应用多元统计分析

应用多元统计分析第1页/共54页12.

MahalDistance马氏距离13.

matrix矩阵['meitriks]14.

maximum最大值['mæksimәn]15.

mean均值16.

meandifference均值差值17.

meansquare均方18.

meansumofsquare均方和19.

measure度量['meʒә]20.

median中位数21.

midpoint中值['mid,pɔint]22.

negativecorrelation负相关['negәtiv]23.

nominalvariable名义变量nominal24.

nonlinearcorrelation非线性相关[nɔn'liniә]25.

nonlinearregression非线性回归26.nonparametricstatistics非参数统计['nɔnpærә'metrik]27.

nonparametrictest非参数检验28.

normaldistribution正态分布['nɔ:ml]第2页/共54页Wehaveseeninthepreviouschaptershowverysimplegraphicaldevicescanhelpinunderstandingthestructureanddependencyofdata.Thegraphicaltoolswerebasedoneitherunivariate(bivariate)datarepresentationsoron“slick”transformationsofmultivariateinformationperceivablebythehumaneye.Mostofthetoolsareextremelyusefulinamodellingstep,butunfortunately,donotgivethefullpictureofthedataset.3MovingtoHigherDimensionsUnivariate[,ju:ni'vєәriәt]Adj.单变量的第3页/共54页Onereasonforthisisthatthegraphicaltoolspresentedcaptureonlycertaindimensionsofthedataanddonotnecessarilyconcentrateonthosedimensionsorsubpartsofthedataunderanalysisthatcarrythemaximumstructuralinformation.InPartIIIofthisbook,powerfultoolsforreducingthedimensionofadatasetwillbepresented.Inthischapter,asastartingpoint,simpleandbasictoolsareusedtodescribedependency.Theyareconstructedfromelementaryfactsofprobabilitytheoryandintroductorystatistics(forexample,thecovarianceandcorrelationbetweentwovariables).3MovingtoHigherDimensions第4页/共54页第5页/共54页第6页/共54页第7页/共54页第8页/共54页第9页/共54页第10页/共54页第11页/共54页第12页/共54页→Thecovarianceisameasureofdependence.→Covariancemeasuresonlylineardependence.→Covarianceisscaledependent.→Therearenonlineardependenciesthathavezerocovariance.→Zerocovariancedoesnotimplyindependence.→Independenceimplieszerocovariance.→Negativecovariancecorrespondstodownward-slopingscatterplots.→Positivecovariancecorrespondstoupward-slopingscatterplots.→ThecovarianceofavariablewithitselfisitsvarianceCov(X,X)=σXX

=σ2X→Forsmalln,weshouldreplacethefactor1/ninthecomputationofthecovarianceby1/n−1.第13页/共54页第14页/共54页第15页/共54页→Thecorrelationisastandardizedmeasureofdependence→Theabsolutevalueofthecorrelationisalwayslessthanone.→Correlationmeasuresonlylineardependence.→Therearenonlineardependenciesthathavezerocorrelation.→Zerocorrelationdoesnotimplyindependence.→Independenceimplieszerocorrelation.→Negativecorrelationcorrespondstodownward-slopingscatterplots.→Positivecorrelationcorrespondstoupward-slopingscatterplots.→Fisher’sZ-transformhelpsusintestinghypothesesoncorrelation.→Forsmallsamples,Fisher’sZ-transformcanbeimprovedbythetransformation第16页/共54页第17页/共54页第18页/共54页→Thecenterofgravityofadatamatrixisgivenbyitsmeanvectorx=n−1

XT1n.→ThedispersionoftheobservationsinadatamatrixisgivenbytheempiricalcovariancematrixS=n−1XTHX.→TheempiricalcorrelationmatrixisgivenbyR=D−1/2

SD−1/2.→AlineartransformationY=XAT

ofadatamatrixXhasmeanAandempiricalcovarianceASXAT.→TheMahalanobistransformationisalineartransformationzi=S−1/2(xi−)whichgivesastandardized,uncorrelateddatamatrixZ.第19页/共54页第20页/共54页第21页/共54页第22页/共54页第23页/共54页第24页/共54页第25页/共54页第26页/共54页第27页/共54页第28页/共54页第29页/共54页第30页/共54页第31页/共54页第32页/共54页第33页/共54页第34页/共54页第35页/共54页第36页/共54页第37页/共54页第38页/共54页第39页/共54页→SimpleANOVAmodelsanoutputYasafunctionofonefactor.→Thereducedmodelisthehypothesisofequalmeans.→Thefullmodelisthealternativehypothesisofdifferentmeans.→TheF-testisbasedonacomparisonofthesumofsquaresunderthefullandthereducedmodels.→Thedegreesoffreedomarecalculatedasthenumberofobservationsminusthenumberofparameters.→TheF-testrejectsthenullhypothesisiftheF-statisticislargerthanthe95%quantileoftheFdf(r)−df(f),df(f)distribution.→TheF-teststatisticfortheslopeofthelinearregressionmodelyi

=α+βxi

+εi

isthesquareofthet-teststatistic.第40页/共54页第41页/共54页第42页/共54页第43页/共54页第44页/共54页CovarianceMatrix第45页/共54页CorrelationMatrix第46页/共54页第47页/共54页第48页/共54页第49页/共54页第50页/共54页4MultivariateDistributionsTheprecedingchaptershowedthatbyusingthetwofirstmomentsofamultivariatedistribution(themeanandthecovariancematrix),alotofinformationontherelationshipbetweenthevariablescanbemadeavailable.Onlybasicstatisticaltheorywasusedtoderivetestsofindependenceoroflinearrelationships.Inthischapterwegiveanintroductiontothebasic