Ballreich Stable and Efficient Cubature-based Filtering in Dynamical Systems

1. Auflage 2017
ISBN: 978-3-319-62130-2
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 167 Seiten

The book addresses the problem of calculation of d-dimensional integrals (conditional expectations) in filter problems. It develops new methods of deterministic numerical integration, which can be used to speed up and stabilize filter algorithms. With the help of these methods, better estimates and predictions of latent variables are made possible in the fields of economics, engineering and physics. The resulting procedures are tested within four detailed simulation studies.

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.

Autoren/Hrsg.

Ballreich, Dominik

Inhaltsverzeichnis

1;Foreword;6
2;Acknowledgments;7
3;Contents;8
4;List of Figures;10
5;List of Tables;11
6;List of Symbols and Abbreviations;12
7;1 Introduction;13
7.1;1.1 Problem Statement and Objective;14
7.2;1.2 Outline;15
8;2 Filtering in Dynamical Systems;17
8.1;2.1 The General Discrete State-Space Model;18
8.2;2.2 The Bayes Filter;18
8.3;2.3 The Kalman Filter;22
8.3.1;2.3.1 The Kalman Filter Algorithm in the Case of the Gaussian Linear Discrete State-Space Model;23
8.3.2;2.3.2 The Nonlinear Kalman Filter and the GaussianAssumption;25
8.4;2.4 Parameter Estimation;33
8.4.1;2.4.1 Maximum Likelihood Estimation;33
8.4.2;2.4.2 Bayesian Parameter Estimation;34
8.5;2.5 Conditional Filtering;43
8.6;2.6 Stabilization of Nonlinear Kalman Filter Algorithms;54
8.7;2.7 Treatment of Missing Data;55
9;3 Deterministic Numerical Integration;58
9.1;3.1 One-Dimensional Deterministic Numerical Integration;59
9.1.1;3.1.1 Lagrange Interpolation;59
9.1.2;3.1.2 Moment Equations for the One-Dimensional Case;60
9.1.3;3.1.3 Gauss Quadrature;64
9.1.4;3.1.4 Clenshaw–Curtis Quadrature;73
9.2;3.2 Multidimensional Deterministic Numerical Integration;76
9.2.1;3.2.1 Stability Factor;77
9.2.2;3.2.2 A Lower Bound for the Number of Abscissae;78
9.2.3;3.2.3 Polynomials in d Dimensions;79
9.2.4;3.2.4 Product Cubature Rules;80
9.2.5;3.2.5 Moment Equations for the d-Dimensional Case;82
9.2.6;3.2.6 Smolyak Cubature;91
9.2.7;3.2.7 Compound Rules;99
9.2.8;3.2.8 Change of Variables;100
10;4 Optimization and Stabilization of Cubature Rules;103
10.1;4.1 Cubature Rules Based on a Least Squares Approach;103
10.2;4.2 Construction of Stabilized Smolyak Cubature Rules;108
10.2.1;4.2.1 Stabilized(1) Rules;109
10.2.2;4.2.2 Stabilized(2) Rules;112
10.2.3;4.2.3 Smolyak Cubature Rules with an Approximate Degree of Exactness;116
11;5 Simulation Studies;119
11.1;5.1 The Univariate Non-Stationary Growth Model;122
11.2;5.2 The Six-Dimensional Coordinated Turn Model;127
11.3;5.3 The Lorenz Model;136
11.4;5.4 The Ginzburg–Landau Model;140
12;6 Results;145
13;A The Conditional Mean;149
14;B The Moments of the Conditional Normal Distribution;151
15;C The Golub–Welsch Algorithm;153
16;D Simplified Multidimensional Moment Equations;158
17;Bibliography;161
18;Index;166