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Bayesian Inference of State Space Models: Kalman Filtering and Beyond

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Bayesian Inference of State Space Models: Kalman Filtering and Beyond offers a comprehensive introduction to Bayesian estimation and forecasting for state space models. The celebrated Kalman filter, with its numerous extensions, takes centre stage in the book. Univariate and multivariate models, linear Gaussian, non-linear and non-Gaussian models are discussed with applications to signal processing, environmetrics, economics and systems engineering.

Over the past years there has been a growing literature on Bayesian inference of state space models, focusing on multivariate models as well as on non-linear and non-Gaussian models. The availability of time series data in many fields of science and industry on the one hand, and the development of low-cost computational capabilities on the other, have resulted in a wealth of statistical methods aimed at parameter estimation and forecasting. This book brings together many of these methods, presenting an accessible and comprehensive introduction to state space models. A number of data sets from different disciplines are used to illustrate the methods and show how they are applied in practice. The R package BTSA, created for the book, includes many of the algorithms and examples presented. The book is essentially self-contained and includes a chapter summarising the prerequisites in undergraduate linear algebra, probability and statistics.

An up-to-date and complete account of state space methods, illustrated by real-life data sets and R code, this textbook will appeal to a wide range of students and scientists, notably in the disciplines of statistics, systems engineering, signal processing, data science, finance and econometrics. With numerous exercises in each chapter, and prerequisite knowledge conveniently recalled, it is suitable for upper undergraduate and graduate courses.

Author(s): Kostas Triantafyllopoulos
Series: Springer Texts in Statistics
Publisher: Springer
Year: 2021

Language: English
Pages: 510
City: Cham

Preface
Acknowledgements
Contents
Acronyms
1 State Space Models
1.1 Introduction
1.1.1 Time Series
1.1.2 Examples of Time Series Data
1.2 Water Tank Dynamics and the State Space Model
1.3 Examples of State Space Models
1.3.1 Forecasting Air-Pollution Levels
1.3.2 Tracking a Ship
1.3.3 Stochastic Volatility
1.3.4 Hookean Spring Force Dynamics
1.4 A Short History of the Kalman Filter
1.5 Layout of the Book
2 Matrix Algebra, Probability and Statistics
2.1 Vectors, Matrices and Basic Operations
2.2 Vector and Matrix Differentiation
2.2.1 Background and Notation
2.2.2 Differentiation of Linear and Quadratic Forms
2.2.3 Differentiation of Determinant and Trace
2.2.4 Optimisation, Integration and Limits
2.3 Probability and Distribution Theory
2.3.1 Random Vectors and Probability Distributions
2.3.2 Common Discrete Distributions
2.3.3 Common Continuous Distributions
2.4 Statistics
2.4.1 Principle Set-Up and Objectives
2.4.2 Maximum Likelihood Estimation: The EM Algorithm
2.4.3 Bayesian Inference
2.5 Exercises
3 The Kalman Filter
3.1 From Regression to the State Space Model
3.1.1 Ordinary Least Squares
3.1.2 Recursive Least Squares
3.1.3 The State Space Model
3.2 Filtering
3.2.1 A First Derivation of the Kalman Filter
3.2.2 A Second Derivation of the Kalman Filter
3.3 Smoothing
3.3.1 Fixed-Interval Smoothing
3.3.2 The Lag-One Covariance Smoother
3.4 Forecasting
3.5 Steady State of the Kalman Filter
3.5.1 Observability
3.5.2 Steady State of the Local Level Model
3.5.3 Steady State of Linear State Space Models
3.6 Exercises
4 Model Specification and Model Performance
4.1 Specification of Model Components
4.1.1 Trend State Space Models
4.1.2 Superposition of State Space Models
4.1.3 Fourier Form Seasonal Models
4.1.4 Trend-Seasonal Models
4.1.5 Time-Varying Regression
4.1.6 Time-Varying Autoregressions
4.2 Decomposition of State Space Models
4.2.1 Historical Note and Motivation
4.2.2 Rational Canonical Form
4.2.3 Decomposition of Linear State Space Models
4.2.4 Turkey Data Revisited
4.3 Estimation of Hyperparameters
4.3.1 Maximum Likelihood Estimation
4.3.2 Specification of Zt Using Discount Factors
4.3.3 Estimation of σ2: Conjugate Bayesian Estimation
4.4 Error Analysis
4.5 Prior Specification
4.5.1 Prior Specification of β0
4.5.2 Prior Specification of σ2
4.6 Automatic Sequential Monitoring
4.6.1 Model Monitoring
4.6.2 Specification of Alternative Models
4.6.3 Monitoring for the Tobacco total sales data—CP6
4.7 Exercises
5 Multivariate State Space Models
5.1 The Kalman Filter
5.2 Model Specification and Design
5.3 Steady State of the Multivariate Local Level Model
5.4 Error Analysis
5.5 Covariance Estimation in State Space Models
5.5.1 Variance Estimation
5.5.2 Covariance Structure and Matrix-Variate Probability Distributions
5.5.3 The Multivariate Scaled Observational Model
5.6 Forecasting Pollution Time Series
5.7 Markov Chain Monte Carlo Inference
5.7.1 Bayesian Inference and the Gibbs Sampler
5.7.2 The Forward Filtering Backward Sampling Scheme
5.7.3 Unknown Variances-Covariances
5.8 Exercises
6 Non-Linear and Non-Gaussian State Space Models
6.1 General Model Formulation
6.2 Dynamic Generalised Linear Models
6.2.1 Model Definition
6.2.2 Count Time Series
6.2.3 Categorical Time Series
6.2.4 Continuous Proportions
6.2.5 Decomposition of Dynamic Generalised Linear Models
6.3 Other Non-Gaussian and Non-linear Models
6.4 Inference for the General State Space Model
6.5 Power Local Level Models
6.5.1 Motivation and Main Model Structure
6.5.2 Poisson-Gamma and Exponential-Gamma Models
6.6 Approximate Inference
6.6.1 Motivation and Methodology
6.6.2 Tracking a Ship
6.6.3 The Extended Kalman Filter
6.6.4 The Unscented Kalman Filter
6.7 Sequential Monte Carlo Inference
6.7.1 Monte Carlo Integration
6.7.2 Importance Sampling
6.7.3 Sequential Importance Sampling
6.7.4 Choice of the Importance Function
6.7.5 Example 1: Multinomial Time Series
6.7.6 Example 2: Bearings-Only Tracking Revisited
6.7.7 Example 3: Non-Linear Time Series
6.7.8 Static Parameter Estimation
6.7.8.1 Introduction and Initial Studies
6.7.8.2 Liu and West Particle Filter
6.7.9 Case Study: Analysis of Asthma Data
6.8 Markov Chain Monte Carlo Inference
6.8.1 Metropolis-Hastings Algorithm
6.8.2 MCMC for Dynamic Generalised Linear Models
6.9 Dynamic Survival Models
6.9.1 Proportional Hazards Model
6.9.2 Dynamic Survival Model
6.10 Exercises
7 The State Space Model in Finance
7.1 Regression with Autocorrelated Errors
7.2 Stationarity and Autoregressive Models
7.2.1 Stationarity and Causality
7.2.2 Stationarity Conditions for AR(2)
7.2.3 Stationarity Conditions for AR(3)
7.3 Univariate Stochastic Volatility Models
7.3.1 Returns and Volatility
7.3.2 Stochastic Volatility Model
7.3.3 MCMC Inference of Stochastic Volatility Models
7.3.4 Particle Filter Inference of Stochastic Volatility Models
7.3.5 Particle Filter Inference of Stochastic Volatility Models with Asymmetric Returns
7.4 Multivariate Stochastic Volatility Models
7.4.1 Motivation and General Overview
7.4.2 Wishart Autoregressive Stochastic Volatility Models
7.4.3 Portfolio Optimisation and Asset Allocation
7.4.3.1 Problem Statement
7.4.3.2 Unconstrained Portfolio Selection
7.4.3.3 Constrained Portfolio Selection
7.5 Pairs Trading
7.5.1 Introduction and Basic Concept
7.5.2 State Space Models for Mean-Reverted Spreads
7.5.3 Time-Varying Autoregressive Models for Trading-Spreads
7.6 Exercises
8 Dynamic Systems and Control
8.1 Dynamic Systems
8.1.1 Basic Principles
8.1.2 Linear Systems
8.1.3 Laplace Transform
8.2 State Space Representation of Dynamic Systems
8.2.1 State Variables and State of a System
8.2.2 Continuous-Time State Space Model
8.2.3 Solution of the State Differential Equation
8.2.4 Discrete-Time State Space Model
8.3 System Stability
8.3.1 Definitions
8.3.2 Stability of Linear Systems
8.3.3 Stability of Non-Linear Systems
8.3.3.1 Lyapunov Indirect Method
8.3.3.2 Lyapunov Direct Method
8.4 Continuous-Time Kalman Filter
8.4.1 Discrete-Time Kalman Filter
8.4.2 Kalman–Bucy Filter
8.4.3 Observability and Convergence
8.4.4 Extended Kalman–Bucy Filter
8.5 Feedback Control
8.5.1 The PID-Controller
8.5.2 Twin Rotor Static Rig for Air-Vehicle Testing
8.6 Exercises
References
Index