1st Edition

Nonlinear Time Series Theory, Methods and Applications with R Examples

    552 Pages 50 B/W Illustrations
    by Chapman & Hall

    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.

    Linear Models
    Stochastic Processes
    The Covariance World
    Linear Processes
    The Multivariate Cases
    Numerical Examples

    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

    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

    Stochastic Recurrence Equations
    The Scalar Case
    The Vector Case
    Iterated Random Function

    Markov Models: Construction and Definitions
    Markov Chains: Past, Future and forgetfulness
    Homogeneous Markov Chain
    Canonical Representation
    Invariant Measures
    Observation-Driven Models
    Iterated Random Functions
    MCMC Methods

    Stability and Convergence
    Uniform Ergodicity
    V-Geometric Ergodicity
    Some Proofs

    Sample Paths and Limit Theorems
    Law of Large Numbers
    Central Limit Theorem
    Deviation Inequalities for Additive Functionals
    Some Proofs

    Inference for Markovian Models
    Likelihood Inference
    MLE: Consistency and Asymptotic Normality
    Observation-Driven Models
    Bayesian Inference
    Some Proofs

    Non-Gaussian and Nonlinear State Space Models

    Definitions and basic properties
    Filtering and smoothing

    Particle Filtering
    Importance sampling
    Sequential importance sampling
    Sampling importance resampling
    Particle filter
    Convergence of the particle filter

    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

    Inference for Nonlinear State Space Models
    Monte Carlo Maximum Likelihood Estimation
    Bayesian Analysis

    Asymptotics of the MLE for NLSS
    Strong Consistency of the MLE
    Asymptotic Normality

    Appendix A: Some Mathematical Background
    Appendix B: Martingales
    Appendix C: Stochastic Approximation
    Appendix D: Data Augmentation


    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