BAYESIAN FILTERING AND SMOOTHING - Aalto

1 This PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and FILTERING ANDSMOOTHINGSimo S arkk aBayesian FILTERING and Smoothinghas beenpublished by Cambridge University Press, asvolume 3 in the IMS Textbooks series. It can bepurchased directly from Cambridge cite this book as:Simo S arkk a (2013). BAYESIAN FILTERING andSmoothing. Cambridge University PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and and abbreviationsxiii1 What are BAYESIAN FILTERING and SMOOTHING ?

2 Of BAYESIAN FILTERING and of BAYESIAN FILTERING and FILTERING and SMOOTHING as BAYESIAN for BAYESIAN FILTERING and of BAYESIAN to maximum likelihood building blocks of BAYESIAN point and recursive BAYESIAN linear linear versus recursive model for linear space model for linear regression with of state space FILTERING equations and exact state space FILTERING filter56vThis PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and and unscented Kalman series Kalman linearized Kalman Gaussian moment Hermite Hermite Kalman cubature Kalman Carlo approximations in BAYESIAN importance importance Blackwellized particle SMOOTHING equations and exact SMOOTHING Tung Striebel and unscented Rauch Tung Striebel linearized Rauch Tung Striebel Rauch Tung Striebel Gaussian Gaussian Rauch Tung Striebel Hermite Rauch Tung Striebel smoother155 This PDF version is made available for personal use.

3 The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and Rauch Tung Striebel fixed-point smoother fixed-lag smoother particle particle particle Blackwellized particle estimation of parameters in state space methods for parameter parameter estimation in state space method should I choose? topics206 AppendixAdditional of Gaussian factorization and its derivatives for the Kalman derivatives for the Gaussian filter214 References219 Index229 This PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and aim of this book is to give a concise introduction to non-linear Kalmanfiltering and SMOOTHING , particle FILTERING and SMOOTHING , and to the re-lated parameter estimation methods.

4 Although the book is intended to bean introduction, the mathematical ideas behind all the methods are care-fully explained, and a mathematically inclined reader can get quite a deepunderstanding of the methods by reading the book. The book is purposelykept short for quick book is mainly intended for advanced undergraduate and graduatestudents in applied mathematics and computer science. However, the bookis suitable also for researchers and practitioners (engineers) who need aconcise introduction to the topic on a level that enables them to implementor use the methods. The assumed background is linear algebra, vector cal-culus, BAYESIAN inference, and MATLABR programming implied by the title, the mathematical treatment of the models andalgorithms in this book is BAYESIAN , which means that all the results aretreated as being approximations to certain probability distributions or theirparameters. Probability distributions are used both to represent uncertain-ties in the models and for modeling the physical randomness.

5 The theo-ries of non-linear FILTERING , SMOOTHING , and parameter estimation are for-mulated in terms of BAYESIAN inference, and both the classical and recentalgorithms are derived using the same BAYESIAN notation and BAYESIAN approach to the topic is far from new. It was pioneered byStratonovich in the 1950s and 1960s even before Kalman s seminal arti-cle in 1960. Thus the theory of non-linear FILTERING has been BAYESIAN fromthe beginning (see Jazwinski, 1970).Chapter 1 is a general introduction to the idea and applications ofBayesian FILTERING and SMOOTHING . The purpose of Chapter 2 is to brieflyreview the basic concepts of BAYESIAN inference as well as the basicnumerical methods used in BAYESIAN computations. Chapter 3 starts witha step-by-step introduction to recursive BAYESIAN estimation via solving aixThis PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a).

6 Commercialreproduction is prohibited, except as authorised by the author and regression problem in a recursive manner. The transition to Bayesianfiltering and SMOOTHING theory is explained by extending and generalizingthe problem. The first Kalman filter of the book is also encountered in BAYESIAN FILTERING theory starts in Chapter 4 where we derive thegeneral BAYESIAN FILTERING equations and, as their special case, the cele-brated Kalman filter. Non-linear extensions of the Kalman filter, the ex-tended Kalman filter (EKF), the statistically linearized filter (SLF), and theunscented Kalman filter (UKF) are presented in Chapter 5. Chapter 6 gen-eralizes these filters into the framework of Gaussian FILTERING . The Gauss Hermite Kalman filter (GHKF) and cubature Kalman filter (CKF) are thenderived from the general framework. Sequential Monte Carlo (SMC) basedparticle filters (PF) are explained in Chapter 7 by starting from the basicSIR filter and ending with Rao Blackwellized particle filters (RBPF).

7 Chapter 8 starts with a derivation of the general (fixed-interval) BAYESIAN SMOOTHING equations and then continues to a derivation ofthe Rauch Tung Striebel (RTS) smoother as their special case. In thatchapter we also briefly discuss two-filter SMOOTHING . The extended RTSsmoother (ERTSS), statistically linearized RTS smoother (SLRTSS),and the unscented RTS smoother (URTSS) are presented in Chapter general Gaussian SMOOTHING framework is presented in Chapter 10,and the Gauss Hermite RTS smoother (GHRTSS) and the cubature RTSsmoother (CRTSS) are derived as its special cases. We also discussGaussian fixed-point and fixed-lag SMOOTHING in the same chapter. InChapter 11 we start by showing how the basic SIR particle filter can beused to approximate the SMOOTHING solutions with a small then introduce the numerically better backward-simulation particlesmoother and the reweighting (or marginal) particle smoother.

8 Finally, wediscuss the implementation of Rao Blackwellized particle 12 is an introduction to parameter estimation in state spacemodels concentrating on optimization and expectation maximization(EM) based computation of maximum likelihood (ML) and maximuma posteriori (MAP) estimates, as well as to Markov chain Monte Carlo(MCMC) methods. We start by presenting the general methods and thenshow how Kalman filters and RTS smoothers, non-linear Gaussian filtersand RTS smoothers, and finally particle filters and smoothers, can beused to compute or approximate the quantities needed in implementationof parameter estimation methods. This leads to, for example, classicalEM algorithms for state space models, as well as to particle EM andThis PDF version is made available for personal use. The copyright in all material rests with the author (Simo S arkk a). Commercialreproduction is prohibited, except as authorised by the author and MCMC methods.

9 We also discuss how Rao Blackwellization cansometimes be used to help parameter 13 is an epilogue where we give some general advice on theselection of different methods for different purposes. We also discuss andgive references to various technical points and related topics that are im-portant, but did not fit into this of the chapters ends with a range of exercises, which give thereader hands-on experience in implementing the methods and in selectingthe appropriate method for a given purpose. The MATLABR source codeneeded in the exercises as well as various other material can be found onthe book s web page book is an outgrowth of lecture notes of courses that I gave duringthe years 2009 2012 at Helsinki University of Technology, Aalto Univer-sity, and Tampere University of Technology, Finland. Most of the text waswritten while I was working at the Department of Biomedical Engineeringand Computational Science (BECS) of Aalto University (formerly HelsinkiUniversity of Technology), but some of the text was written during my visitto the Department of Engineering at the University of Cambridge, UK.

10 Iam grateful to the former Centre of Excellence in Computational ComplexSystems Research of the Academy of Finland, BECS, and Aalto UniversitySchool of Science for providing me with the research funding which madethis book would like to thank Jouko Lampinen and Aki Vehtari from BECS forgiving me the opportunity to do the research and for co-operation which ledto this book. Arno Solin, Robert Pich e, Juha Sarmavuori, Thomas Sch on,Pete Bunch, and Isambi S. Mbalawata deserve thanks for careful checkingof the book and for giving a lot of useful suggestions for improving the am also grateful to Jouni Hartikainen, Ville V a an anen, Heikki Haario,and Simon Godsill for research co-operation that led to improvement ofmy understanding of the topic as well as to the development of some of themethods which now are explained in this book. I would also like to thankDiana Gillooly from Cambridge University Press and series editor SusanHolmes for suggesting the publication of my lecture notes in book , I am grateful to my wife Susanne for her support and patienceduring the writing of this S arkk aVantaa, FinlandThis PDF version is made available for personal use.