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1、UniversityofSydneySchoolofInformationTechnologiesSimulationofNetworksBasicsofConfidenceIntervals–averypragmaticapproachSimulationresultsaremeaninglessifwedonothavesomeideaoftheiraccuracy,andfurthermore,eachsimulationrunwillgiveadifferentestimate.Howdoweknowwhatthecorrectvalue
2、is?Generallyspeaking,wecan’tknow–butwecanusethesimulationdatatomakeanestimate.IndependentSamplesLet’sfirstassumewehaveasequenceofndatapoints{z1,z2,,zn}whichareknowntobeindependent.Weestimatethemeanas:nzizˆ=i=1nandthevarianceas:n()2(ˆ)2zi-nzVˆ=i=1n(n-1)Wecannowconstructacon
3、fidenceintervalforthemean.Anapproximatewaytodothisisto:·AssumethatthereareenoughsamplessothatwecanusetheCentralLimitTheorem(i.e.approxGaussiandistributed)·Makeahypothesistest–with(say)95%probability,thetruemeanlieswithinsomeinterval·FromaGaussiandistribution,theconfidenceinte
4、rvalistheestimatedmean,±1.96estimatedstandarddeviations(standarddeviationisthesquarerootofthevariance)·Thenwesaythat,withprobability95%,thetruemeanlieswithintherange[zˆ-1.96Vˆ,zˆ+1.96VˆConfidenceInterval,withprobabilityx%,EstimatedMeantruemeanlieswithinthisrangeEffectofCorrel
5、ationOfcourse,simulationdataisneverindependent–thereisalwayscorrelationbetweensuccessivesamples.Itisalwaysagoodideatoestimatethecorrelation–theautocovarianceoflagkisdefinedas:R(k)=E{(zi-z)(zi-k-z)}wherezisthetruemeanIftheautocovariancesarelow,thenanapproximationofindependence
6、isreasonable,otherwiseweneedtodobetter.Acoupleofsimplemethodsarelistedbelow–therearemanymoreintheliterature.1Estimatesbasedonestimatesofautocovariance(seeRipley)ns2UseVˆ=1-(cs-zˆ)(n-L)(n-L-1)ns7、roforlagsgreaterthanL.2MethodofBatchMeans(seeRipley,Pawlikowski)Theideahereistogroupthedataintobatches,tothatsuccessivebatchesareapproximatelyindependent.Forexample,ifwehavendatapoints{z1,z2,,zn},groupthemintombatchesofsizep(assumen=mp):{z1,z2,,zp}{zp+1,z2,,z2p}{zn-p+1,z2
8、,,zn}p2pnzizizii=1i=p+1i=n-p+1Nowtakethebatchmeans:x1=,x2=,,xm=
