Multivariate Time Series With Linear State Space Structure PDF
Preview Multivariate Time Series With Linear State Space Structure
Víctor Gómez Multivariate Time Series With Linear State Space Structure Multivariate Time Series With Linear State Space Structure Víctor Gómez Multivariate Time Series With Linear State Space Structure 123 VíctorGómez MinisteriodeHaciendayAdministraciones Públicas DirecciónGral.dePresupuestos SubdirecciónGral.deAnálisisyP.E. Madrid,Spain ISBN978-3-319-28598-6 ISBN978-3-319-28599-3 (eBook) DOI10.1007/978-3-319-28599-3 LibraryofCongressControlNumber:2016938930 MathematicsSubjectClassification(2010):37M10,62-XX,62M10,93E11,62M20,60Gxx,65Fxx ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland To mywifeMaríaC. and mydaughterBerta Preface The subject of this book is the estimation of random vectors given observations ofa relatedrandomprocessassumingthatthere isa linearrelationbetweenthem. Sincetheclassoflinearmodelsisveryrich,werestrictourattentiontothosehaving a state space structure. The origin of this topic can be traced back to illustrious researchessuch as Laplace, Gauss, and Legendreand, more recently,to H. Wold, A.N.Kolmogorov,andN.Wienerinthelate1930sandearlier1940s. Thetheoryreceivedagreatimpulsewiththeincorporationofstatespacemodels. The main contributor to this development was R.E. Kalman, who also made importantrelated contributionsto linear systems, optimalcontrol,stability theory, etc.Thesubjectmatterofstatespacemodelshasexpandedalotinrecentyearsand todayincludesnonlinearaswellasnon-Gaussianmodels.Wehavelimitedthescope ofthebooktolinearstatespacemodels,however,becauseotherwiseitssizewould havebeenexcessive. In this book, the emphasis is on the development of the theory of least- squares estimation for finite-dimensional linear systems with two aims: firstly, that the foundations are solidly laid and, secondly, that efficient algorithms are simultaneously given to perform the necessary computations. For this reason, the theoryispresentedinallgeneralitywithoutfocusingonspecificstatespacemodels from the beginning, as is the case in, for example, the books by Harvey (1989) or Durbin & Koopman (2012) with regard to structural time series models. The theorydevelopedinthisbookcoversmostaspectsofwhatisgenerallyknownasthe Wiener–KolmogorovandKalmanfilteringtheory. The book is intended for students and researchers in many fields, including statistics,economicsandbusiness,engineering,medicine,biology,sociology,etc.It assumesfamiliaritywithbasicconceptsoflinearalgebra,matrixtheory,andrandom processes.Someappendicesatthe endofseveralchaptersprovidethe readerwith backgroundmaterialinsomeoftheseorrelatedareas. Chapters 1 and 2 deal with the definition of orthogonal projection and the introductionof manytopics associated with it, includingstate space and VARMA models.InChap.3,stationaryprocessesandtheirpropertiesareconsidered.Chap- ter 4 is dedicatedto the generalstate space model,includingmanyalgorithmsfor vii viii Preface filtering and smoothing. General Markov processes are also included. Chapter 5 contains the development of special features associated with time invariant state space models. In Chap.6, time invariant state space models with inputs are considered. The Wiener–Kolmogorov theory is developed in Chap.7. First for infinite and then for finite samples. Finally, the SSMMATLAB software package isdescribedinChap.8. Madrid,Spain VíctorGómez November2015 Computer Software Manyofthealgorithmspresentedinthisbookhavebeenimplementedbytheauthor inMATLABinasoftwarepackagecalledSSMMATLAB(Gómez, 2014,2015). A brief description of the SSMMATLAB package is given in Chap.8. In this chapter,alistofthemostimportantSSMMATLABfunctionsisgiven,togetherwith referencestothesectionsofthisbookwithwhichthefunctionsareconnected. As describedin the SSMMATLAB manual,there are many examplesand case studiestakenfromtimeseriesbooks,suchasBox&Jenkins(1976),Reinsel(1997), Lütkepohl(2007),andDurbin&Koopman(2012),publishedtimeseriesarticles,or simulatedseries. ix
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