogramswithx0=137.65(upperleft),x0=137.75(lowerleft),withx0=137.85(upperright),andx0=137.95(lowerright).Allthegraphshavebeenscaledequallyonthey-axistoallowcomparison.Oneseesthatdespitethefixedbinwidthhtheinterpretationisnotfacilitated(fsiliteitidvt.使容易).Theshiftoftheoriginx0(to4differentlocations)created4differenthistograms.Thispropertyofhistogramsstronglycontradictsthegoalofpresentingdatafeatures.,Modesofthedensityaredetectedwithahistogram.Modescorrespondtostrongpeaksinthehistogram.Histogramswiththesamehneednotbeidentical.Theyalsodependontheoriginx0ofthegrid.Theinfluenceoftheoriginx0isdrastic.Changingx0createsdifferentlookinghistograms.Theconsequenceofanhthatistoolargeisanunstructuredhistogramthatistooflat.Abinwidthhthatistoosmallresultsinanunstablehistogram.Thereisan“optimal”h=(24/n)1/3.Itisrecommendedtouseaveragedhistograms.Theyarekerneldensities.,Summary,1.4Scatterplots,Scatterplotsarebivariateortrivariateplotsofvariables(vribl)againsteachother.Theyhelpusunderstandrelationshipsamongthevariablesofadataset.Adownward-sloping(slupi)scatterindicatesthatasweincreasethevariableonthehorizontalaxis,thevariableontheverticalaxisdecreases(di:kri:svt.减少).Ananalogous(nlgsadj.类似的)statementcanbemadeforupward-slopingscatters.,Figure1.12plotsthe5thcolumn(upperinnerframe)ofthebankdataagainstthe6thcolumn(diagonal).Thescatterisdownward-sloping.Asweal