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Stable and Efficient Cubature-based Filtering in Dynamical Systems 1st ed. 2017 [Kõva köide]

  • Formaat: Hardback, 160 pages, kõrgus x laius: 235x155 mm, kaal: 3967 g, 14 Illustrations, color; 5 Illustrations, black and white; XVII, 160 p. 19 illus., 14 illus. in color., 1 Hardback
  • Ilmumisaeg: 06-Sep-2017
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319621297
  • ISBN-13: 9783319621296
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Stable and Efficient Cubature-based Filtering in Dynamical Systems 1st ed. 2017Suurem pilt
  • Formaat: Hardback, 160 pages, kõrgus x laius: 235x155 mm, kaal: 3967 g, 14 Illustrations, color; 5 Illustrations, black and white; XVII, 160 p. 19 illus., 14 illus. in color., 1 Hardback
  • Ilmumisaeg: 06-Sep-2017
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319621297
  • ISBN-13: 9783319621296
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319621296.html
Märksõnad:
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
1 Introduction
1(4)
1.1 Problem Statement and Objective
2(1)
1.2 Outline
3(2)
2 Filtering in Dynamical Systems
5(42)
2.1 The General Discrete State-Space Model
6(1)
2.2 The Bayes Filter
6(4)
2.3 The Kalman Filter
10(11)
2.3.1 The Kalman Filter Algorithm in the Case of the Gaussian Linear Discrete State-Space Model
11(2)
2.3.2 The Nonlinear Kalman Filter and the Gaussian Assumption
13(8)
2.4 Parameter Estimation
21(10)
2.4.1 Maximum Likelihood Estimation
21(1)
2.4.2 Bayesian Parameter Estimation
22(9)
2.5 Conditional Filtering
31(11)
2.6 Stabilization of Nonlinear Kalman Filter Algorithms
42(1)
2.7 Treatment of Missing Data
43(4)
3 Deterministic Numerical Integration
47(46)
3.1 One-Dimensional Deterministic Numerical Integration
48(17)
3.1.1 Lagrange Interpolation
48(1)
3.1.2 Moment Equations for the One-Dimensional Case
49(4)
3.1.3 Gauss Quadrature
53(9)
3.1.4 Clenshaw-Curtis Quadrature
62(3)
3.2 Multidimensional Deterministic Numerical Integration
65(28)
3.2.1 Stability Factor
66(1)
3.2.2 A Lower Bound for the Number of Abscissae
67(1)
3.2.3 Polynomials in d Dimensions
68(1)
3.2.4 Product Cubature Rules
69(2)
3.2.5 Moment Equations for the d-Dimensional Case
71(9)
3.2.6 Smolyak Cubature
80(8)
3.2.7 Compound Rules
88(1)
3.2.8 Change of Variables
89(4)
4 Optimization and Stabilization of Cubature Rules
93(16)
4.1 Cubature Rules Based on a Least Squares Approach
93(5)
4.2 Construction of Stabilized Smolyak Cubature Rules
98(11)
4.2.1 Stabilized(1) Rules
99(3)
4.2.2 Stabilized(2) Rules
102(4)
4.2.3 Smolyak Cubature Rules with an Approximate Degree of Exactness
106(3)
5 Simulation Studies
109(26)
5.1 The Univariate Non-Stationary Growth Model
112(5)
5.2 The Six-Dimensional Coordinated Turn Model
117(9)
5.3 The Lorenz Model
126(4)
5.4 The Ginzburg-Landau Model
130(5)
6 Results
135(4)
A The Conditional Mean 139(2)
B The Moments of the Conditional Normal Distribution 141(2)
C The Golub-Welsch Algorithm 143(6)
D Simplified Multidimensional Moment Equations 149(4)
Bibliography 153(6)
Index 159
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.