1st Edition
Nonlinear Time Series Theory, Methods and Applications with R Examples
Designed for researchers and students, Nonlinear Times Series: Theory, Methods and Applications with R Examples familiarizes readers with the principles behind nonlinear time series models—without overwhelming them with difficult mathematical developments. By focusing on basic principles and theory, the authors give readers the background required to craft their own stochastic models, numerical methods, and software. They will also be able to assess the advantages and disadvantages of different approaches, and thus be able to choose the right methods for their purposes.
The first part can be seen as a crash course on "classical" time series, with a special emphasis on linear state space models and detailed coverage of random coefficient autoregressions, both ARCH and GARCH models. The second part introduces Markov chains, discussing stability, the existence of a stationary distribution, ergodicity, limit theorems, and statistical inference. The book concludes with a self-contained account on nonlinear state space and sequential Monte Carlo methods. An elementary introduction to nonlinear state space modeling and sequential Monte Carlo, this section touches on current topics, from the theory of statistical inference to advanced computational methods.
The book can be used as a support to an advanced course on these methods, or an introduction to this field before studying more specialized texts. Several chapters highlight recent developments such as explicit rate of convergence of Markov chains and sequential Monte Carlo techniques. And while the chapters are organized in a logical progression, the three parts can be studied independently.
Statistics is not a spectator sport, so the book contains more than 200 exercises to challenge readers. These problems strengthen intellectual muscles strained by the introduction of new theory and go on to extend the theory in significant ways. The book helps readers hone their skills in nonlinear time series analysis and their applications.
FOUNDATIONS
Linear Models
Stochastic Processes
The Covariance World
Linear Processes
The Multivariate Cases
Numerical Examples
Exercises
Linear Gaussian State Space Models
Model Basics
Filtering, Smoothing, and Forecasting
Maximum Likelihood Estimation
Smoothing Splines and the Kalman Smoother
Asymptotic Distribution of the MLE
Missing Data Modifications
Structural Component Models
State-Space Models with Correlated Errors
Exercises
Beyond Linear Models
Nonlinear Non-Gaussian Data
Volterra Series Expansion
Cumulants and Higher-Order Spectra
Bilinear Models
Conditionally Heteroscedastic Models
Threshold ARMA Models
Functional Autoregressive Models
Linear Processes with Infinite Variance
Models for Counts
Numerical Examples
Exercises
Stochastic Recurrence Equations
The Scalar Case
The Vector Case
Iterated Random Function
Exercises
MARKOVIAN MODELS
Markov Models: Construction and Definitions
Markov Chains: Past, Future and forgetfulness
Kernels
Homogeneous Markov Chain
Canonical Representation
Invariant Measures
Observation-Driven Models
Iterated Random Functions
MCMC Methods
Exercises
Stability and Convergence
Uniform Ergodicity
V-Geometric Ergodicity
Some Proofs
Endnotes
Exercises
Sample Paths and Limit Theorems
Law of Large Numbers
Central Limit Theorem
Deviation Inequalities for Additive Functionals
Some Proofs
Exercises
Inference for Markovian Models
Likelihood Inference
MLE: Consistency and Asymptotic Normality
Observation-Driven Models
Bayesian Inference
Some Proofs
Endnotes
Exercises
STATE SPACE AND HIDDEN MARKOV MODELS
Non-Gaussian and Nonlinear State Space Models
Definitions and basic properties
Filtering and smoothing
Endnotes
Exercises
Particle Filtering
Importance sampling
Sequential importance sampling
Sampling importance resampling
Particle filter
Convergence of the particle filter
Endnotes
Exercises
Particle Smoothing
Poor man’s Smoother Algorithm
FFBSm Algorithm
FFBSi Algorithm
Smoothing Functionals
Particle Independent Metropolis-Hastings
Particle Gibbs
Convergence of the FFBSm and FFBSi Algorithms
Endnotes
Exercises
Inference for Nonlinear State Space Models
Monte Carlo Maximum Likelihood Estimation
Bayesian Analysis
Endnotes
Exercises
Asymptotics of the MLE for NLSS
Strong Consistency of the MLE
Asymptotic Normality
Endnotes
Exercises
APPENDICES
Appendix A: Some Mathematical Background
Appendix B: Martingales
Appendix C: Stochastic Approximation
Appendix D: Data Augmentation
References
Biography
Randal Douc, Eric Moulines, David Stoffer
"This book is very suitable for mathematicians requiring a very rigorous and complete introduction to nonlinear time series and their applications in several fields."
—Zentralblatt MATH 1306"This book focuses on theory and methods, with applications in mind. It is quite theory-heavy, with many rigorously established theoretical results. …It is also very timely and covers many recent developments in nonlinear time series analysis… readers can get a very up-to-date view of the current developments in nonlinear time series analysis from this book."
—Journal of the American Statistical Association, December 2014"… the book will definitely help readers who are very mathematically inclined and keen on rigour and interested in further pursuing the probabilistic aspects of nonlinear time series. I have no doubt the book will be useful and timely, and I have no hesitation in recommending the book … ."
—T. Subba Rao, Journal of Time Series Analysis, 2014
