A NOTE ON PARAMETER DIFFERENTIATION OF MATRIX EXPO

ANOTEONPARAMETERDIFFERENTIATIONOFMATRIXEXPONENTIALS,WITHAPPLICATIONSTOCONTINUOUS-TIMEMODELING(SHORTRUNNINGTITLE:PARAMETERDIFFERENTIATIONOFMATRIXEXPONENTIALS)HENGHSIUTSAIInstituteofStatisticalScience,AcademiaSinica,Taipei,Taiwan115,[e-mail]cs&ActuarialScience,UniversityofIowa,IowaCity,IA52242,[e-mail]27,2001SummaryWehavederivedanewanalyticformulaforevaluatingthederivativesofamatrixexponential.Incontrasttosomeexistingmethods,eigenvaluesandeigenvectorsdonotappearexplicitlyintheformulae,althoughweshowthatanecessaryandsucientconditionforthevalidityoftheformulaeisthatthematrixhasdistincteigenvalues.Thenewformulaexpressesthederivativesofamatrixexponentialintermsofminors,polynomials,exponentialofthematrixaswellasmatrixinver-1sion,andhenceisalgebraicallymoremanageable.Forsparsematrices,theformulacanbefurthersimplied.Twoexamplesarediscussedinsomedetails.Forthecompanionmatrixofacontinuous-timeautoregressivemovingaverageprocess,thederivativesoftheexponentialofthecompanionmatrixcanbecomputedre-cursively.WeillustrateanuseoftheseformulaeintheconditionalleastsquareestimationofaCAR(p)modelthatleadstoanumericallystableestimationpro-cedure.Thesecondexampleconcernstheexponentialofthetridiagonaltransitionintensitymatrixofanite-state-spacecontinuous-timeMarkovchainwhosein-stantaneoustransitionsmustbebetweenadjacentstates.Keywords:Cayley-HamiltonTheorem;CARMAmodels;companionmatrix;nite-state-spacecontinuous-timeMarkovprocesses;maximumlikelihoodestimation;minimalpolynomial;tridiagonalmatrix.1IntroductionVariousmethodsofparameterdierentiationofamatrixexponentialhavebeenstudiedinstatisticalmechanicsandquantumtheory(see,e.g.,Wilcox,1967),aswellasinthemathematics,economicsandstatisticsliterature,seee.g.,JennrichandBright(1976),VanLoan(1978),KalbeischandLawless(1985),Graham(1986),HornandJohnson(1991),ChanandMunoz-Hernandez(1997),ChenandZadrozny(2001).Forcontinuous/discretestatespacemodelling(see,e.g.,Jazwin-ski,1970andSinger,1995),parameterdierentiationofamatrixexponentialisneededforcomputingtheanalyticalscorefunction.Forcontinuous-timeMarkovmodeling,ecientalgorithmforthecomputationofthetransitionprobabilityma-trixanditsderivativeswithrespecttothetransitionintensityparametersisneededformaximumlikelihoodestimation.Forexample,seeKalbeischandLawless(1985)foranapproachofanalyzingapanelofcategoricaldatabyassumingthatthedataareobtainedfromsamplingalatentcontinuous-timenite-state-spaceMarkovprocess.Weproposeinthisnoteanalternativemethodforcomputingthederivativesofamatrixexponential.Incontrasttosomeexistingmethods,eigenvaluesandeigenvectorsdonotappearexplicitlyintheformulae,althoughweshowthatanec-2essaryandsucientconditionforthevalidityoftheformulaeisthatthematrixhasdistincteigenvalues.Thenewformulaexpressesthederivativesofamatrixexponentialintermsofminors,polynomials,exponentialofthematrixaswellasmatrixinversion,andhenceisalgebraicallymoremanageable.Whenthema-trixhasrepeatedeigenvalues,itseemshardtoextendtheresults.Seetheendofsection3fordiscussion.Fortunately,inmoststatisticalapplicationsthatinvolvematrixexponentials,thedistincteigenvalueassumptionoftenholds.Forexam-ple,incontinuous-timeMarkovchainmodelling,formostmodelsofinterest,thetransitionintensitymatrixhasdistincteigenvaluesforalmostallparametervalues(see,e.g.,KalbeischandLawless,1985).Thisnoteisorganizedasfollows.Inx2,wederivethenewformulaforcomputingthederivativesofamatrixexponentialandanecessaryandsucientconditionforthevalidityoftheformula.Forsparsematrices,theformulamaybefurthersimplied.Twointerestingexamplesaretheexponentialofthecompanionmatrixarisingfromacontinuous-timeautoregressivemovingaverageprocessandthatofthetridiagonaltransitionintensitymatrixarisingfromacontinuous-timeMarkovchainwhoseinstantaneoustransitionsmustbejumpsbetweenadjacentcategories.Thesimpliedformulaeforthesetwoexamplesaregiveninx3.2MainresultsLetA=[aij]beappmatrixwhoseelementsarefunctionsof#=(#1;:::;#r)0.Byequation(2.1)ofWilcox(1967),wehavethat,fori=1;:::;r,@etA@#i=Zt0e(tu)A@A@#i!euAdu:(1)Alternatively,ifweassumeAhasdistincteigenvaluesd1;


;dpandXistheppmatrixwhosejthcolumnisarighteigenvectorcorrespondingtodj,thenA=XDX1,whereD=diag(d1;


;dp).ThenetA=Xdiag(ed1t;


;edpt)X1,and@etA@#u=XVuX1;u=1;


;r;(2)whereVuisappmatrixwith(i;j)entryg(u)ij(editedjt)=(didj);i6=j;3g(u)iitedit;i=j;andg(u)ijisthe(i;j)entryinG(u)=X1(@A=@u)X.SeeKalbeischandLawless(1985)fortheaboveformulaandrelateddiscussions.SeealsoJennrichandBright(1976)andChanandMunos-Hernandez(1997).WhenAhasrepeatedeigenvalues,ananalogousdecompositionofAtoJordancanonicalformispossible(seechapter4ofCoxandMiller,1965).ButaspointedoutbyKalbeischandLawless(1985),thisisrarelynecessary,sinceformostmodelsofinterestincontinuous-timeMarkovmodelling,Ahasdistincteigenvaluesforalmostallparameters.Oneofthemainresultsofthispaperistoderiveanotherclosedformsolutionfor@etA=@#i.Forr=1;:::;p,denertobeap1vectorwith1inpositionrand0elsewhere.For1i;jp,letBij=i0j,anddeneij=Zt0e(