E-raamat: Stable and Efficient Cubature-based Filtering in Dynamical Systems

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.

Contents viList of Figures viiiList of Tables ixList of symbols and abbreviations x1 Introduction 11.1 Problem statement and objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Filtering in dynamical systems 52.1 The general discrete state-space model . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Bayes lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The Kalman lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 The Kalman lter algorithm in the case of the Gaussian linear discrete statespacemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 The nonlinear Kalman lter and the Gaussian assumption . . . . . . . . . . 122.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1 Max

imum Likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Bayesian parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Conditional ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Stabilization of nonlinear Kalman lter algorithms . . . . . . . . . . . . . . . . . . . 372.7 Treatment of missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Deterministic numerical Integration 413.1 One-dimensional deterministic numerical integration . . . . . . . . . . . . . . . . . . 423.1.1 Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 Moment equations for the one-dimensional case . . . . . . . . . . . . . . . . 433.1.3 Gauss quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1.4 Clenshaw-Curtis quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Multidimensional determin

istic numerical integration . . . . . . . . . . . . . . . . . 563.2.1 Stability Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.2 A lower bound for the number of abscissae . . . . . . . . . . . . . . . . . . . 583.2.3 Polynomials in d dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.4 Product cubature rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.5 Moment equations for the d-dimensional case . . . . . . . . . . . . . . . . . 613.2.6 Smolyak cubature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.7 Compound rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2.8 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 Optimization and stabilization of cubature rules 784.1 Cubature rules based on a least squares approach . . . . . . . . . . . . . . . . . . . 784.2 Construction of stabilized Smolyak cubatu

re rules . . . . . . . . . . . . . . . . . . . 834.2.1 Stabilized(1) rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2 Stabilized(2) rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.3 Smolyak cubature rules with an approximate degree of exactness . . . . . . . 905 Simulation studies 935.1 The univariate non-stationary growth model . . . . . . . . . . . . . . . . . . . . . . 965.2 The six-dimensional coordinated turn model . . . . . . . . . . . . . . . . . . . . . . 1015.3 The Lorenz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 The Ginzburg-Landau model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 Results 117Appendix A The conditional mean 120Appendix B The moments of the conditional normal distribution 122Appendix C The Golub-Welsch algorithm 124Appendix D S