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.
Contents vi
List of Figures viii
List of Tables ix
List of symbols and abbreviations x
1 Introduction 1
1.1 Problem statement and objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Filtering in dynamical systems 5
2.1 The general discrete state-space model . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The Bayes lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The Kalman lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 The Kalman lter algorithm in the case of the Gaussian linear discrete statespace
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 The nonlinear Kalman lter and the Gaussian assumption . . . . . . . . . . 12
2.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Maximum Likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Bayesian parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Conditional ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Stabilization of nonlinear Kalman lter algorithms . . . . . . . . . . . . . . . . . . . 37
2.7 Treatment of missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Deterministic numerical Integration 41
3.1 One-dimensional deterministic numerical integration . . . . . . . . . . . . . . . . . . 42
3.1.1 Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Moment equations for the one-dimensional case . . . . . . . . . . . . . . . . 43
3.1.3 Gauss quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.4 Clenshaw-Curtis quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Multidimensional deterministic numerical integration . . . . . . . . . . . . . . . . . 56
3.2.1 Stability Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.2 A lower bound for the number of abscissae . . . . . . . . . . . . . . . . . . . 58
3.2.3 Polynomials in d dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.4 Product cubature rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.5 Moment equations for the d-dimensional case . . . . . . . . . . . . . . . . . 61
3.2.6 Smolyak cubature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.7 Compound rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2.8 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Optimization and stabilization of cubature rules 78
4.1 Cubature rules based on a least squares approach . . . . . . . . . . . . . . . . . . . 78
4.2 Construction of stabilized Smolyak cubature rules . . . . . . . . . . . . . . . . . . . 83
4.2.1 Stabilized(1) rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.2 Stabilized(2) rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.3 Smolyak cubature rules with an approximate degree of exactness . . . . . . . 90
5 Simulation studies 93
5.1 The univariate non-stationary growth model . . . . . . . . . . . . . . . . . . . . . . 96
5.2 The six-dimensional coordinated turn model . . . . . . . . . . . . . . . . . . . . . . 101
5.3 The Lorenz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 The Ginzburg-Landau model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Results 117
Appendix A The conditional mean 120
Appendix B The moments of the conditional normal distribution 122
Appendix C The Golub-Welsch algorithm 124
Appendix D Simplied multidimensional moment equations 128
Index 130
Bibliography
