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kalman filtering and neural networks.pdf
Kalman Filtering & Neural Networks
Copyright
Contents
Preface
Contributors
Adaptive & Learning Systems for Signal Processing, Communications & Control
Ch1 Kalman Filters
Ch2 Parameter-Based Kalman Filter Training: Theory & Implementation
Ch3 Learning Shape & motion from Image Sequences
Ch4 Chaotic Dynamics
Ch5 Dual Extended Kalman Filter Methods
Ch6 Leaning Nonlinear Dynamical Systems using Expectation-Maximization Algorithm
Ch7 Unscented Kalman Filter
Index
KALMAN FILTERING AND
NEURAL NETWORKS
KALMAN FILTERING AND
NEURAL NETWORKS
Edited by
Simon Haykin
Communications Research Laboratory,
McMaster University, Hamilton, Ontario, Canada
New York = Chichester = Weinheim = Brisbane = Singapore = Toronto
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS, INC.
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and registration.
Copyright 2001 by John Wiley & Sons, Inc.. All rights reserved.
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addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue,
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This publication is designed to provide accurate and authoritative information in regard to
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engaged in rendering professional services. If professional advice or other expert
assistance is required, the services of a competent professional person should be sought.
ISBN 0-471-22154-6
This title is also available in print as ISBN 0-471-36998-5.
For more information about Wiley products, visit our web site at www.Wiley.com.
CONTENTS
Preface
Contributors
1 Kalman Filters
Simon Haykin
Introduction = 1
1.1
1.2 Optimum Estimates = 3
1.3 Kalman Filter = 5
1.4 Divergence Phenomenon: Square-Root Filtering = 10
1.5 Rauch–Tung–Striebel Smoother = 11
1.6 Extended Kalman Filter = 16
1.7 Summary = 20
References = 20
2 Parameter-Based Kalman Filter Training:
Theory and Implementation
Gintaras V. Puskorius and Lee A. Feldkamp
Introduction = 23
2.1
2.2 Network Architectures = 26
2.3 The EKF Procedure = 28
2.3.1 Global EKF Training = 29
2.3.2 Learning Rate and Scaled Cost Function = 31
2.3.3 Parameter Settings = 32
2.4 Decoupled EKF (DEKF) = 33
2.5 Multistream Training = 35
xi
xiii
1
23
v
vi
CONTENTS
2.5.1 Some Insight into the Multistream Technique = 40
2.5.2 Advantages and Extensions of Multistream
Training = 42
2.6 Computational Considerations = 43
2.6.1 Derivative Calculations = 43
2.6.2 Computationally Efficient Formulations for
Multiple-Output Problems = 45
2.6.3 Avoiding Matrix Inversions = 46
2.6.4 Square-Root Filtering = 48
2.7 Other Extensions and Enhancements = 51
2.7.1 EKF Training with Constrained Weights = 51
2.7.2 EKF Training with an Entropic Cost Function = 54
2.7.3 EKF Training with Scalar Errors = 55
2.8 Automotive Applications of EKF Training = 57
2.8.1 Air=Fuel Ratio Control = 58
2.8.2 Idle Speed Control = 59
2.8.3 Sensor-Catalyst Modeling = 60
2.8.4 Engine Misfire Detection = 61
2.8.5 Vehicle Emissions Estimation = 62
2.9 Discussion = 63
2.9.1 Virtues of EKF Training = 63
2.9.2 Limitations of EKF Training = 64
2.9.3 Guidelines for Implementation and Use = 64
References = 65
3 Learning Shape and Motion from Image Sequences
69
Gaurav S. Patel, Sue Becker, and Ron Racine
Introduction = 69
3.1
3.2 Neurobiological and Perceptual Foundations of our Model = 70
3.3 Network Description = 71
3.4 Experiment 1 = 73
3.5 Experiment 2 = 74
3.6 Experiment 3 = 76
3.7 Discussion = 77
References = 81
4 Chaotic Dynamics
Gaurav S. Patel and Simon Haykin
CONTENTS
vii
83
Introduction = 83
4.1
4.2 Chaotic (Dynamic) Invariants = 84
4.3 Dynamic Reconstruction = 85
4.4 Modeling Numerically Generated Chaotic Time Series = 87
4.4.1 Logistic Map = 87
4.4.2 Ikeda Map = 91
4.4.3 Lorenz Attractor = 99
4.5 Nonlinear Dynamic Modeling of Real-World
Time Series = 106
4.5.1 Laser Intensity Pulsations = 106
4.5.2 Sea Clutter Data = 113
4.6 Discussion = 119
References = 121
5 Dual Extended Kalman Filter Methods
123
Eric A. Wan and Alex T. Nelson
Introduction = 123
5.1
5.2 Dual EKF – Prediction Error = 126
5.2.1 EKF – State Estimation = 127
5.2.2 EKF – Weight Estimation = 128
5.2.3 Dual Estimation = 130
5.3 A Probabilistic Perspective = 135
5.3.1 Joint Estimation Methods = 137
5.3.2 Marginal Estimation Methods = 140
5.3.3 Dual EKF Algorithms = 144
5.3.4 Joint EKF = 149
5.4 Dual EKF Variance Estimation = 149
5.5 Applications = 153
5.5.1 Noisy Time-Series Estimation and Prediction = 153
5.5.2 Economic Forecasting – Index of Industrial
Production = 155
5.5.3 Speech Enhancement = 157
5.6 Conclusions = 163
Acknowledgments = 164
viii
CONTENTS
Appendix A: Recurrent Derivative of the Kalman Gain = 164
Appendix B: Dual EKF with Colored Measurement Noise = 166
References = 170
6 Learning Nonlinear Dynamical System Using the
Expectation-Maximization Algorithm
175
Sam T. Roweis and Zoubin Ghahramani
6.1 Learning Stochastic Nonlinear Dynamics = 175
6.1.1 State Inference and Model Learning = 177
6.1.2 The Kalman Filter = 180
6.1.3 The EM Algorithm = 182
6.2 Combining EKS and EM = 186
6.2.1 Extended Kalman Smoothing (E-step) = 186
6.2.2 Learning Model Parameters (M-step) = 188
6.2.3 Fitting Radial Basis Functions to Gaussian
Clouds = 189
6.2.4 Initialization of Models and Choosing Locations
for RBF Kernels = 192
6.3 Results = 194
6.3.1 One- and Two-Dimensional Nonlinear State-Space
Models = 194
6.3.2 Weather Data = 197
6.4 Extensions = 200
6.4.1 Learning the Means and Widths of the RBFs = 200
6.4.2 On-Line Learning = 201
6.4.3 Nonstationarity = 202
6.4.4 Using Bayesian Methods for Model Selection and
Complexity Control = 203
6.5 Discussion = 206
6.5.1 Identifiability and Expressive Power = 206
6.5.2 Embedded Flows = 207
6.5.3 Stability = 210
6.5.4 Takens’ Theorem and Hidden States = 211
6.5.5 Should Parameters and Hidden States be Treated
Differently? = 213
6.6 Conclusions = 214
Acknowledgments = 215
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