Application of one‐step method to parameter estimation in ODE models Itai Dattner.pdf

该【Application of one‐step method to parameter estimation in ODE models Itai Dattner 】是由【周瑞】上传分享，文档一共【31】页，该文档可以免费在线阅读，需要了解更多关于【Application of one‐step method to parameter estimation in ODE models Itai Dattner 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..126StatisticaNeerlandica(2018),,–156doi:-stepmethodtoparameterestimationinODEmodelsItaiDattnerDepartmentofStatistics,UniversityofHaifa,199AbaKhoushyAve.,MountCarmel,Haifa3498838,IsraelShotaGugushvili*MathematicalInstitute,LeidenUniversity,,2300RALeiden,herlandsInthispaper,westudyapplicationofLeCam’sone--linearleastsquaresesti-mator,whichtypicallyrequirestheuseofamultistepiterativealgo--stepmethodstartsfromapreliminary√-stepmethodstartsfromapreliminary√n-consistentestimatoroftheparameterofinterestandnextturnsitintoanasymptotic(asthesamplesizen→∞)-stepestimatorviaexten--√naryn-consistentestimatorthatweusedependsonnon-parametricsmoothing,andweprovideadata--:non-linearleastsquares,ordinarydiffer-entialequations,smoothandmatchestimator,integralestimator,Levenberg–Marquardtalgorithm,one-:Primary:62F12,Secondary:62G08,62G201IntroductionSystemsofordinarydifferentialequations(ODEsinshort)monlyusedforthemathematicalmodellingoftherateofchangeofdynamicprocesses(-*shota.******@,whichpermitsuse,distributionandreproductioninanymedium,providedtheoriginalworkisproperlycited.?&SonsLtdonbehalfofVVS.:..EstimationinODEmodels127icalbiology,seeEdelstein-Keshet,2005;works,seeFeinberg,1979andSontag,2001;andinbiochemistry,seeVoit,2000).StatisticalinferenceforODEsisnotatrivialtask,becausenumericalevaluationofstandardesti-mators,likethemaximumlikelihoodortheleastsquaresestimators,,overthelastfewdecades,irstinthenumericalanalysisandmathematicalbiologyliteratureandlatelyalsointhestatisticalliter-ature,variousalternative,primarilynon-parametricsmoothing-basedmethodshavebeenproposedtotackletheproblem,see,(1971),Varah(1982),VoitandSavageau(1982),Ramsayetal.(2007),Hooker(2009),Hookeretal.(2011),GugushviliandKlaassen(2012),CampbellandLele(2014),Vujaciˇc′etal.(2015),Dat-tner(2015),DattnerandKlaassen(2015),,{′x(t)=F(x(t),θ),t∈[0,1],(1)x(0)=ξwherex(t)=(x(t),…,x(t))trisad-dimensionalstatevariable,θ=(θ,…,θ)tr1d1pdenotesap-dimensionalparameter,whilethecolumnd-vectorx(0)=∶=(ξ,θ)anddenotethesolutiontoEquation1correspondingtotheparameterbyx(,t)∶=(x(,t),…,x(,t)),theseparametersaffectthequali-tativepropertiesofthesystem,,inpractice,,=(ξ0,θ0)bethe‘true’,…,tn(notnecessarilyequallyspaced)istheadditivemeasurementerrormodel,Yij=xi(η0,tj)+?ij,i=1,…,d,j=1,…,n,(2)wheretherandomvariablesijareindependentmeasurementerrors(notnecessarilyGaussian).Basedonobservationpairs(tj,Yij),i=1,…,d,j=1,…,n,-linearleastsquares(NLS)-linearregressionproblem,wheretheregressionfunction?&SonsLtdonbehalfofVVS.:..(,?)?n=(ξ?n,θ?n)ofη0isdeinedasaminimizeroftheleastsquarescriterionfunctionRn(?),∑d∑nη?=(ξ?,θ?)=argmin(Y?x(η,t))2nnnηijiji=1j=1(3)=∶argminηRn(η).Thestrongestjustiicationfortheuseoftheleastsquaresestimatorliesinitsattrac-tiveasymptoticproperties;see,(1969)andWu(1981).Inmostpracticalapplications,thesolutionx(η,?)toEquation1isnon-linearintheparameterη,andtherefore,?nand,then,proceedbyconstruct-essiveapproximationstotheleastsquaresestimator(inadirectionguidedbythegradientofthecriterionfunction,whenagradient-basedoptimizationmethod,–Marquardtmethod,isused).However,thenoisyandnon-linearcharacteroftheoptimizationproblemmayleadfortheproceduretoendupinalocalminimumoftheleastsquarescriterionfunction,,inmostoftheinterestingapplica-tions,thesystem(Equation1)isnon-linearanddoesnothaveaclosed-,ateverystepoftheiterativeprocedure,onehastonumericallyintegrateEquation1(aswellasthesystemoftheassociatedsensitivityequationsinordertocomputethegradientofthecriterionfunction,incaseagradient-basedoptimizationmethodisused).Becausethenumberofiterationsmadeuntilconvergenceofthealgo-rithmcanbeascertainedisusuallylarge,inmostcases,,whereahighlynon-linearcharacterofdependenceofthesolutionx(,?)ontheparameterηleadsto‘stiff’,see,.(2007)andVoitandAlmeida(2004).AlthoughNLSalgorithmsandODEintegrationroutinesareconstantlyimproving,putationalpower,admittedlymuchtimeandeffortcanbesavedwithalternative,putationallyintenseapproaches,seeVoitandAlmeida(2004).Inthispaper,weexploreapplicationofLeCam’sone-stepestimator(see,,1998)(1975),al.(1992),FieldandWiens(1994),Caietal.(2000),Delecroixetal.(2003)andRieder(2012).Inparticular,ourmaingoalistoshowthattheone-parablewithNLS,-stepmethodisnotsimplyanumericalapproximationtoanalgorithmusedfornumericalevaluationofNLS::(i)Smoothing-basedparameterestima-tionmethodsforODEsystemscanbeupgradedtohavestatisticaleficiencyofNLSputationallysimpleone-stepmethod.(ii)IfonewantstoavoidusingNLS(asisoftenthecaseintheappliedliterature,see,.,i?&SonsLtdonbehalfofVVS.:..EstimationinODEmodels129etal.,2016),onecanstilldothis,whilenotlosingstatisticaleficiencyofNLSandcomputationalpropertiesofsmoothing-basedmethods.(iii)Weshowhowtoperformsmoothinginadata-drivenmannerandprovidetheorysupportingourdata-drivenalgorithm.(iv)Wepointoutaverysimpleschemeforimplementingtheone-stepestimator,whichisreadilyavailableinanysoftwarethatimplementsNewton-typeoptimizationalgorithms,suchasR,seeRCoreTeam(2017),andMatlab,seeTheMathworks,Inc.(2017).Pertainingtopoint(i)mentionedearlier,wehighlighttheextentoflossofeficiencyofsmoothing-paredwiththeNLSandtheone-stepmethod,,dense-in-timedata,ingincreasinglyavailableinpractice,speciicallyinmolecularbiol-ogy(seeVoitandAlmeida,2004;Goeletal.,2008),andthatwouldallowanin-depthstudyofunderlyingbiologicalprocesses,suchastatisticaleficiencylossisclearlyunde-,currentODEinferencealgorithmsmustalsomeetchal-(ii),asnotedinChouandVoit(2009),thatfarnoparameterestima-tiontechniqueforODEshasarisenasaclearwinnerintermsofeficiency,,additionoftheone-stepmethod(thatsharessomeofthebetterpropertiesofboththesmoothing-basedmethodsandNLS)toapractitioner’(iii),wenotethatmuchoftheliteraturedealingwithsmoothing--,concerningourcontribution(iv),wepointoutanimportantrelationbetweentheone-stepestimatorandtheLevenberg–Marquardtalgorithm,whichleadstoaverypracticalandstraightforwardimplementationofthemethod:putationaltimeisanissue,oursimulationsandtheoryjustifytheuseoftheLevenberg–Marquardtmethodwithoneiteration,provideditisinitializedatanappropriatesmoothing-basedparameterestimator,becausethisreducestotheone-:inSection2,wedescribetheone-,-stepmethod,withfurtherexamplesinSection5,whileSection6containsnumer-,-stepestimateforordinarydifferentialequationsWhenoneadoptsanasymptoticpointofviewonstatistics,√howonceapreliminaryn-consistentestimatorη?noftheparameterηisavailable(see?&SonsLtdonbehalfofVVS.:..),onecanobtainanasymptoticallyequivalentestimatortotheleastsquaresestimatorinjustoneextrastep,referredtoastheone-stepmethodinthestatisticalliterature,see,(1998)∑nΨn(η)=η(tj,Yj),(4)j=1whereψ(t,y)=(x′(η,t))tr(y?x(η,t)),(5)ηηwithx′(η,t)denotingthederivativeofx(η,t),theithrowofx′(,t)isthegradientofx(η,t)-stepestimatorη?nofη0isdeinedasasolutioninηoftheequationdΨn(η?n)+Ψn(η?n)(η?η?n)=(η?)isinvertible,theestimatorη?canbeexpressedasdηnnn()?1dη?n=η?n?Ψn(η?n)Ψn(η?n).(6)dηInordertoimplementtheestimatorjustdeined,thetwoessentialstepsthathavetobedoneare(i)evaluationofapreliminaryestimatorη?n,and(ii)evaluationofΨn(η?n)andthederivativematrixdΨ(η?).,asmentionedinSection1,step(i)isveryfast,whenasmoothing-basedesti-matorisused,,step(ii)reducestorequiringjustonenumericalintegrationofthesensitivityandvariationalequationsassociatedwiththesystem(Equation1),-dardinthenumericalanalysisandODEliterature(,2002;RamsayandHooker,2017)butperhapslessfamiliartostatisticians,,wewritetheright-handsideFofEquation1asF(x(η,t),η).DifferentiatingbothsidesofEquation1withrespecttoηandinterchangingtheorderofat-derivativewithanη-derivative,weget{dx(η,t)=F′(x(η,t),η)x(η,t)+F′(x(η,t),η),dtxηηtr(7)ηx(η,0)=(1,0),where1and0intheinitialconditionshereandinEquations8–eedingtextshouldbeunderstoodasvectorsof1′sand0′(Equation7)isamatrixdifferentialequationandisusuallyreferredtointhe?&SonsLtdonbehalfofVVS.:..EstimationinO