Bayesian Filtering and Smoothing

252 Pages·2013·4.29 MB·English

by  Simo Särkkä

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BAYESIAN FILTERING AND
SMOOTHING
SimoSa¨rkka¨
Bayesian Filtering and Smoothing has been
published by Cambridge University Press, as
volume 3 in the IMS Textbooks series. It can be
purchased directly from Cambridge University
Press:www.cambridge.org/sarkka.
Pleasecitethisbookas:
Simo Sa¨rkka¨ (2013). Bayesian Filtering and
Smoothing.CambridgeUniversityPress.
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Contents
Preface ix
Symbolsandabbreviations xiii
1 WhatareBayesianfilteringandsmoothing? 1
1.1 ApplicationsofBayesianfilteringandsmoothing 1
1.2 OriginsofBayesianfilteringandsmoothing 7
1.3 OptimalfilteringandsmoothingasBayesianinference 8
1.4 AlgorithmsforBayesianfilteringandsmoothing 12
1.5 Parameterestimation 14
1.6 Exercises 15
2 Bayesianinference 17
2.1 PhilosophyofBayesianinference 17
2.2 Connectiontomaximumlikelihoodestimation 17
2.3 ThebuildingblocksofBayesianmodels 19
2.4 Bayesianpointestimates 20
2.5 Numericalmethods 22
2.6 Exercises 24
3 BatchandrecursiveBayesianestimation 27
3.1 Batchlinearregression 27
3.2 Recursivelinearregression 30
3.3 Batchversusrecursiveestimation 31
3.4 Driftmodelforlinearregression 34
3.5 Statespacemodelforlinearregressionwithdrift 36
3.6 Examplesofstatespacemodels 39
3.7 Exercises 46
4 Bayesianfilteringequationsandexactsolutions 51
4.1 Probabilisticstatespacemodels 51
4.2 Bayesianfilteringequations 54
4.3 Kalmanfilter 56
v
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vi Contents
4.4 Exercises 62
5 ExtendedandunscentedKalmanfiltering 64
5.1 Taylorseriesexpansions 64
5.2 ExtendedKalmanfilter 69
5.3 Statisticallinearization 75
5.4 Statisticallylinearizedfilter 77
5.5 Unscentedtransform 81
5.6 UnscentedKalmanfilter 86
5.7 Exercises 92
6 GeneralGaussianfiltering 96
6.1 Gaussianmomentmatching 96
6.2 Gaussianfilter 97
6.3 Gauss–Hermiteintegration 99
6.4 Gauss–HermiteKalmanfilter 103
6.5 Sphericalcubatureintegration 106
6.6 CubatureKalmanfilter 110
6.7 Exercises 114
7 Particlefiltering 116
7.1 MonteCarloapproximationsinBayesianinference 116
7.2 Importancesampling 117
7.3 Sequentialimportancesampling 120
7.4 Sequentialimportanceresampling 123
7.5 Rao–Blackwellizedparticlefilter 129
7.6 Exercises 132
8 Bayesiansmoothingequationsandexactsolutions 134
8.1 Bayesiansmoothingequations 134
8.2 Rauch–Tung–Striebelsmoother 135
8.3 Two-filtersmoothing 139
8.4 Exercises 142
9 Extendedandunscentedsmoothing 144
9.1 ExtendedRauch–Tung–Striebelsmoother 144
9.2 StatisticallylinearizedRauch–Tung–Striebelsmoother 146
9.3 UnscentedRauch–Tung–Striebelsmoother 148
9.4 Exercises 152
10 GeneralGaussiansmoothing 154
10.1 GeneralGaussianRauch–Tung–Striebelsmoother 154
10.2 Gauss–HermiteRauch–Tung–Striebelsmoother 155
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Contents vii
10.3 CubatureRauch–Tung–Striebelsmoother 156
10.4 Generalfixed-pointsmootherequations 159
10.5 Generalfixed-lagsmootherequations 162
10.6 Exercises 164
11 Particlesmoothing 165
11.1 SIRparticlesmoother 165
11.2 Backward-simulationparticlesmoother 167
11.3 Reweightingparticlesmoother 169
11.4 Rao–Blackwellizedparticlesmoothers 171
11.5 Exercises 173
12 Parameterestimation 174
12.1 Bayesianestimationofparametersinstatespacemodels 174
12.2 Computationalmethodsforparameterestimation 177
12.3 Practicalparameterestimationinstatespacemodels 185
12.4 Exercises 202
13 Epilogue 204
13.1 WhichmethodshouldIchoose? 204
13.2 Furthertopics 206
Appendix Additionalmaterial 209
A.1 PropertiesofGaussiandistribution 209
A.2 Choleskyfactorizationanditsderivative 210
A.3 ParameterderivativesfortheKalmanfilter 212
A.4 ParameterderivativesfortheGaussianfilter 214
References 219
Index 229
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Preface
Theaimofthisbookistogiveaconciseintroductiontonon-linearKalman
filtering and smoothing, particle filtering and smoothing, and to the re-
lated parameter estimation methods. Although the book is intended to be
an introduction, the mathematical ideas behind all the methods are care-
fullyexplained,andamathematicallyinclinedreadercangetquiteadeep
understandingofthemethodsbyreadingthebook.Thebookispurposely
keptshortforquickreading.
The book is mainly intended for advanced undergraduate and graduate
studentsinappliedmathematicsandcomputerscience.However,thebook
is suitable also for researchers and practitioners (engineers) who need a
conciseintroductiontothetopiconalevelthatenablesthemtoimplement
orusethemethods.Theassumedbackgroundislinearalgebra,vectorcal-
culus,Bayesianinference,andMATLAB(cid:13)R programmingskills.
As implied by the title, the mathematical treatment of the models and
algorithms in this book is Bayesian, which means that all the results are
treatedasbeingapproximationstocertainprobabilitydistributionsortheir
parameters. Probability distributions are used both to represent uncertain-
ties in the models and for modeling the physical randomness. The theo-
ries of non-linear filtering, smoothing, and parameter estimation are for-
mulated in terms of Bayesian inference, and both the classical and recent
algorithms are derived using the same Bayesian notation and formalism.
This Bayesian approach to the topic is far from new. It was pioneered by
Stratonovichinthe1950sand1960s–evenbeforeKalman’sseminalarti-
clein1960.Thusthetheoryofnon-linearfilteringhasbeenBayesianfrom
thebeginning(seeJazwinski,1970).
Chapter 1 is a general introduction to the idea and applications of
Bayesian filtering and smoothing. The purpose of Chapter 2 is to briefly
review the basic concepts of Bayesian inference as well as the basic
numerical methods used in Bayesian computations. Chapter 3 starts with
a step-by-step introduction to recursive Bayesian estimation via solving a
ix
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x Preface
linearregressionprobleminarecursivemanner.ThetransitiontoBayesian
filteringandsmoothingtheoryisexplainedbyextendingandgeneralizing
theproblem.ThefirstKalmanfilterofthebookisalsoencounteredinthis
chapter.
The Bayesian filtering theory starts in Chapter 4 where we derive the
general Bayesian filtering equations and, as their special case, the cele-
brated Kalman filter. Non-linear extensions of the Kalman filter, the ex-
tendedKalmanfilter(EKF),thestatisticallylinearizedfilter(SLF),andthe
unscentedKalmanfilter(UKF)arepresentedinChapter5.Chapter6gen-
eralizesthesefiltersintotheframeworkofGaussianfiltering.TheGauss–
HermiteKalmanfilter(GHKF)andcubatureKalmanfilter(CKF)arethen
derivedfromthegeneralframework.SequentialMonteCarlo(SMC)based
particle filters (PF) are explained in Chapter 7 by starting from the basic
SIRfilterandendingwithRao–Blackwellizedparticlefilters(RBPF).
Chapter 8 starts with a derivation of the general (fixed-interval)
Bayesian smoothing equations and then continues to a derivation of
the Rauch–Tung–Striebel (RTS) smoother as their special case. In that
chapter we also briefly discuss two-filter smoothing. The extended RTS
smoother (ERTSS), statistically linearized RTS smoother (SLRTSS),
and the unscented RTS smoother (URTSS) are presented in Chapter 9.
The general Gaussian smoothing framework is presented in Chapter 10,
and the Gauss–Hermite RTS smoother (GHRTSS) and the cubature RTS
smoother (CRTSS) are derived as its special cases. We also discuss
Gaussian fixed-point and fixed-lag smoothing in the same chapter. In
Chapter 11 we start by showing how the basic SIR particle filter can be
used to approximate the smoothing solutions with a small modification.
We then introduce the numerically better backward-simulation particle
smootherandthereweighting(ormarginal)particlesmoother.Finally,we
discusstheimplementationofRao–Blackwellizedparticlesmoothers.
Chapter 12 is an introduction to parameter estimation in state space
models concentrating on optimization and expectation–maximization
(EM) based computation of maximum likelihood (ML) and maximum
a posteriori (MAP) estimates, as well as to Markov chain Monte Carlo
(MCMC) methods. We start by presenting the general methods and then
show how Kalman filters and RTS smoothers, non-linear Gaussian filters
and RTS smoothers, and finally particle filters and smoothers, can be
used to compute or approximate the quantities needed in implementation
of parameter estimation methods. This leads to, for example, classical
EM algorithms for state space models, as well as to particle EM and
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Preface xi
particleMCMCmethods.WealsodiscusshowRao–Blackwellizationcan
sometimesbeusedtohelpparameterestimation.
Chapter 13 is an epilogue where we give some general advice on the
selection of different methods for different purposes. We also discuss and
give references to various technical points and related topics that are im-
portant,butdidnotfitintothisbook.
Each of the chapters ends with a range of exercises, which give the
readerhands-onexperienceinimplementingthemethodsandinselecting
theappropriatemethodforagivenpurpose.TheMATLAB(cid:13)R sourcecode
needed in the exercises as well as various other material can be found on
thebook’swebpageatwww.cambridge.org/sarkka.
ThisbookisanoutgrowthoflecturenotesofcoursesthatIgaveduring
the years 2009–2012 at Helsinki University of Technology, Aalto Univer-
sity,andTampereUniversityofTechnology,Finland.Mostofthetextwas
writtenwhileIwasworkingattheDepartmentofBiomedicalEngineering
andComputationalScience(BECS)ofAaltoUniversity(formerlyHelsinki
UniversityofTechnology),butsomeofthetextwaswrittenduringmyvisit
to the Department of Engineering at the University of Cambridge, UK. I
amgratefultotheformerCentreofExcellenceinComputationalComplex
SystemsResearchoftheAcademyofFinland,BECS,andAaltoUniversity
SchoolofScienceforprovidingmewiththeresearchfundingwhichmade
thisbookpossible.
I would like to thank Jouko Lampinen and Aki Vehtari from BECS for
givingmetheopportunitytodotheresearchandforco-operationwhichled
to this book. Arno Solin, Robert Piche´, Juha Sarmavuori, Thomas Scho¨n,
PeteBunch,andIsambiS.Mbalawatadeservethanksforcarefulchecking
ofthebookandforgivingalotofusefulsuggestionsforimprovingthetext.
I am also grateful to Jouni Hartikainen, Ville Va¨a¨na¨nen, Heikki Haario,
and Simon Godsill for research co-operation that led to improvement of
myunderstandingofthetopicaswellastothedevelopmentofsomeofthe
methodswhichnowareexplainedinthisbook.Iwouldalsoliketothank
Diana Gillooly from Cambridge University Press and series editor Susan
Holmes for suggesting the publication of my lecture notes in book form.
Finally, I am grateful to my wife Susanne for her support and patience
duringthewritingofthisbook.
SimoSa¨rkka¨
Vantaa,Finland
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