Multivariate Time Series With Linear State Space Structure

553 Pages·2016·4.817 MB·English

by  Víctor Gómez (auth.)

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
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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
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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.
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