Sequential Monte Carlo Methods in Practice

Transcription

1 Arnaud Doucet Nando de Freitas Neil Gordon Editors Sequential Monte Carlo Methods in Practice Foreword by Adrian Smith With 168 Illustrations Springer

2 Contents Foreword Acknowledgments Contributors v vii xxi I Introduction 1 1 An Introduction to Sequential Monte Carlo Methods 3 Arnaud Doucet, Nando de Freitas, and Neil Gordon 1.1 Motivation Problem statement Monte Carlo methods Perfect Monte Carlo sampling Importance sampling The Bootstrap filter Discussion 13 II Theoretical Issues 15 2 Particle Filters - A Theoretical Perspective 17 Dan Crisan 2.1 Introduction Notation and terminology Markov chains and transition kernels The filtering problem Convergence of measure-valued random variables Convergence theorems The fixed observation case The random observation case Examples of particle filters Description of the particle filters 25

3 x Contents Branching mechanisms Convergence of the algorithm Discussion Appendix Conditional probabilities and conditional expectations The recurrence formula for the conditional distribution of the signal 38 3 Interacting Particle Filtering With Discrete Observations 43 Pierre Del Moral and Jean Jacod 3.1 Introduction Nonlinear filtering: general facts An interacting particle system under Case A Subcase Al Subcase A An interacting particle system under Case B Subcase Bl Subcase B Discretely observed stochastic differential equations Case A Case B 73 III Strategies for Improving Sequential Monte Carlo Methods 77 4 Sequential Monte Carlo Methods for Optimal Filtering 79 Christophe Andrieu, Arnaud Doucet, and Elena Punskaya 4.1 Introduction Bayesian filtering and sequential estimation Dynamic modelling and Bayesian filtering Alternative dynamic models Sequential Monte Carlo Methods Methodology A generic algorithm Convergence results Application to digital communications Model specification and estimation objectives SMC applied to demodulation Simulations 93

4 Contents xi 5 Deterministic and Stochastic Particle Filters in State- Space Models 97 Erik B0lviken and Geir Storvik 5.1 Introduction General issues Model and exact filter Particle filters Gaussian quadrature Quadrature niters Numerical error A small illustrative example Case studies from ecology Problem area and models Quadrature filters in practice Numerical experiments Concluding remarks Appendix: Derivation of numerical errors RES AMPLE-MOVE Filtering with Cross-Model Jumps 117 Carlo Berzuini and Walter Gilks 6.1 Introduction Problem statement The RESAMPLE-MOVE algorithm Comments Central limit theorem Dealing with model uncertainty Illustrative application Applying RESAMPLE-MOVE Simulation experiment Uncertainty about the type of target Conclusions Improvement Strategies for Monte Carlo Particle Filters 139 Simon Godsill and Tim Clapp 7.1 Introduction General sequential importance sampling Markov chain moves The use of bridging densities with MCMC moves Simulation example: TVAR model in noise Particle filter algorithms for TVAR models Bootstrap (SIR) filter Auxiliary particle filter (APF) MCMC resampling Simulation results Summary 157

5 xii Contents 7.6 Acknowledgements Approximating and Maximising the Likelihood for a General State-Space Model 159 Markus Hiirzeler and Hans R. Kiinsch 8.1 Introduction Bayesian methods Pointwise Monte Carlo approximation of the likelihood Examples Approximation of the likelihood function based on filter samples Approximations based on smoother samples Approximation of the likelihood function Stochastic EM-algorithm Comparison of the methods AR(1) process Nonlinear example, 3 parameters Nonlinear model, 5 parameters Discussion Recursive estimation Monte Carlo Smoothing and Self-Organising State-Space Model 177 Genshiro Kitagawa and Seisho Sato 9.1 Introduction General state-space model and state estimation The model and the state estimation problem Non-Gaussian filter and smoother Monte Carlo filter and smoother Approximation of non-gaussian distributions Monte Carlo filtering Derivation of the Monte Carlo filter...: Monte Carlo smoothing Non-Gaussian smoothing for the stochastic volatility model Nonlinear Smoothing Self-organising state-space models Likelihood of the model and parameter estimation Self-organising state-space model Examples Self-organising smoothing for the stochastic volatility model Time series with trend and stochastic volatility Conclusion 195

6 Contents xiii 10 Combined Parameter and State Estimation in Simulation- Based Filtering 197 Jane Liu and Mike West 10.1 Introduction and historical perspective General framework Dynamic model and analysis perspective Filtering for states Filtering for states and parameters The treatment of model parameters Artificial evolution of parameters Kernel smoothing of parameters Reinterpreting artificial parameter evolutions A general algorithm Factor stochastic volatility modelling Discussion and future directions A Theoretical Framework for Sequential Importance Sampling with Resampling 225 Jun S. Liu, Rong Chen, and Tanya Logvinenko 11.1 Introduction Sequential importance sampling principle Properly weighted sample Sequential build-up Operations for enhancing SIS Reweighting, resampling and reallocation Rejection control and partial rejection control Marginalisation Monte Carlo filter for state-space models The general state-space model Conditional dynamic linear model and the mixture Kalman filter Some examples A simple illustration Target tracking with MKF Discussion Acknowledgements Improving Regularised Particle Filters 247 Christian Musso, Nadia Oudjane, and Francois Le Gland 12.1 Introduction Particle filters The (classical) interacting particle filter (IPF) Regularised particle filters (RPF) Progressive correction Focus on the correction step 256

7 xiv Contents Principle of progressive correction Adaptive choice of the decomposition The local rejection regularised particle filter (L2RPF) Description of the filter Computing the coefficient c[ l) (a t ) Applications to tracking problems Range and bearing Bearings-only Multiple model particle filter (MMPF) Auxiliary Variable Based Particle Filters 273 Michael K. Pitt and Neil Shephard 13.1 Introduction Particle filters The definition of particle filters Sampling the empirical prediction density Weaknesses of particle filters Auxiliary variable The basics A generic SIR based auxiliary proposal Examples of adaption Fixed-lag filtering Reduced random sampling Basic ideas Simple outlier example Conclusion Acknowledgements Improved Particle Filters and Smoothing 295 Photis Stavropoulos and D.M. Titterington 14.1 Introduction The methods The smooth bootstrap Adaptive importance sampling The kernel sampler of Hiirzeler and Kiinsch Partially smooth bootstrap Roughening and sample augmentation Application of the methods in particle filtering and smoothing Application of smooth bootstrap procedures to a simple control problem Description of the problem An approach to the continuous-time version of the problem An adaptation of Titterington's method 310

8 Contents xv Probabilistic criterion Probabilistic criterion 2: working directly with the cost Unknown variances :7 Resampling implementation Simulation results Further work on this problem 317 IV Applications Posterior Cramer-Rao Bounds for Sequential Estimation 321 Niclas Bergman 15.1 Introduction Review of the posterior Cramer-Rao bound Bounds for sequential estimation Estimation model Posterior Cramer-Rao bound Relative Monte Carlo evaluation Example - terrain navigation Conclusions Statistical Models of Visual Shape and Motion 339 Andrew Blake, Michael Isard, and John MacCormick 16.1 Introduction Statistical modelling of shape Statistical modelling of image observations Sampling methods ; Modelling dynamics Learning dynamics Particle filtering Dynamics with discrete states Conclusions Sequential Monte Carlo Methods for Neural Networks 359 N de Freitas, C Andrieu, P H0jen-S0rensen, M Niranjan, and A Gee 17.1 Introduction Model specification MLP models for regression and classification Variable dimension RBF models Estimation objectives General SMC algorithm Importance sampling step Selection step 368

9 xvi Contents MCMC Step Exact step On-line classification Simple classification example An application to fault detection in marine diesel engines An application to financial time series Conclusions Sequential Estimation of Signals under Model Uncertainty 381 Petar M. Djuric 18.1 Introduction The problem of parameter estimation under uncertainty Sequential updating of the solution Sequential algorithm for computing the solution A Sequential-Importance-Resampling scheme Sequential sampling scheme based on mixtures Example Conclusions Acknowledgment Particle Filters for Mobile Robot Localization 401 Dieter Fox, Sebastian Thrun, Wolfram Burgard, and Frank Dellaert 19.1 Introduction Monte Carlo localization Bayes filtering Models of robot motion and perception Implementation as particle filters Robot results Comparison to grid-based localization MCL with mixture proposal distributions The need for better sampling An alternative proposal distribution The mixture proposal distribution Robot results Multi-robot MCL Basic considerations Robot results Conclusion Self-Organizing Time Series Model 429 Tomoyuki Higuchi 20.1 Introduction 429

10 Contents xvii Generalised state-space model Monte Carlo filter Self-organizing time series model Genetic algorithm filter Self-organizing state-space model Resampling scheme for filtering Selection scheme Comparison of performance: simulation study Application Time-varying frequency wave in small count data Self-organizing state-space model for time-varying frequency wave Results Conclusions Sampling in Factored Dynamic Systems 445 Daphne Roller and Uri Lerner 21.1 Introduction Structured probabilistic models Bayesian networks Hybrid networks Dynamic Bayesian networks Particle filtering for DBNs Experimental results Conclusions In-Situ Ellipsometry Solutions Using Sequential Monte Carlo 465 Alan D. Marrs 22.1 Introduction Application background State-space model Ellipsometry measurement model System evolution model Particle filter Results Conclusion Acknowledgments Manoeuvring Target Tracking Using a Multiple-Model Bootstrap Filter 479 Shaun McGinnity and George W. Irwin 23.1 Introduction Optimal multiple-model solution The IMM algorithm 483

11 xviii Contents 23.4 Multiple model bootstrap filter Example 486 ' 23.5 Target tracking examples Target scenarios Linear, Gaussian tests Polar simulation results Conclusions Acknowledgments Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks 499 Kevin Murphy and Stuart Russell 24.1 Introduction RBPF in general How do we choose which nodes to sample? The RBPF algorithm in detail Application: concurrent localisation and map learning for a mobile robot Results on a one-dimensional grid Results on a two-dimensional grid Conclusions and future work Particles and Mixtures for Tracking and Guidance 517 David Salmond and Neil Gordon 25.1 Introduction Guidance as a stochastic control problem Information state Dynamic programming and the dual effect Separability and certainty equivalence Sub-optimal strategies Derivation of control laws from particles Certainty equivalence control A scheme based on open-loop feedback control Guidance in the presence of intermittent spurious objects and clutter Introduction Problem formulation Simulation example Guidance results Monte Carlo Techniques for Automated Target Recognition. 533 Anuj Srivastava, Aaron D. Lanterman, Ulf Grenander, Marc Loizeaux, and Michael I. Miller 26.1 Introduction 533

12 Contents xix The Bayesian posterior Inference engines Jump-diffusion sampling Diffusion Processes Jump processes Jump-diffusion algorithm Sensor models Experiments Acknowledgments 552 Bibliography 553 Index 577