Stable and Efficient Cubature-based Filtering in Dynamical Systems (eBook)

Dominik Ballreich is a research assistant at the Chair for Applied Statistics and Methods of Empirical Social Research at the University of Hagen. His research interests lie in the fields of recursive Bayesian estimation, numerical integration and heuristic optimization.

Foreword 6
Acknowledgments 7
Contents 8
List of Figures 10
List of Tables 11
List of Symbols and Abbreviations 12
1 Introduction 13
1.1 Problem Statement and Objective 14
1.2 Outline 15
2 Filtering in Dynamical Systems 17
2.1 The General Discrete State-Space Model 18
2.2 The Bayes Filter 18
2.3 The Kalman Filter 22
2.3.1 The Kalman Filter Algorithm in the Case of the Gaussian Linear Discrete State-Space Model 23
2.3.2 The Nonlinear Kalman Filter and the GaussianAssumption 25
2.4 Parameter Estimation 33
2.4.1 Maximum Likelihood Estimation 33
2.4.2 Bayesian Parameter Estimation 34
2.5 Conditional Filtering 43
2.6 Stabilization of Nonlinear Kalman Filter Algorithms 54
2.7 Treatment of Missing Data 55
3 Deterministic Numerical Integration 58
3.1 One-Dimensional Deterministic Numerical Integration 59
3.1.1 Lagrange Interpolation 59
3.1.2 Moment Equations for the One-Dimensional Case 60
3.1.3 Gauss Quadrature 64
3.1.4 Clenshaw–Curtis Quadrature 73
3.2 Multidimensional Deterministic Numerical Integration 76
3.2.1 Stability Factor 77
3.2.2 A Lower Bound for the Number of Abscissae 78
3.2.3 Polynomials in d Dimensions 79
3.2.4 Product Cubature Rules 80
3.2.5 Moment Equations for the d-Dimensional Case 82
3.2.6 Smolyak Cubature 91
3.2.7 Compound Rules 99
3.2.8 Change of Variables 100
4 Optimization and Stabilization of Cubature Rules 103
4.1 Cubature Rules Based on a Least Squares Approach 103
4.2 Construction of Stabilized Smolyak Cubature Rules 108
4.2.1 Stabilized(1) Rules 109
4.2.2 Stabilized(2) Rules 112
4.2.3 Smolyak Cubature Rules with an Approximate Degree of Exactness 116
5 Simulation Studies 119
5.1 The Univariate Non-Stationary Growth Model 122
5.2 The Six-Dimensional Coordinated Turn Model 127
5.3 The Lorenz Model 136
5.4 The Ginzburg–Landau Model 140
6 Results 145
A The Conditional Mean 149
B The Moments of the Conditional Normal Distribution 151
C The Golub–Welsch Algorithm 153
D Simplified Multidimensional Moment Equations 158
Bibliography 161
Index 166