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1、POLYNOMIALCHAOSEXPANSIONFORGENERALMULTIVARIATEDISTRIBUTIONSWITHCORRELATEDVARIABLESMARIANAVARRO,JEROENWITTEVEEN,JOKEBLOM?Abstract.Recently,theuseofPolynomialChaosExpansion(PCE)hasbeenin-creasingtostudytheuncertaintyinmathematicalmodelsforawiderangeofappli-cationsandseveralextensionsoftheorig
2、inalPCEtechniquehavebeendevelopedtodealwithsomeofitslimitations.ButasoftodatePCEmethodsstillhavetherestrictionthattherandomvariableshavetobestatisticallyindependent.Thispaperpresentsamethodtoconstructabasisoftheprobabilityspaceoforthogonalpoly-nomialsforgeneralmultivariatedistributionswithc
3、orrelationsbetweentherandominputvariables.Weshowthat,asforthecurrentPCEmethods,thestatisticslikemean,varianceandSobol'indicescanbeobtainedatnosigni cantextrapostprocess-ingcosts.WestudythebehavioroftheproposedmethodforarangeofcorrelationcoecientsforanODEwithmodelparametersthatfollowabivari
4、atenormaldis-tribution.Inallcasestheconvergencerateoftheproposedmethodisanalogoustothatfortheindependentcase.Finally,weshow,foracanonicalenzymaticreaction,howtopropagateexperimentalerrorsthroughtheprocessof ttingparameterstoaprobabilisticdistributionofthequantitiesofinterest,andwedemonstrat
5、ethesignif-icantdi erenceintheresultsassumingindependenceorfullcorrelationcomparedtotakingintoaccountthetruecorrelation.Keywords.Uncertaintyquanti cation?polynomialchaosexpansion?correlatedinputsCWI,SciencePark123,1098XGAmsterdam,theNetherlands?Correspondingauthor.Tel.:+31205924263Emailaddr
6、esses.maria.navarro@cwi.nl,jeroen.witteveen@cwi.nl,joke.blom@cwi.nl1.IntroductionarXiv:1406.5483v1[math.NA]20Jun2014Todescribereal-lifephenomenaweoftenmakeuseofdeterministicmathematicalmod-els.Typicalconstituentsofsuchmodelsareassignedade nitevalueandweseekadeterministicsolutiontotheproblem
7、.Inreality,however,thosephenomenawillal-mostalwayshaveuncertaincomponents:unknownparameters,impreciseexperimentaldata,etc..Aprecisemathematicaldescriptionshouldre
ectsuchuncertainties.Inotherwords,theparametersofamathematicalmodelofareal-lifeproblemposse
